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arxiv | 1912.12092v1 | Bremsstrahlung emission from nuclear reactions in compact stars | Bremsstrahlung emission of photons during nuclear reactions inside dense stellar medium is investigated in the paper. For that, a new model of nucleus is developed, where nuclear forces combine nucleons as bound system in dependence on deep location inside compact star. A polytropic model of stars at index $n=3$ with densities characterized from white dwarf to neutron star is used. Bremsstrahlung formalism and calculations are well tested on existed experimental information for scattering of protons of light nuclei in Earth. We find the following. (1) In neutron stars a phenomenon of dissociation of nucleus is observed --- its disintegration on individual nucleons, starting from some critical distance between this nucleus and center of star with high density. We do not observe such a phenomenon in white dwarfs. (2) In the white dwarfs, influence of stellar medium imperceptibly affects on bremsstrahlung photons. Also, we have accurate description of bremsstrahlung photons in nuclear reactions in Sun. (3) For neutron stars, influence of stellar medium is essentially more intensive and it crucially changes the bremsstrahlung spectrum. The most intensive emission is from bowel of the star, while the weakest emission is from periphery. | Sergei P. Maydanyuk, Kostiantyn A. Shaulskyi | nucl-th, astro-ph, astro-ph.HE, astro-ph.SR, hep-ph | https://arxiv.org/abs/1912.12092v1 | arXiv:1912.12092v1 [nucl-th] 27 Dec 2019
Bremsstrahlung emission from nuclear reactions in compact stars
Sergei P. Maydanyuk(1)∗and Kostiantyn A. Shaulskyi(2)†
(1)Institute for Nuclear Research, National Academy of Sciences of Ukraine, Kyiv, 03680, Ukraine and
(2)Taras Shevchenko National University of Kyiv, Ukraine
(Dated: November 27, 2024)
Bremsstrahlung emission of photons during nuclear reactions inside dense stellar medium is inves-
tigated in the paper. For that, a new model of nucleus is developed, where nuclear forces combine
nucleons as bound system in dependence on deep location inside compact star. A polytropic model
of stars at index n = 3 with densities characterized from white dwarf to neutron star is used.
Bremsstrahlung formalism and calculations are well tested on existed experimental information for
scattering of protons of light nuclei in Earth. We find the following. (1) In neutron stars a phe-
nomenon of dissociation of nucleus is observed — its disintegration on individual nucleons, starting
from some critical distance between this nucleus and center of star with high density. We do not
observe such a phenomenon in white dwarfs. (2) In the white dwarfs, influence of stellar medium im-
perceptibly affects on bremsstrahlung photons. Also, we have accurate description of bremsstrahlung
photons in nuclear reactions in Sun. (3) For neutron stars, influence of stellar medium is essentially
more intensive and it crucially changes the bremsstrahlung spectrum. The most intensive emission
is from bowel of the star, while the weakest emission is from periphery.
Keywords:
bremsstrahlung, photon, binding energy, ground state properties of finite nuclei, polytropic star,
white dwarf, neutron star, scattering
I.
INTRODUCTION
Light from stars provides us a main information about them. Here, photons emitted due to processes with partic-
ipation of elementary particles in stellar medium have been studied the most intensively. But, we know that stars
have nuclei from the lightest up to heavy (for example, see Chapter IV in book [1] for nuclei in Sun). Stability of
calculations of the bremsstrahlung spectra is achieved, if to take into account space regions up to atomic shells and
larger, even for low energies. This indicates on importance to study nucleus as unite system of evolving nucleons,
which should be studied via solution of many body quantum mechanical problem, where interactions cannot be small
and study cannot be reduced to perturbation approaches. So, important ingredient is emission of photons from stars
produced in nuclear reactions with participation of nuclei, from the lightest up to heavy. Bremsstrahlung emission of
photons during nuclear reactions in dense stellar medium has not been studied deeply and, so, it is a topic of current
paper.
Study of nuclear forces [2, 3] in extreme conditions enlarges our possibilities to understand them deeper, where stars
is a good laboratory for investigations [4–6]. Quantum nature of nuclear interactions can be displayed in possibility
of nucleons to combine as bound system, i.e. atomic nucleus, characterized via binding energy. Many-nucleon unified
theories of nucleus and nuclear reactions (for example, see microscopic cluster models based on developments of
resonating group method [7–9] and generator coordinate method [10]), shell models, collective models, relativistic
mean-field (RMF) theory [11–23], Ab initio calculations theory [24], QCD approaches for systems of nucleons, quark
models (quark-meson models [26], potential models) have been constructed for study of nuclei. In stars, a main focus is
given to obtain equation of state of matter (for example, see [25]). Naturally, many from models above are generalized
for such a task (for example, see Refs. [27–30] for RMF theories, see Ref. [25] for Ab initio calculations theory, etc.).
Today, one of the most accepted models is APR (Akmal, Pandharipande, Ravenhall) model [31] constructed in some
variants (see also Refs. [32–35]). It is based on principles of quantum mechanics and can be related with many-nucleon
models indicated above.
In this paper we are interesting in different question, that is about ability of nuclear forces to keep nucleons as bound
system inside star at high densities of stellar medium. In particular, it would be interesting to see how such property
of nuclear forces is changed in dependence on location of nucleus inside star, and at change of density (gravity) of star.
As it was demonstrated in Ref. [36, 37], even for the same full wave functions with the same boundary conditions there
are different nuclear processes (with the same nuclei and energies) where difference in cross-sections of them can reach
∗Electronic address: maidan@kinr.kiev.ua
†Electronic address: konstiger1998@live.com
2
up to 3-4 times. RMF theories cannot explain such fully quantum phenomenon, which is not small and important for
understanding nuclear processes. By such a motivation, we would like to use basis of quantum mechanics for analysis
in this paper.1 We use compact stars (stars at densities from white dwarf to neutron star [39]) in analysis. It turns
out that model of deformed oscillators shells [40–43] is enough convenient for such a research (see straightforward
investigations in many nucleon formalism [44–48] for basics of the model for the binary cluster configurations for light
nuclei, [48–51] for its extensions to describe binary clusters coupled to collective channels, [52–55, 57] for three-cluster
configurations, [58–60] for newest developments). After calculations, solution can be obtained in exact analytical form
with included additional influence of stellar medium. We apply this simplified model for estimations of emission of
bremsstrahlung photons during scattering of protons offnuclei in compact stars. We study how emission is changed
in dependence of characteristics of stellar medium.
The paper is organized by the following way. In Sec. II and III we present model in estimating binding energy for
light nuclei, and influence of stellar medium in frameworks of polytropic model of star. Analysis of change of binding
energy of nuclei in stars is given in Sec. IV. In Sec. V we study emission of bremsstrahlung photons during scattering
of protons offnuclei from stellar medium of compact stars. We summarize conclusions in Sect. VI. Derivation of
correction of energy of nucleus due to influence of stellar medium is presented in Appendix A.
II.
MODEL OF DEFORMED OSCILLATOR SHELLS
A.
Potential energies of nuclear and Coulomb forces and kinetic energy of nucleus
We define Hamiltonian of system of A nucleons as [40, 41, 43]
ˆHDOS = ˆT −ˆTcm +
A
X
i>j=1
ˆV (ij) +
A
X
i>j=1
e2
|ri −rj|.
(1)
We determine potential energy of two-nucleon nuclear forces for nucleus and potential energy of Coulomb forces
between protons on the basis of the matrix elements [see Eqs. (2.5), (2.6) in [43], also Eq. (15) in [40]]:
Unucl =
D
Ψ(1 . . . A)
A
P
i<j
ˆVij
Ψ(1 . . . A)
E
=
=
Z
Fp(n; r1, r1) Fp(n; r2, r2) 3V33(r12) + V13(r12)
2
dr1 dr2 −
Z Fp(n; r1, r2)
2 3V33(r12) −V13(r12)
2
dr1 dr2 +
+
Z
Fp(n; r1, r1) Fn(n; r2, r2) 3V33(r12) + 3V31(r12) + V13(r12) + V11(r12)
2
dr1 dr2 −
−
Z
Fp(n; r1, r2) Fn(n; r2, r1) 3V33(r12) −3V31(r12) −V13(r12) + V11(r12)
2
dr1 dr2 +
+
Z
Fn(n; r1, r1) Fn(n; r2, r2) 3V33(r12) + V13(r12)
2
dr1 dr2 −
Z Fn(n; r1, r2)
2 3V33(r12) −V13(r12)
2
dr1 dr2,
(2)
and
UCoul =
D
Ψ(1 . . . A)
ZP
i>j=1
e2
r12
Ψ(1 . . . A)
E
=
= 2
Z
Fp(n; r1, r1) Fp(n; r2, r2) e2
r12
dr1 dr2 −
Z Fp(n; r1, r2)
2 e2
r12
dr1 dr2,
(3)
where proton density (for nuclei with even number of protons) is
Fp(n; ri, rj) =
z/2
X
s=1
exp
h
−1
2
x2
i
a2 + y2
i
b2 + z2
i
c2
i
exp
h
−1
2
x2
j
a2 + y2
j
b2 + z2
j
c2
i
π3/2 abc
q
2nxi+nyi +nzi+nxj +nyj +nzj nxi!nyi!nzi!nxj!nyj!nzj!
×
× Hnxi
xi
a
Hnyi
yi
b
Hnzi
zi
c
· Hnxj
xj
a
Hnyj
yj
b
Hnzj
zj
c
,
(4)
1 There are indications on importance to implement quantum mechanical nuclear models to bremsstrahlung theory in study of emission
of photons in stars during proton-capture reactions by nuclei and other nuclear processes [68].
3
where summation is performed over all states of needed configuration, Hn(x) are Hermitian polynomials [we use
definition from Ref. [61], p. 749, (,6)], a, b, c are oscillator parameters along axes x, y, z. Neutron density Fn(n; ri, rj)
is obtained after change of proton configuration and numbers of states on the corresponding neutron characteristics.
According to definitions (3) and (4), proton density is the same for different isotopes. So we obtain the same energy
of Coulomb forces for different isotopes. For two-nucleon potentials we shall use (see Eq. (1) in Ref. [40]):
V31(r) = −3V33(r) = −Vt exp
−r2
µ2
t
, V13(r) = −1/3V11(r) = −Vs exp
−r2
µ2s
,
(5)
where Vt = 72.5 MeV, µt = 1.47 fm, Vs = 39, 15 MeV, µs = 1, 62 fm.
We define kinetic energy of system of nucleons (in center-of-mass frame), according to Eq. (2.4) in Ref. [43] :
Tfull =
D
Ψ(1 . . . A)
−¯h2
2m
A
X
i=1
∇2
i +
¯h2
2Am
A
X
i=1
∇i
2 Ψ(1 . . . A)
E
=
= A −1
4
¯h2
m
1
a2 + 1
b2 + 1
c2
+ ¯h2
m
Z/2
X
s=1
nx,s
a2 + ny,s
b2 + nz,s
c2
+
N/2
X
s′=1
nx,s′
a2
+ ny,s′
b2
+ nz,s′
c2
.
(6)
B.
Proton and neutron densities
From Eq. (4) we calculate the proton and neutron densities for isotopes of 4,6,8He and 8,10Be:
Fp(r1, r2)(4He) = Fn(r1, r2)(4He) = F0(r1, r2),
Fp(ri, rj)(6He) = F0(ri, rj),
Fn(ri, rj)(6He) = F0(ri, rj) ·
n
1 + 2xixj
a2
o
,
Fp(ri, rj)(8He) = F0(ri, rj),
Fn(ri, rj)(8He) = F0(ri, rj) ·
n
1 + 2 xixj
a2
+ 2 yiyj
b2
o
,
Fp(ri, rj)(8Be) = Fn(ri, rj)(8Be) = F0(ri, rj) ·
n
1 + 2xixj
a2
o
,
Fp(ri, rj)(10Be) = Fp(ri, rj)(8Be), Fn(ri, rj)(10Be) = Fn(ri, rj)(8Be) + F0(ri, rj) · 2 yiyj
b2
,
(7)
where
F0(ri, rj) =
exp
h
−1
2
x2
i
a2 + y2
i
b2 + z2
i
c2
i
exp
h
−1
2
x2
j
a2 + y2
j
b2 + z2
j
c2
i
π3/2 abc
.
(8)
One can see that the proton and neutron densities are the same for nuclei 4He, 8Be, they are different for nuclei 6He,
8He, 10Be. Also we have properties:
F0(r1, r2) = F0(r2, r1) = F ∗
0 (r1, r2) = F ∗
0 (r2, r1),
(9)
F0(r1, r1) · F0(r2, r2) = F 2
0 (r1, r2).
(10)
Z
F 2
0 (r1, r2) · exp
−r2
12
µ2
· x2
1
a2 dr1dr2 =
Z
F 2
0 (r1, r2) · exp
−r2
12
µ2
· x2
2
a2 dr1dr2,
Z
F 2
0 (r1, r2) · exp
−r2
12
µ2
· xn
1
an
ym
2
bm dr1dr2 =
Z
F 2
0 (r1, r2) · exp
−r2
12
µ2
· xn
2
an
ym
1
bm dr1dr2,
(11)
C.
Potential energy of nuclear two-nucleon interactions
We shall find potential energy of two-nucleon nuclear interactions for nucleus 6He on the basis of matrix element
in Eq. (2) and found densities (7). After calculations, we obtain:
Unucl(4He) = −3
Vt Nt + Vs Ns
,
Unucl(6He) = U (sym)
nucl (6He) + U (asym)
nucl
(6He),
U (sym)
nucl (6He) = Unucl(4He),
Unucl(8He) = U (sym)
nucl (8He) + U (asym)
nucl
(8He),
U (sym)
nucl (8He) = Unucl(4He),
Unucl(8Be) = U (sym)
nucl (8Be) + U (asym)
nucl
(8Be),
U (sym)
nucl (8Be) = 2 Unucl(4He),
Unucl(10Be) = U (sym)
nucl (10Be) + U (asym)
nucl
(10Be), U (sym)
nucl (10Be) = U (sym)
nucl (8Be),
(12)
4
where
U (asym)
nucl
(6He) = −
n
3 Vt Nt
a2
t
1 + 2 a2
t
+ Vs Ns
1 + 3 a2
s
1 + 2a2
s
+ 3 a4
s
1 + 2 a2s
2
o
,
(13)
U (asym)
nucl
(8He) = −
n
Vt Nt
h
3 a2
t
1 + 2 a2
t
+
3 b2
t
1 + 2 b2
t
−1 + a2
t
1 + 2 a2
t
1 + b2
t
1 + 2 b2
t
+
a2
tb2
t
(1 + 2 a2
t) (1 + 2 b2
t)
i
+
+ Vs Ns
h1 + 3 a2
s
1 + 2 a2s
+ 1 + 3 b2
s
1 + 2 b2s
+ 1 + a2
t
1 + 2 a2
t
1 + b2
t
1 + 2 b2
t
+
a2
sb2
s
(1 + 2 a2s) (1 + 2 b2s) +
3 a4
s
(1 + 2 a2s)2 +
3 b4
s
(1 + 2 b2s)2
i o
,
(14)
U (asym)
nucl
(8Be) = −9
n
Vt Nt
a4
t
(1 + 2a2
t)2 + Vs Ns
a4
s
(1 + 2a2s)2
o
,
(15)
U (asym)
nucl
(10Be) = U (asym)
nucl
(8Be) −
−
n
3 Nt
b2
t
1 + 2 b2
t
+ 4 Ns
b2
s
1 + 2 b2s
+ 3 Ns
b4
s
(1 + 2 b2s)2
o
−3 ·
n
Nt a2
tb2
t
(1 + 2 a2
t) (1 + 2 b2
t) +
Ns a2
sb2
s
(1 + 2 a2s) (1 + 2 b2s)
o
,
(16)
and we use notations (with change of indexes t and s):
at = a/µt, bt = b/µt, ct = c/µt, as = a/µs, bs = b/µs, cs = c/µs,
(17)
Nt =
1
p
1 + 2a2
t
p
1 + 2b2
t
p
1 + 2c2
t
.
(18)
From solutions above one can see that (1) only 4He is spherical in the ground state, while other nuclei are deformed;
(2) nuclei 6He, 8Be are axially symmetric in the ground state, while 8He, 10Be are fully deformed. Further calculations
of minima of full energy of these nuclei confirm such a logic.
In spherically symmetric approximation (a = b = c), we obtain
Unucl(6He) = −3 Vt N (sph)
t
n
1 +
a2
t
1 + 2 a2
t
o
−Vs N (sph)
s
3 +
1 + 3 a2
s
1 + 2a2
s
+ 3 a4
s
1 + 2 a2s
2
,
Unucl(8He) = Unucl(4He) −
n
Vt N (sph)
t
h6 a2
t −1
1 + 2 a2
t
i
+ Vs N (sph)
s
h
3 +
8 a4
s
(1 + 2 a2s)2
i o
,
Unucl(8Be) = −3
n
Vt N (sph)
t
h
2 +
3 a4
t
(1 + 2a2
t)2
i
+ Vs N (sph)
s
h
2 +
3 a4
s
(1 + 2a2s)2
io
,
Unucl(10Be) = Unucl(8Be) −
n
3 Nt
a2
t
1 + 2 a2
t
+ 3 Nt
a4
t
(1 + 2 a2
t)2 + 4 Ns
a2
s
1 + 2 a2s
+ 6 Ns
a4
s
(1 + 2 a2s)2
o
.
(19)
where
N (sph)
t
= (1 + 2a2
t)−3/2.
(20)
D.
Kinetic energy of nuclei
We calculate kinetic energy of nuclei, according to Eq. (6), and obtain:
Tfull(4He)
= 3
4
¯h2
m
1
a2 + 1
b2 + 1
c2
,
Tfull(6He)
= 5
3 · Tfull(4He) +
¯h2
m a2 ,
Tfull(8He)
= 7
3 · Tfull(4He) + ¯h2
m
1
a2 + 1
b2
,
Tfull(8Be)
= 7
3 · Tfull(4He) + 2 ¯h2
m a2 = Tfull(8He) + ¯h2
m
1
a2 −1
b2
,
Tfull(10Be) = 9
3 · Tfull(4He) + ¯h2
m
2
a2 + 1
b2
.
(21)
5
In the spherically symmetric case (a = b = c) we obtain:
T sph
full (4He) = 9
4
¯h2
m a2 ,
Tfull(6He) = 19
9 · Tfull(4He),
Tfull(8He) = 29
9 · Tfull(4He),
Tfull(8Be) = Tfull(8He), Tfull(10Be) = 31
9 Tfull(4He) = 31
29 Tfull(8Be).
(22)
III.
ENERGY OF NUCLEUS INSIDE COMPACT STARS
A.
Polytropic stars
Star without rotation and magnetic field has spherical form. Its equilibrium is determined by balance of forces of
gravity and gradient of pressure. Nuclear reactions take place in stars and there is radiation from their surfaces. For
relativistic stars on last stage of evolution, pressure P depends on density ρ only and can be described by equation of
state of P = P(ρ). At some approximation, star under conditions above can be described by Lane-Emden equation
(see Ref. [4], p. 19):
d
dξ
ξ2 dθ
dξ
= −ξ2 θn
(23)
with boundary conditions of θ(0) = 1, dθ(0)/dξ = 0. Here, ξ is a dimensionless distance from center of star and θ is
related density. n is the polytropic index that appears in the polytropic equation of state:
P = K · ργ, γ = 1 + 1
n,
(24)
where P and ρ are the pressure and density, K is constant of proportionality. Density ρ(r) inside star at distance r
from center of star is derived as
ρ(r) = ρc · θn, r = R0 · ξ, R2
0 = (n + 1) K
4π G
ρ
1
n −1
c
,
(25)
where ρc is pressure at center of star, R0 is parameter. In frameworks of such a model, radius and mass of star are
calculated as (see Ref. [4], p. 22):
rR = ξR · R0, M =
rR
Z
0
4πρ r2dr = 4π
h(n + 1) K
4π G
i3/2
ρ
3 −n
2n
c
ξ1
Z
0
θnξ2 dξ.
(26)
ξR is dimensionless radius of star defined from condition θ(ξR) = 0. In particular, mass of star at n = 3 does not
depend on density ρc.
In this paper we will be interesting in what happens with nucleus in dependence on depth of its location inside
star of such a type. Step-by-step, we will change a distance from center of star to this nucleus and analyze how much
strong are forces keeping nucleons of this nucleus as bound quantum system. Clear understanding can be obtained
from binding energy of nucleus as system of nucleons. We calculate binding energy as summation of potential energy
of nuclear forces, Coulomb forces, kinetic energy of nucleons described above. It turns out, that the simplest case of
n = 3 allows to obtain a clear picture (other cases add more technical derivations, so we will omit them in this paper).
B.
Quantum mechanics of nucleus under influence of stelar medium
Let us write full hamiltonian of nucleus with additional influence of medium of star on nucleons of this nucleus as
ˆH = −¯h2
2m
A
X
i=1
∇2
i +
A
X
i,j=1
VDOS(|ri −rj|) +
A
X
i,j=1
Vstar(ri, rj).
(27)
In the first approximation, we shall assume that influence of stellar medium on nucleons of the studied nucleus is
homogeneous. Force F of such an influence should depend on distance R between center of star and center of mass
6
of the studied nucleus. Potential of such an influence should depend on relative distances between nucleons of the
studied nucleus.2 Such a formalism is given in quantum mechanics (see Ref. [61], p. 100–102, for details), therefore,
we define it as3
Vstar(R, ri, rj) = + |FP (R) · (ri −rj)|.
(29)
Corresponding correction ∆Estar to the full energy of nucleus due to inclusion of influence of star on nucleons of
nucleus can be defined as
∆Estar =
D
Ψ(1 . . . A)
A
X
i,j=1
Vstar(R, ri, rj)
Ψ(1 . . . A)
E
.
(30)
Calculations of such a matrix element are presented in Appendix A. For even-even nuclei (at Z = N) we obtain:
⟨Ψ(1 . . . A) | ˆV (ri, rj)| Ψ(1 . . . A)⟩=
1
A · (A −1)
A
X
k=1
A
X
m=1,m̸=k
D
ϕ0(ri) ϕ0(rj)
ˆV (ri, rj)
ϕ0(ri) ϕ0(rj)
E
=
=
D
ϕ0(ri) ϕ0(rj)
ˆV (ri, rj)
ϕ0(ri) ϕ0(rj)
E
.
(31)
In particular, for 4He we have (see Eq. (A11) at a = b = c)
∆Estar(4He) = 12 · FP (R) ·
Z
F 2
0 (r1, r2) (r1 −r2) dr1 dr2 = 12 · 23/2 a
π1/2
· FP (R).
(32)
where FP (R) = |FP (R)|. Even without numerical estimations, now picture of influence of stellar medium on the
studied nucleus has became clear. Forces of stellar medium press on nucleons of nucleus. The deeper this nucleus is
located in star, the stronger such forces press on nucleus. However, binding energy (it is negative for nucleus in the
external layer of star) is increased at deeper location of this nucleus in star. Starting from some critical distance from
nucleus to center of star, the binding energy becomes positive. This means that full energy of individual nucleons of
the studied nucleus is already larger than mass of this nucleus, i.e. we obtain unbound system of nucleons and nucleus
is disintegrated on nucleons. Now it could be interesting to estimate if such a phenomenon is appeared in white dwarfs
and neutron stars in frameworks of the model above. The kinetic energy is increased at deeper location of nucleus in
star. At decreasing distance from the studied nucleus to center of star, change of kinetic energy is unlimited, while
change of nuclear energy is limited. So, ratio between kinetic energy of nucleons of nucleus and nuclear energy of
nucleus is changed also.
IV.
ANALYSIS
One of objects, where polytropic model is successfully applied, is wight dwarf (see indications in Ref. [5], p. 364–370;
Ref. [6], p. 213–233, p. 475–496). Thus, according to Ref. [4] (see Fig. 2.2 in that book, p. 33; also Fig. 103 in Ref. [5],
p. 365), density in center of such a star is in the region of 10+5g cm−3 – 1.4 · 10+9 g cm−3 . So, let us start analysis
from such a type of star.
At first, let us see how density is dependent on distance to center of star in frameworks of such a model.
In
Fig. 1 (a) one cab see solution of Lane-Emden equation (23) at n = 3 by the finite-difference method. Radius of star
2 Gradient of potential U with opposite sign is force FP acting on particle with mass m: FP ≡−∇U.
One can clarify that, if to
analyze action on particle in quantum mechanics in semiclassical approximation (see Ref. [61], p. 209). Operator of velocity in quantum
mechanics as m ˆ˙v = −∇U (see (19.3), Ref. [61], p. 82) indicates on such a relation between U and FP also. From here, one can obtain
potential. In particular, for homogeneous force one can find:
U(r) = −FP
Z
dr = −FP r.
(28)
3 In order to understand, which sign should be used in this formula, we return back to logics in Ref. [61] (see p. 100 in this book). In
particular, at increasing of distances between nucleons, |ri −rj|, potential of influence stelar medium should suppress (i.e. not reinforce)
relative leaving of nucleons from the nucleus. Therefore, the potential should increase (not decrease) at increasing of |ri −rj|. I.e. sign
if Eq. (29) is chosen correctly.
7
0
2
4
6
8
10
0,0
0,2
0,4
0,6
0,8
1,0
n = 3
density,
distance from center of star,
5000
20000
35000
50000
65000
10
5
10
6
10
7
10
8
10
9
n = 3
Density at center
of star,
cr
[g cm
-3
]
Radius of star,
R
star
(km)
FIG. 1: (Color online) Panel (a): Solution of Lane-Emden equation (23) at n = 3 by the finite-difference method [parameters
of calculations: boundary conditions are θ(0) = 1, dθ(0)/dξ = 0 ]. In figure one can see monotonous decreasing of density θ at
increasing of distance ξ. According to the model, internal region of starcorresponds to condition θ(ξ) ≥0, and radius of star,
ξr, is found from condition of θ(ξr) = 0. Panel (b): Radius of star in dependence on density in its center (densities are chosen
for white dwarfs, at n = 3).
is determined from condition of θ(ξr) = 0, we obtain ξr = 6.881. In Fig. 1 (b) one can see radius of star in dependence
on its density at center for such a model. One can see that such a model gives white dwarfs with radiuses in region
from 3011.28 kilometres (at ρcr = 1.4 · 109 g cm−3) to 72 576.27 kilometres (at ρcr = 105 g cm−3).
We shall analyze, how the density ρ in star is changed in dependence on distance from center of star. Results of
such calculations at n = 3 are presented in Fig. 2 (a). As nest step, we shall estimate pressure in star in dependence
0
1
2
3
4
5
6
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
(a)
Density ,
[g/ sm
3
]
Distance from center of star,
rho
cr
= 10
6
g/sm
3
rho
cr
= 10
5
g/sm
3
rho
cr
= 10
8
g/sm
3
rho
cr
= 10
9
g/sm
3
0
1
2
3
4
5
6
10
-11
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
(b)
Pressure , P [MeV
4
]
Distance from center of star,
rho
cr
= 10
5
g/sm
3
rho
cr
= 10
6
g/sm
3
rho
cr
= 10
8
g/sm
3
rho
cr
= 10
9
g/sm
3
FIG. 2: (Color online) Density ρ (a) and pressure P (b) inside star in dependence on distance ξ from center of star at n = 3
[parameters of calculations: density is defined in Eq. (25), pressure is defined in Eq. (24) ].
of distance between nucleus (its center of mass) and center of mass of star. Using Eqs. (24), at n = 3 we have (see
Eq. (2.3), p. 32 in Ref. [4]):
γ = 4
3, Pn=3 = K · ρ4/3, Kn=3 = (3 π2)1/3
4
¯hc (µe mp)−4/3 = 3.384 782 · 10−5 MeV−4/3.
(33)
Results of such calculations are presented in Fig. 2 (b).
Now let us estimate force acting on the studied nucleus in star in result of influence of stellar medium. We shall
find it on the basis of pressure stellar medium. According to definition, pressure is force applied perpendicular to the
surface of an object per unit area over which that force is distributed. So, pressure acting on selected layer inside star,
represents force, acting on unit area of this layer. Then, force acting on full such a layer can be found as the pressure
8
multiplied on the full area of this layer. Let us consider nucleus with surface Snucl inside star. One can determine
force acting on this nucleus as pressure multiplied on area of surface of this nucleus as
FR(R) = P(R) · Snucl.
(34)
In the simplest approximation, the studied nucleus can be considered in the spherical form where its area is Snucl =
4π R2
nucl, Rnucl = a · A1/3. Results of calculations of such a force acting on nucleus 4He in star are presented in
Fig. 3 (a), if pressure is shown in Fig. 2 (b)4.
0
1
2
3
4
5
6
10
-21
10
-17
10
-13
10
-9
10
-5
10
-1
10
3
10
7
(a)
Force , F
R
[MeV
2
]
Distance from center of star,
rho
cr
= 10
5
g/sm
3
rho
cr
= 10
9
g/sm
3
rho
cr
= 10
14
g/sm
3
rho
cr
= 10
16
g/sm
3
0
1
2
3
4
5
6
10
-20
10
-18
10
-16
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
Energy ,
E
star
[MeV]
Distance from center of star,
(b)
rho
cr
= 10
5
g/sm
3
rho
cr
= 10
9
g/sm
3
rho
cr
= 10
14
g/sm
3
rho
cr
= 10
16
g/sm
3
FIG. 3: (Color online) Panel (a): Force FR, acting on nucleons of nucleus 4He, in dependence on its distance ξ to the center of
star (at n = 3) [Force is defined in Eq. (34), oscillator parameter a = 1.05 fm is fixed for estimations, that is close to minimum
of full energy of 4He in natural conditions (in Earth)]. Panel (b): Correction ∆E to energy of nucleus 4He, in result of influence
of stellar medium , in dependence on distance to center of star (at n = 3) [Correction of energy ∆E is defined in Eq. (32)].
After obtaining force, we shall find correction to full energy of nucleus from such an influence. Results of such
calculations for nucleus 4He are presented in Fig. 3 (b).
From this figure one can see that in the white dwarfs
(corresponding to densities in region of 105 g cm−3 – 1.4 · 109 g cm−3) nucleus cannot be disintegrated (in this model).
However, we see that for more high densities this phenomenon really happens, starting from some critical distances
from center of stars [see upper brown dashed line (at ρcr = 1016 g cm−3) and green dash-double dotted line (at
ρcr = 1014 g cm−3) in figure]. This case corresponds to the neutron stars.
Now we shall analyze possibility of nucleus to disintegrate on individual nucleons in the neutron star. Different
types of energy for nucleus 4He concerning to its full energy are shown in Fig. 4. Also one can see that full energy
of nucleus is already positive. This means that system of nucleons representing the nucleus is not bound system (i.e.
nucleus is disintegrated on nucleons). Also one can see that parameter a, corresponding to the minimum of the full
energy of system of nucleons, is decreased in comparison with its value for nucleus in usual conditions (i.e. outside
from star). So, relative distances between nucleons of 4He is decreased. This is explained by influence of pressure of
stellar medius on these nucleons. The similar tendencies we obtain for other isotopes of He and Be considered above.
After analysis above, we would like to estimate where inside star there is such a phenomenon of disintegration of
nucleus. Such calculations are presented in Fig. 5 in dependence on density of stellar medium at center of neutron
star. In particular, one can see that this model describes that for more compact stars dissosoiation of nuclei happens
closer to external surface.
V.
BREMSSTRAHLUNG EMISSION OF PHOTONS DURING SCATTERING OF PROTONS OFF
NUCLEI IN STELLAR MEDIUM OF COMPACT STARS AT n = 3
We analyze emission of photons in nuclear reactions inside compact stars. We will focus on question, how a dense
medium of star influences on emission of photons. Note that such a question has not been studied else. Note that
4 For simplicity of presentation, in this paper we use units for force in MeV (that is used in computer calculations and allows to study
physical process inside distances of nuclei and stars, at the same time). This do not forbid to perform comparable analysis.
9
0,5
1,0
1,5
2,0
-400
-300
-200
-100
0
100
200
300
400
500
600
calc
= 3.73
cr
=10
14
g/sm
3
(a)
4
He
Energy , E [MeV]
Oscillat. parameter,
a
x
(fm)
energy full
energy kinetic
energy nuclear
energy Coulomb
energy star
0,6
0,8
1,0
1,2
1,4
1,6
-50
-40
-30
-20
-10
0
10
20
30
40
calc
= 2.42
cr
=10
14
g cm
-3
(b)
4
He
Energy , E [MeV]
Oscillat. parameter,
a
x
(fm)
full energy of nucleus inside star
full energy of nucleus in Earth
FIG. 4: (Color online) Panel (a): Energy of “nucleus” 4He inside star at distance ξ = 2.42 from center of star with density at
center ρcr = 1014g cm−3 (at n = 3). At minimum of the full energy of nucleus, we obtain: a = 0.8374 fm, Efull = 5.038 MeV,
Efull per nucl = 1.327 MeV, Ekin = 138.096 MeV, ECoul = 1.363 MeV, Enucl = −164.492 MeV, Estar = 30.313 MeV. Panel (b):
Full energy of nucleus inside star in comparison with full energy of this nucleus in Earth.
2,0
2,5
3,0
3,5
4,0
4,5
5,0
10
14
10
15
10
16
4
He
Distance from center of star,
Density ,
c
[g/ sm
3
]
(a)
FIG. 5: (Color online) Critical distance ξ from center of star in dependence on its density at center, where disintegration of
nucleus 4He on nucleons takes place.
some calculations were done for proton-capture reactions [63], that is enough popular for stars. However, in those
calculations nucleus was considered as stable, without influence of stellar medium. Now we will take into account
change of nucleus due to influence of stellar medium. To be close to that analysis, we will choose scattering of protons
of nuclei.
For scattering of proton offnucleus, we can rewrite hamiltonian (27) as
ˆH = −¯h2
2m
A+1
X
i=1
∇2
i +
A+1
X
i,j=1
VDOS(|ri −rj|) +
A+1
X
i,j=1
Vstar(|ri −rj|).
(35)
Inclusion of emission of bremsstrahlung photons can be described via the following hamiltonian:
ˆHfull = −¯h2
2m
A+1
X
i=1
∇2
i +
A+1
X
i,j=1
VDOS(|ri −rj|) +
A+1
X
i,j=1
Vstar(|ri −rj|) + ˆHγ,
(36)
where ˆHγ is a new operator which describes emission of bremsstrahlung photons for the studied reaction inside star.
A concrete form of this operator should be defined explicitly.
10
Emission of bremsstrahlung photons without influence of stellar medium [i.e. without last term in Eq. (36)] for
scattering of protons offnuclei in conditions of Earth was studied enough often by different researchers.
Here,
agreement between theory and existed experimental information has been obtained with the highest precision for this
reaction in frameworks of approach [62, 63] (this is data [79] for p + 208Pb at proton energy beam of Ep = 145 MeV,
data [78] for p + 12C, p + 58Ni, p + 107Ag, p + 197Au at proton energy beam of Ep = 190 MeV and corresponding
calculations in Figs. 5–8 in Refs. [63]). Therefore, we will generalize bremsstrahlung formalism for these reactions
from above-zero energies up to intermediate energies in stars, basing on formalism and results of papers [62, 63] (see
improvements of formalism in Refs. [64–68], for other reactions see Refs. [68–77]).
According to such an approach, for reaction in conditions of Earth in laboratory frame we define cross-section of
bremsstrahlung emission of photons in frameworks of papers [62], where the full matrix element of emission of photons
is defined as
⟨Ψf| ˆHγ| Ψi⟩0
=
s
2π c2
¯hwph
n
MP + M (E)
p
+ M (M)
p
+ M∆E + M∆M + Mk
o
,
(37)
matrix elements have form
M (E, dip,0)
p
= i¯h2 (2π)3 e
µc Z(dip,0)
eff
X
α=1,2
e(α) · I1,
M (M, dip,0)
p
= −¯h (2π)3 1
µ M(dip,0)
eff
X
α=1,2
h
I1 × e(α)i
,
M∆E
= 0,
M∆M
= i ¯h (2π)3 f1 · |kph| · zA · I2,
Mk
= fk
f1
· M∆M,
(38)
coefficient are defined as
f1 = A −1
2A
µ(an)
pn , fk
f1
= −¯hA
A −1
(39)
and integrals are defined as
I1 =
Φp−nucl,f(r)
e−i kphr d
dr
Φp−nucl,i(r)
,
I2 =
D
Φp−nucl,f(r)
ei cp kphr Φp−nucl,i(r)
E
.
(40)
Here, r is radius vector from center-of-mass on nucleus to the scattered proton, µ = mp mA/(mp + mA) is reduced
mass, A is number of nucleons in nucleus, cp = mp/(mp + mA), e(α) are unit vectors of polarization of the photon
emitted [e(α),∗= e(α)], kph is wave vector of the photon and wph = kphc =
kph
c, Eph = ¯hwph is energy of photon.
Vectors e(α) are perpendicular to kph in Coulomb calibration. We have two independent polarizations e(1) and e(2)
for the photon with impulse kph (α = 1, 2). µ(an)
pn
= µ(an)
p
+ µ(an)
n
, µ(an)
p
and µ(an)
n
are anomalous magnetic moments
of proton and neutron.
The matrix elements M (E, dip,0)
p
and M (M, dip,0)
p
describe coherent bremsstrahlung emission of photons of electric
and magnetic types, the matrix elements M∆E and M∆M describe incoherent bremsstrahlung emission of photons
of electric and magnetic types. MP is related with motion of full nuclear system, which we neglect in this paper.
Effective electric charge and magnetic moment of system in dipole approximation (i.e. at kphr →0) are
Z(dip,0)
eff
= mA zp −mp zA
mp + mA
, M(dip,0)
eff
= −
mp
mp + mA
MA.
(41)
mp and zp are mass and charge of proton, mA and zA are mass and charge of nucleus. Here, we introduced magnetic
moment of nucleus MA (without inclusion of characteristics of photons emitted):
MA =
A
X
j=1
D
ψnucl,f(βA)
µ(an)
j
mAj σ
ψnucl,i(βA)
E
,
(42)
where µ(an)
j
is anomalous magnetic moment of proton or neutron in nucleus, mAj is mass of nucleon with number j
in nucleus, σ is operator of spin (acting on wave function of nucleon of nucleus).
11
For first estimations of bremsstrahlung emission for nuclear reactionbs in stellar medium, we shall us perturbation
theory. We will take into account influence of stellar medium on emission as
ˆHγ new = ˆHγ0 + ∆ˆHγ, ∆ˆHγ =
A+1
X
i,j=1
Vstar(|ri −rj|),
(43)
From here we find the matrix element of emission for reaction inside star as
⟨Ψf| ˆHγ| Ψi⟩star
=
⟨Ψf| ˆHγ| Ψi⟩0 + ⟨Ψf| ∆ˆHγ| Ψi⟩.
(44)
According to perturbation theory, for determination of the first correction we should use wave functions of unperturbed
system, i.e. we take wave functions as in matrix element (37):
⟨Ψf| ∆ˆHγ| Ψi⟩=
s
2π c2
¯hwph
· Mstar(Eph), Mstar(Eph) = N · FP (R) ·
Z
ϕ2
p−nucl(r, kf) ϕ2
0(r) |r| dr,
(45)
where
FP (R) = P(R) = K · ργ(R).
(46)
Here ϕp−nucl(r, kf) is wave function of scattering of proton offnucleus (energy has continuous spectrum, as emitted
photon takes some energy of proton-nucleus system), ϕ2
0(r) is wave function of nucleus (energy has only discrete levels,
nucleons are only in bound states). We additionally renormalize wave function of scattering of proton offnucleus5.
For 4He we have
ϕ0(r) = ϕnx=0(x) · ϕny=0(y) · ϕnz=0(z), ϕnx=0(x) =
exp
−x2
2 a2
π1/4 √
2nx nx! √a
· Hnx=0
x
a
=
exp
−x2
2 a2
π1/4 √a
, N = 12.
(47)
Substituting Eq. (47) to (45) for 4He (at a = b = c), we obtain:
Mstar(Eph) = FP (R) ·
N
π3/2 a3
Z
ϕ2
p−nucl(r, kf) exp
−r2
a2
r dr,
(48)
We calculate the wave functions ϕp−nucl numerically concerning the chosen potential of interaction between the
proton and the spherically symmetric core. For description of proton-nucleus interaction we use the potential as
V (r) = vc(r) + vN(r) + vso(r) + vl(r), where vc(r), vN(r), vso(r) and vl(r) are Coulomb, nuclear, spin-orbital and
centrifugal components defined in Ref. [81].
Results of calculation of spectrum of emision of bremsstrahlung photons in scattering of protons offnuclei in stars
on the basis of such an approach are shown in Fig. 6. From this figure we conclude the following.
• In the white dwarfs, according to Fig. 4, influence of stellar medium on emission is not larger than 0.1 MeV 2.
This means that influence of stellar medium imperceptibly affects on emission of bremsstrahlung photons. In
particular, such a conclusion can be formulated for nuclear reactions inside Sun.
I.e., it turns out that we
have enough accurate description of emission of bremsstrahlung photons during nuclear reactions in Sun, white
dwarfs and similar stars.
• For neutron stars, influence of stellar medium is essentially more intensive and it crucially changes shape of
the spectrum of the bremsstrahlung photons (see Fig. 6). In the simplest approximation, one can find that
maximum of probability of the emitted photons is for their energy, which is half of energy of the scattered
protons: Eph ≃Ep/2. One can see that the most intensive emission is created in the bowel of the star, while
the weakest emission is from the periphery (for the same energy of the scattered proton).
5 This is caused by that normalization of wave function of photon is determined by factor
s
2π c2
¯hwph
at exponent of vector potential of
electromagnetic field A in QED [see representation (5) in Ref. [62]], that in principle is different from normalization of wave functions
of nucleons in bound states and states of scattering in quantum mechanics.
12
0
20
40
60
80
100
10
-12
10
-11
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
E
p
= 100 MeV
p +
4
He
d
/ dE
(b / MeV )
Photon energy,
E
(MeV)
F=0.1 MeV
2
F=1.0 MeV
2
F=10 MeV
2
F=100 MeV
2
F=1000 MeV
2
0
20
40
60
80
100
10
-11
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
p +
4
He
E
p
= 100 MeV
d
/ dE
(b / MeV)
Photon energy, E
(MeV)
F=0.1 MeV
2
F=1.0 MeV
2
F=10 MeV
2
F=100 MeV
2
F=1000 MeV
2
FIG. 6: (Color online) Bremsstrahlung emission of photons in scattering of protons offnuclei 4He inside star at energy of
protons of Ep = 100 MeV [we calculate spectrum on the basis of the leading matrix element M (E, dip,0)
p
, which gives the
largest contribution to full spectrum, according to analysis in Refs. [62, 63] ]. Contribution on the basis of matrix element
⟨Ψf| ∆ˆHγ| Ψi⟩in Eq. (44) (a), and full spectrum on the basis of matrix element ⟨Ψf| ˆHγ| Ψi⟩star in Eq. (44) (b) are shown in
these figures.
0
10
20
30
40
50
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
a=0.95fm
a=0.85fm
E
p
= 50 MeV
(a)
p +
4
He
d
/ dE
(b / MeV )
Photon energy, E
(MeV)
FIG. 7: (Color online) Cross-sections of bremsstrahlung photons of electric type emitted during the scattering of protons off
4He inside star (red dashed line) and in vacuum (blue solid line) [parameters of calculations: rmax = 20000 fm, 10000 intervals
of integration of matrix elements of emission ]. One can see small difference between the spectra at high energy part of photons.
In Fig. 7 one can see spectra of the bremsstrahlung photons emitted in p + 4He is star in comparison with the
same reaction in vacuum. In such calculations, we just use deformation of nucleus due to influence of stellar medium
(for convenience we chose stage before disintegration of nucleus, and obtain parameter a = 0.85 fm, in vacuum we
have a = 0.95 fm) without inclusion of contribution ⟨Ψf| ∆ˆHγ| Ψi⟩. One can see that such an influence is almost
neglectable in comparison with influence of term ⟨Ψf| ∆ˆHγ| Ψi⟩given in Fig. 6.
VI.
CONCLUSIONS
In this paper we investigate ability (role) of nuclear forces to combine nucleons as bound nuclear system in de-
pendence on its deep location inside the compact star. In order to perform such a research, we generalize the model
of deformed oscillator shells [40, 41, 43] with two nucleon forces with new inclusion of additional influence of stellar
medium. We have obtained new simple exact formulas of energy for the lightest even-even nuclei, that is convenient
for analysis of stellar influence on binding energy of nuclei. As studied star, polytropic stars at n = 3 with densities
13
characterized from white dwarf to neutron star were included to analysis.
We observe a phenomenon of dissociation of nucleus — its disintegration on individual nucleons, starting from
some critical distance between this nucleus and center of star with high density. We explain this phenomenon by the
following logic. Forces of stellar medium press on nucleons of nucleus. The deeper this nucleus is located in star, the
stronger such forces press on nucleus. However, binding energy (it is negative for nucleus in the external layer of star)
is increased at deeper location of this nucleus in star. Starting from some critical distance from nucleus to center
of star, the binding energy becomes positive (see Fig. 4). This means that full energy of individual nucleons of the
studied nucleus is already larger than mass of this nucleus, i.e. we obtain unbound system of nucleons and nucleus is
disintegrated on nucleons. According to estimations, we observe such a phenomenon in neutron stars, while in white
dwarfs its is not observed. We have estimated such a critical distance for nucleus 4He in dependence on density at
center of neutron star (see Fig. 5), where disintegration of this nucleus on nucleons takes place. The kinetic energy
is increased at deeper location of nucleus in star. At decreasing distance from the studied nucleus to center of star,
change of kinetic energy is unlimited, while change of nuclear energy is limited. So, ratio between kinetic energy of
nucleons of nucleus and nuclear energy of nucleus is changed also.
Basing on the model above, we have generalized the bremsstrahlung formalism [62, 63] (see also improvements of
this formalism in Refs. [64–68]) for scattering of protons offnuclei in compact stars. Using such a new model, we find
the following. (1) In the white dwarfs, influence of stellar medium imperceptibly affects on emission of bremsstrahlung
photons. This means that we have enough accurate description of emission of bremsstrahlung photons during nuclear
reactions in Sun, white dwarfs and similar stars. (2) For neutron stars, influence of stellar medium is essentially more
intensive and it crucially changes shape of the spectrum of the bremsstrahlung photons. Maximum of probability
of the emitted photons is for their energy, which is half of energy of the scattered protons: Eph ≃Ep/2. The most
intensive emission is created in the bowel of the star, while the weakest emission is from the periphery (for the same
energy of the scattered proton).
Summarizing, we find the model of deformed oscillator shells as convenient and not complicated technically basis
for obtaining clear understanding about different forces and emission of bremsstrahlung photons in nuclear reactions
in compact stars.
Acknowledgements
S. P. M. is highly appreciated to Dr.
A. I. Steshenko for deep insight to the DOS model and help.
Authors
are highly appreciated to Profs. V. S. Vasilevsky, M. I. Gorenstein, A. V. Nesterov for useful recommendations and
interesting discussions concerning to modern many-nucleons nuclear models and physics of nuclear processes inside
dense stellar medium. Authors also highly appreciated to Prof. Janos Balog for interesting discussions concerning to
physics of nucleon-nucleon interactions, and Prof. Zhigang Xiao for interesting discussions concerning to emission of
bremsstrahlung in heavy-ion collisions.
Appendix A: Correction of energy of nucleus due to influence of stellar medium
In this Section we shall find correction of energy of nucleus due to influence of stellar medium (30):
∆Estar =
D
Ψ(1 . . . A)
A
X
i,j=1
Vstar(R, ri, rj)
Ψ(1 . . . A)
E
.
(A1)
Substituting Eq. (29) to this formula and taking into account the same action of force FP (R) for each nucleon, we
obtain:
∆Estar = FP (R) ·
A
X
i,j=1
D
Ψ(1 . . . A)
ri −rj
Ψ(1 . . . A)
E
.
(A2)
We use property:
⟨Ψf(1 · · · A) | ˆV (ri, rj)| Ψi(1 · · · A)⟩=
=
1
A (A −1)
A
X
k=1
A
X
m=1,m̸=k
⟨ψk(i) ψm(j)| ˆV (ri, rj)| ψk(i) ψm(j)⟩−⟨ψk(i) ψm(j)| ˆV (ri, rj)| ψm(i) ψk(j)⟩
.
(A3)
14
Here, summation is performed over all states for the given configuration of nucleus (they are denoted by indexes m
and k). All nuclerons are numeberd by indexes i and j. We use representation for one-nucleon wave function:
ψλs(s) = ϕns(rs)
σ(s)τ (s)
,
(A4)
where ϕns is space function of the nucleon with number s, ns is number of state of the space function of the nucleon
with number s,
σ(s)τ (s)
is spin-isospin function of the nucleon with number s. For operator ˆV (ri, rj) acting on
space functions for two nucleons only, we calculate matrix element:
⟨Ψf(1 · · · A) | ˆV (ri, rj)| Ψi(1 · · · A)⟩=
1
A (A −1)
A
X
k=1
A
X
m=1,m̸=k
D
ϕk(ri) ϕm(rj)
ˆV (ri, rj)
ϕk(ri) ϕm(rj)
E
−
−
D
ϕk(ri) ϕm(rj)
ˆV (ri, rj)
ϕm(ri) ϕk(rj)
E
σ(k)(i)
σ(m)(i)
σ(m)(j)
σ(k)(j)
τ (k)(i)
τ (m)(i)
τ (m)(j)
τ (k)(j)
,
(A5)
where orthogonalization properties of spin and isospin functions are used:
σ(k)(i)
σ(k)(i)
= 1,
τ (k)(i)
τ (k)(i)
= 1.
(A6)
In particular, for 4He Eqs. (A6) are simplified:
σ(k)(i)
σ(m)(i)
= δkm,
τ (k)(i)
τ (m)(i)
= δkm,
(A7)
and we obtain
⟨Ψ(4He) | ri −rj| Ψ(4He)⟩=
Z
F 2
0 (ri, rj) (ri −rj) dr1 dr2,
(A8)
where Eq. (8) for F0(ri, rj) is used. For correction of energy, from (A1) we obtain:
∆Estar(4He) = FP (R) ·
A=4
X
i,j=1
D
Ψ(4He)
ri −rj
Ψ(4He)
E
= 12 · FP (R) ·
Z
F 2
0 (r1, r2) (r1 −r2) dr1 dr2.
(A9)
In the spherically symmetric case (a = b = c), we calculate integral:
Z
F 2
0 (r1, r2) |r1 −r2| dr1 dr2 =
1
π3 a6
Z
exp
h
−x2
1 + x2
2 + y2
1 + y2
2 + z2
1 + z2
2
a2
i
· r12 dr1 dr2 = 23/2 a
π1/2
(A10)
and obtain solution:
∆Estar(4He) = 12 · FP (R) ·
Z
F 2
0 (r1, r2) |r1 −r2| dr1 dr2 = 12 · 23/2 a
π1/2
· FP (R).
(A11)
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arxiv | 0812.4649v3 | Towards a warped inflationary brane scanning | We present a detailed systematics for comparing warped brane inflation with the observations, incorporating the effects of both moduli stabilization and ultraviolet bulk physics. We explicitly construct an example of the inflaton potential governing the motion of a mobile D3 brane in the entire warped deformed conifold. This allows us to precisely identify the corresponding scales of the cosmic microwave background. The effects due to bulk fluxes or localized sources are parametrized using gauge/string duality. We next perform some sample scannings to explore the parameter space of the complete potential, and first demonstrate that without the bulk effects there can be large degenerate sets of parameters with observationally consistent predictions. When the bulk perturbations are included, however, the observational predictions are generally spoiled. For them to remain consistent, the magnitudes of the bulk effects need to be highly suppressed via fine tuning. | Heng-Yu Chen, Jinn-Ouk Gong | hep-th, astro-ph, gr-qc, hep-ph | https://arxiv.org/abs/0812.4649v3 | arXiv:0812.4649v3 [hep-th] 31 Aug 2009
MAD-TH-08-16
Towards a warped inflationary brane scanning
Heng-Yu Chen∗
Department of Physics, University of Wisconsin-Madison, Madison, WI 53706-1390, USA
Jinn-Ouk Gong†
Instituut-Lorentz for Theoretical Physics, Universteit Leiden, 2333 CA Leiden, The Netherlands
We present a detailed systematics for comparing warped brane inflation with the observations, incorporating
the effects of both moduli stabilization and ultraviolet bulk physics. We explicitly construct an example of the
inflaton potential governing the motion of a mobile D3 brane in the entire warped deformed conifold. This
allows us to precisely identify the corresponding scales of the cosmic microwave background. The effects due
to bulk fluxes or localized sources are parametrized using gauge/string duality. We next perform some sample
scannings to explore the parameter space of the complete potential, and first demonstrate that without the bulk
effects there can be large degenerate sets of parameters with observationally consistent predictions. When the
bulk perturbations are included, however, the observational predictions are generally spoiled. For them to remain
consistent, the magnitudes of the additional bulk effects need to be highly suppressed.
PACS numbers: 98.80.Cq, 11.25.Mj
CONSTRUCTING A POTENTIAL FOR WARPED BRANE
INFLATION
The inflationary paradigm [1] addresses a number of fine
tuning problems of the standard hot big bang cosmology, such
as the horizon and the flatness problems. It also predicts a
nearly scale invariant power spectrum of the curvature pertur-
bation, which has been verified to high accuracies by the ob-
servation of the thermal fluctuations in the cosmic microwave
background (CMB) and the large scale structure of the uni-
verse [2, 3]. Numerous models of inflation based on effective
field theory have been proposed, however, distinct predictions
of a given model crucially depend on its ultraviolet comple-
tion. To construct a truly predictive inflationary model, it is
clearly important to embed it into a consistent microscopic
theory of quantum gravity such as string theory.
During the past few years, our understanding of the various
ingredients for obtaining string inflation has been significantly
expanded, and many models with increasing sophistication
and striking signatures have been proposed (For recent devel-
opments, see Ref. [4] and references therein). In the com-
ing decade, beyond the ongoing Sloan Digital Sky Survey [2]
and the Wilkinson Microwave Anisotropy Probe [3], vastly
improved cosmological data will become available from the
advanced CMB observations [5], the CMB polarization ex-
periments [6], the dark energy surveys [7] as well as the map
of large scale structure [8]. They will allow us to constrain
the parameter spaces of these models, and possibly to even
rule out some of them. It is therefore of timely interest to per-
form a thoroughly updated and complete case study in such
a direction1. In this paper we shall focus on one of the most
1 What we mean by “complete” should become clear momentarily.
developed string inflation models in the literature, usually re-
ferred to as “brane inflation”.
The original setup of brane inflation first introduced in
Ref. [9] is to consider a pair of spacetime-filling D3-D3
branes, separated at some distance greater than the local string
length on a compact six manifold. As D3 and D3 move to-
wards each other under the Coulombic attraction, the canoni-
cal inflaton is then identified as the separation between them.
Unfortunately in such a simple setup, the Coulombic attrac-
tion is too strong for the slow-roll inflation to persist. To over-
come this obstacle, the authors of Ref. [10] considered instead
placing the D3-D3 pair in a locally warped deformed conifold
throat developed in a compact Calabi-Yau orientifold by back-
ground fluxes. The D3 is then stabilized at the tip of deformed
conifold, and D3 is attracted weakly by the warped down D3-
D3 potential given by
VD3D3(τ,σ) = D0
U2
1 −
3D0
16π2T 2
3 |y −¯y|4
!
.
(1)
Here D0 = 2T3a4
0, T3 = 1/[(2π)3gs(α′)2] is the D3 brane ten-
sion, a0 = exp[−2πK/(3gsM)] is the warp factor at the tip of
deformed conifold with −K and M being the quanta of NSNS
and RR three form fluxes, and |y−¯y| is the D3-D3 separation.
Furthermore, the factor U is the universal Kähler modulus,
whose role we shall discuss in detail later. Notice that the first
term in (1) gives the positive contribution and uplifts the total
potential energy, whereas the Coulombic attraction is highly
suppressed by a8
0 and only becomes dominating near the tip
of deformed conifold. Eventually when |y−¯y| becomes com-
parable to warped string length ∼a0ls, D3 and D3 annihilate
through open string tachyon condensation, rapidly terminates
the brane inflation.
There are two important further ingredients that have so far
been missing in our discussion, and they are crucial for obtain-
ing the inflationary phase and making detailed comparisons
2
with observational data. The first ingredient is the stabiliza-
tion of both closed and open sting moduli. They are usually
stabilized by the perturbative flux potential [11] and the non-
perturbative superpotential generated by wrapped branes [12].
The second ingredient is the ultraviolet corrections arising
from embedding the warped throat into a compact Calabi-
Yau orientifold. The bulk fluxes and the distant branes, or
additional supersymmetry breaking and moduli stabilization
sources can give significant perturbations to the inflaton po-
tential derived from the local sources and geometry.
The warped deformed conifold offers us an ideal venue
to analyze these two ingredients. Its explicit metric [13] al-
lows for studying the moduli stabilization and the construc-
tion of the inflaton potential valid for the entire evolution, in-
cluding precise identification of where inflation ends. Fur-
thermore while bulk physics is largely unknown, the spec-
trum of supergravity states in singular conifold has been tab-
ulated in Ref. [14]. The gauge/string duality then allows us
to parametrize these symmetry breaking bulk perturbations to
the inflaton potential by coupling dual (approximately) con-
formal field theory to these bulk modes [15]. Combining these
with the D3-D3 interaction, we can schematically parametrize
the total potential of the inflaton φ including both local and
bulk effects, experienced by a mobile D3 brane in the warped
deformed conifold as
V(φ) = VD3D3(φ)+Vstab.(φ)+Vbulk(φ).
(2)
Here Vstab.(φ) arises from moduli stabilization and Vbulk(φ)
encodes all other possible perturbations from bulk physics2.
Most of the quantities specifying Vstab.(φ) are exclusively re-
lated to the local geometry of the throat, e. g. the warp factor
a0. However Vstab.(φ) typically also depends on other quan-
tities controlled by bulk physics, such as the one loop deter-
minant of the non-perturbative superpotential. The quantities
controlling Vstab.(φ) are usually treated as free parameters and
yield a landscape of possible inflaton potentials.
The current observational data [2, 3] enable us to make
comparisons with the predictions yielded by different param-
eter sets and constrain their allowed values. In order for this
exercise to be instructive, it is crucial to include all the signif-
icant contributions to the inflaton potential. The existing liter-
ature in this direction [16, 17] has mostly focused on the first
two contributions in (2) without taking into account Vbulk(φ).
However in light of recent results in Ref. [15] indicating that
Vbulk(φ) can be generically comparable to Vstab.(φ), it is clearly
necessary to apply such general results and scan the enlarged
parameter space to include the bulk corrections.
2 Although we follow similar scheme as in Ref. [15], we do not partition
Vstab.(φ) such that the inflaton mass term ∼H2φ2 is singled out. As H2 is
usually a combination of microscopic parameters, for the purpose of full
parameter scanning we shall calculate Vstab.(φ) in full detail and express it
explicitly in terms of the microscopic parameters while treating Vbulk(φ) as
further perturbations.
Here we aim to provide some initial steps toward a com-
plete systematic parameter scanning for warped brane infla-
tion. We shall first consider a specific brane configuration and
stabilize explicitly both universal Kähler modulus and some
of the angular moduli of D3. We then construct an example
of VD3D3(φ)+Vstab.(φ) valid for entire warped deformed coni-
fold throat. This potential should be regarded as the infrared
completion to the model obtained in Ref. [18] under the sin-
gular conifold limit (See also [19, 20] for related work). It
allows us to identify the end point of inflation, hence extrap-
olate precisely to the CMB scale. Next we shall briefly re-
view the parametrization of the bulk effects Vbulk(φ) given in
Ref. [15], and discuss the microscopic and observational con-
straints on the inflationary parameter scanning. Finally we
shall first present different degenerate parameter sets such that
the resultant VD3D3(φ) +Vstab.(φ) yields observationally con-
sistent curvature power spectrum PR and the corresponding
spectral index nR . We next demonstrate that the perturba-
tions due to Vbulk(φ) can have significant impact and need to
be small to preserve these seemingly optimal parameter sets.
These sample scannings aim to highlight the possible degen-
eracies and the important role of bulk effects.
ENUMERATING THE INFLATON POTENTIAL
Moduli stabilization potential from warped throat
In this section, we shall explicitly consider the effects of
moduli stabilization on the mobile D3 brane in the entire de-
formed conifold. This is important for accurate comparisons
between the predictions and the observational data. In partic-
ular, this allows us to identify precisely the end point of infla-
tion φe, defined to be the point where the slow-roll parameter
ε ≡2M2
Pl
H′(φ)
H(φ)
2
≈M2
Pl
2
V′(φ)
V(φ)
2
(3)
becomes 1 so that the universe ceases accelerated expansion.
Here, MPl = (8πG)−1/2, H(φ) is the field dependent Hubble
parameter and a prime denotes a derivative with respect to
φ. Note that the second approximation holds under the slow-
roll limit. It is crucial to properly take into account the late
evolution of the universe during inflation for making correct
inflationary predictions. The form of the potential near the
end of inflation can substantially lower the inflationary en-
ergy scale [21], and the light fields other than the canonical
inflaton can completely dominate the curvature power spec-
trum PR and the corresponding spectral index nR [22]. Fur-
thermore, the post-inflationary evolution can also modify the
spectral index at an observationally detectable level [23].
In the context of warped brane inflation, ε(φ) is driven to-
wards 1 mostly by the D3-D3 Coulombic attraction, which is
exponentially suppressed and only becomes significant inside
the deformed conifold. Moreover, as shown in Ref. [24] when
Coulombic attraction is insignificant, ε(φ) remains small all
3
the way to the tip for generic parameter sets. We therefore
expect inflation persists well into the deformed conifold re-
gion, despite only a proportionally small number of e-folds
is expected to be generated there. Moreover as some of the
inflationary predictions are already tightly constrained by ob-
servations to high degrees of accuracies, e.g. PR and nR , it
is important to take into account such infrared completion in
constraining the parameter space of brane inflation.
The key component capturing the moduli stabilization ef-
fects on the mobile D3 is the N = 1 supergravity F-term
scalar potential,
VF(zα, ¯zα,ρ, ¯ρ) = eκ2K
K ΣΩDΣWDΩW −3κ2|W|2
,
(4)
where κ2 = M−2
Pl . Let us discuss various contributions to (4) in
turn. In the presence of a D3 brane, the universal Kähler mod-
ulus U(z,ρ) depends on the brane position {zα , ¯zα} and the
usual complex bulk Kähler modulus ρ = σ + iχ. The indices
in (4) therefore runs over {ρ,zα} and total Kähler potential is
the given by [25]
κ2K (zα, ¯zα,ρ, ¯ρ) =−3log[ρ+ ¯ρ−γk(zα, ¯zα)]
≡−3logU(z,ρ).
(5)
Here, k(zα, ¯zα) is the geometric Kähler potential of the metric
on the Calabi-Yau, and γ = σ0T3/(3M2
P) is the normalization
constant with σ0 being the value of σ when the D3 brane is at
its stabilized configuration [18], including the uplifting poten-
tials3.
To stabilize some of the geometric and Kähler moduli, we
need to consider the total superpotential W(zα,ρ) consisting
of two contributions as
W(zα,ρ) = W0 + A(zα)e−aρ .
(6)
The first term W0 =
R G3 ∧Ω3 is the perturbative flux super-
potential [11], which allows us (at least in principle) to sta-
bilize the complex structure moduli and dilaton-axion. One
mechanism for stabilizing ρ and some of the mobile D3 brane
position moduli is to include non-perturbative effects through
gaugino condensation on a stack of space-filling D7 branes
(or a Euclidean D3 brane instanton), as appears in the second
term of (6). The prefactor A(zα) is a holomorphic function of
the D3 brane moduli and can be written as [28]
A(zα) = A0
f(zα)
f(0)
1/n
.
(7)
Here A0 is a complex constant whose exact value depends on
the stabilized complex structure moduli, and n is the number
of D7 (or n = 1 for Euclidean D3) giving the gaugino conden-
sate (or instanton correction). The parameter a in (6) is given
by 2π/n. The explicit dependence on the position of mobile
D3 brane appears through the holomorphic embedding func-
tion f(zα) = 0 of the four cycle in the Calabi-Yau space.
Substituting the total superpotential (6) as well as the ex-
pression for the inverse metric K ΣΩsolved in Ref. [19] into
(4), the explicit form of VF(zα, ¯zα,ρ, ¯ρ) is given by
VF(zα, ¯zα,ρ, ¯ρ)
=
κ2
3[U(z,ρ)]2
nh
U(z,ρ)+ γkγ¯δkγk¯δ
i
|W,ρ|2 −3
WW,ρ + c.c.
o
+
κ2
3[U(z,ρ)]2
kα¯δk¯δW ,¯ρW,α + c.c.
+ 1
γ kα¯βW,αW ,¯β
.
(8)
Here, the subscript of a letter with a comma denotes a partial
differentiation with respect to the corresponding component.
Specifically for a deformed conifold defined by the complex
embedding equation ∑4
α=1(zα)2 = ε2 with zα ∈C, the Kähler
potential is given by
k(τ) = ε4/3
21/3
Z
τ dτ′
sinh(2τ′)−2τ′1/3 .
(9)
In writing (9), we have also used the standard relation
∑4
α=1 |zα|2 = ε2 coshτ (See Refs. [13, 29] for the explicit met-
ric in terms of τ and angular coordinates). To apply the general
formula (8), we note the inverse metric k¯ij is given by
k¯ij = r3
k′′
R¯ij + cothτ
k′′
k′ −cothτ
L¯ij
, (i, ¯j = 1,2,3)
(10)
where k′ = dk/dτ and k′′ = d2k/dτ2, and the 3 × 3 matrices
R¯ij and L¯ij in (10) are, respectively,4
R¯ij =δ¯ij −zi¯zj
r3 ,
(11)
L¯ij =
1 −ε4
r6
δ¯ij + ε2
r3
zizj + ¯zi¯zj
r3
−zi¯zj + ¯zizj
r3
.
(12)
We can now readily calculate various terms depending on the
inverse deformed conifold metric k¯ij in the F-term scalar po-
tential (8). First we notice that L¯i j has the property k¯iL¯i j =
L¯ijkj = 0; therefore, the norm k¯i jk¯ikj is given by
k¯i jk¯ikj = 3
4
ε4/3
21/3
[sinh(2τ)−2τ]4/3
sinh2 τ
.
(13)
Similarly, we can calculate that
4
3 Strictly speaking, the derivation of (5) given in Ref. [25] is invalid for the
warped background, hence raises the question about the validity of (5) itself
in the warped deformed conifold. However, some interesting new develop-
ment in Ref. [26] about the universal Kähler modulus indicates that (5) can
remain valid in the warped background. It would be useful to verify this by
combining the earlier work on the dynamics of warped compactification,
e.g. Ref. [27] with the recent results, Ref. [26]. We thank Bret Underwood
for discussing with us about this issue.
4 Here, we have made the substitution z4 = ±
q
ε2 −(z2
1 +z2
2 +z2
3).
k¯ijk¯iWj =3
4
coshτ
sinh3 τ [sinh(2τ)−2τ]
3
∑
j=1
zj −¯zj
ε2
r3
A je−aρ ,
(14)
k¯ijW ¯iWj =
3
2 ·22/3ε2/3 coshτ
sinhτ2 [sinh(2τ)−2τ]2/3
R¯ijW ¯iWj +
2
3
sinh(2τ)
sinh(2τ)−2τ −coth2 τ
× L¯ijW ¯iWj
,
(15)
where
R¯ijW ¯iWj =e−2aσ
"
3
∑
i=1
|Ai|2 −1
r3
3
∑
i=1
zi ¯Ai
!
3
∑
j=1
¯zjA j
!#
,
(16)
L¯ijW ¯iWj =e−2aσ
(
3
∑
i=1
1 −ε4
r6
|Ai|2 −1
r3
3
∑
i,j=1
¯Ai
zi¯zj + zj¯zi −ε2
r3 (zizj + ¯zi¯zj)
A j
!)
.
(17)
Putting various components together, we can obtain the gen-
eral expression of the F-term scalar potential in deformed
conifold. We shall see that for a specific D7 embedding given
in Ref. [30], the resultant expression nicely simplifies along
its angular stable trajectory.
A case study
As an explicit example, we consider specifically the D7
brane embedding given by [30]
f(zα) = z1 −µ,
(18)
from which we can easily find that
A(zα) =A0
1 −z1
µ
1/n
,
(19)
Ai(zα) =−A0
nµ
1 −z1
µ
1/n−1
δi1 .
(20)
Without lost of generality, we shall take µ ∈R+. The em-
bedding (18) is highly symmetric, and preserves SO(3) sub-
group of the full SO(4) continuous isometry group of the
deformed conifold.5
Substituting Ai(zα) and A j(zα) into
the earlier expressions derived for the F-term scalar poten-
tial, the dependence on the D3 brane position now only ap-
pears through the combinations (z1 + ¯z1) and |z1|2. The re-
sultant expression therefore has the functional dependence
VF = VF
z1 + ¯z1,|z1|2,τ,σ,χ
.
5 The actual angular stable trajectory however only preserves SO(2) sub-
group of SO(3).
In addition, we also need to stabilize some of the moduli
appearing in VF
z1 + ¯z1,|z1|2,τ,σ,χ
following the standard
procedure outlined in Ref. [18]. First the axion of the com-
plex Kähler modulus χ can be stabilized by rotating the phase
of the flux induced superpotential W0 ∈R−, and making the
replacement exp(iaχ)/A(zα) →1/|A(zα)|. As the isometry of
the deformed conifold is partially broken by D7 branes, some
of the angular coordinates of the mobile D3 can also be sta-
bilized by the resultant F-term scalar potential. In Ref. [24],
such specific angular stable trajectory for the D7 embedding
(18) for the entire deformed conifold is derived to be
z1 = −εcosh
τ
2
.
(21)
We refer the readers to Ref. [24] for the derivation of this tra-
jectory and the discussion about its stability6. The resultant
two-field scalar potential VF(τ,σ), for such an angular stable
trajectory, is thus given by
VF(τ,σ) =2a2κ2|A0|2e−2aσ
[U(τ,σ)]2
|g(τ)|2/n
×
U(τ,σ)
6
+ 1
a
1 −|W0|
|A0|
eaσ
[g(τ)]1/n
+ F(τ)
,
(22)
6 Furthermore, as the angular dependences are only encoded in the F-term
scalar potential (at least for the region where most of e-folds occur), we
expect including the uplifting term does not affect the stability analysis.
5
where various functions in VF are
U(τ,σ) =2σ−γk(τ),
(23)
g(τ) =1 + ε
µ cosh
τ
2
,
(24)
F(τ) =ε4/3γ
h
K(τ)sinh
τ
2
i2
×
K(τ)cosh
τ
2
−
ε/µ
4πε4/3γg(τ)
2
,
(25)
K(τ) =[sinh(2τ)−2τ]1/3
21/3sinhτ
.
(26)
One can check that (22) smoothly interpolates to the two-field
potential derived in Ref. [18] in the large τ limit ε2 coshτ ≈
ε2eτ/2 ≈r3, where r is the usual radial coordinate of the sin-
gular conifold7.
Having obtained the two-field F-term scalar potential
VF(τ,σ), the canonical inflaton can be derived from the DBI
action of a mobile D3 brane moving in the full deformed coni-
fold metric as the following integral expression;
φ(τ) =
r
T3
6 ε2/3
Z
τ
dτ′
K(τ′) .
(27)
Here, we have used the explicit deformed conifold metric
given in terms of radial and angular coordinates (see, for ex-
ample, Refs. [29, 31]), and one can see this definition has the
asymptotic limits
φ(τ) →
r
3
2T3r,
(τ ≫1)
√T3
25/631/6 ε2/3τ,
(τ ≪1)
(28)
where we have used the definition r3 = ε2 coshτ to rewrite the
τ ≫1 limit. The expressions of the canonical inflaton in the
large and small τ limits have been used in Refs. [18] and [24],
respectively.
As the deformed conifold throat is attached to a compact
Calabi-Yau at some finite ultraviolet radius rUV, it is impor-
tant to stabilize the volume modulus σ, which controls the
overall size. Within the adiabatic approximation proposed in
Ref. [18], such that σ is stabilized at an instantaneous mini-
mum as the radial coordinate τ varies, this amounts to solving
the equation
∂(VF +Vuplift)(τ,σ)
∂σ
σ⋆[φ(τ)]
= 0.
(29)
Here,
we have included the positive definite potential
Vuplift(τ,σ) = (D0 + Dothers)/[U(τ,σ)]2, which is required
7 However, we have checked that once the volume modulus σ is stabilized in
the adiabatic approximation we shall discuss next, there are deviations in
resultant single field potentials, due to different radial dependence of σ(τ).
to uplift the total energy and to obtain a de Sitter phase.
Vuplift(τ,σ) can include the first term of VD3D3(φ) given by
(1) and other supersymmetry breaking sources in the bulk as
encoded in Dothers/[U(τ,σ)]2, which can be generated by dis-
tant D3 or wrapped D7 with supersymmetry breaking world
volume fluxes [32]8. One can also parametrize the uplifting
potential by defining the uplifting ratio s as
s = Vuplift(0,σF)
|VF(0,σF)| ,
(30)
with σF being given by ∂σVF(0,σ)|σ=σF = 0. The distant
sources are essentially needed for a small positive cosmolog-
ical constant at the end of inflation after D3-D3 annihilation.
Combining this fact with the requirement that s ≲3 during
inflation for avoiding runaway decompactification, one can
deduce that Dothers should typically dominate over D0. Al-
ternatively, one can also argue that as the distant sources are
located in the unwarped region, it should naturally dominate
over the D3 localized at the tip of the highly warped deformed
conifold [24].
Equation (29) is transcendental and is usually solved nu-
merically. However to get a qualitative understanding, we
can adopt a semi-analytic approach given in Ref. [18], where
one sets the σ dependence in U(τ,σ) equals to large fixed
value σ0 and treat (29) as a quadratic equation of the variable
exp[−aσ⋆(φ)]. A double expansion in 1/σ0 and φ(τ) around
the tip region, such that φ(τ) is approximated by the τ ≪1
limit of (28), then yields at leading order correction
σ⋆(φ) ≈σ0
(
1 +
1
aσF
"
1
3 + (2/3)2/3α
8n(1 + α)β
# φ
MPl
2)
. (31)
In deriving the above expression we have used the approxima-
tion aσ0 ≈aσF + s/(aσF) given in Ref. [18]9. Note that we
have introduced two important dimensionless parameters
α = ε
µ ,
(32)
β =
r
T3
6
ε2/3
MPl
.
(33)
8 The precise U(τ,σ) dependence in fact varies for different distant super-
symmetry breaking sources: for D3 the potential ∼U(τ,σ)−2 and for D-
term uplifting [32] induced by D7 carrying supersymmetry breaking flux,
it is ∼U(τ,σ)−3. Here in the limit U(τ,σ) ≫1, we merely keep the most
dominant contribution.
9 Let us comment on the difference between the expression for σ⋆(φ)
in Ref. [18],
which was given schematically by σ⋆(φ) ≈σ0[1 +
b3/2(φ/MPl)3/2], and our expression (31). In Ref. [18], VF(τ,σ) was calcu-
lated exclusively for the large radius, singular conifold limit. The authors
of Ref. [18] then expanded in canonical inflaton φ ≈
p
3T3/2r around the
near tip region of deformed conifold to extract the radial dependence of the
stabilized volume. Here we improved upon such calculation, using VF(τ,σ)
for the entire deformed conifold and expanding near the tip of the deformed
conifold using the small radius limit of the canonical inflaton (28) to obtain
the expression (31).
6
Geometrically, α measures the depth which D7 branes extend
into deformed conifold, and β is proportional to the warp fac-
tor a0 at the tip. Of course the analytic approximation for the
stabilized volume σ⋆(φ) only gives a qualitative understand-
ing, and is expected to deviate from the actual behavior at
large radius. For full quantitative parameter scanning how-
ever, the numerical solution to (29) can also be readily imple-
mented.
Combining our expression for the stabilized volume σ⋆(φ)
given by (31), the potentialVD3D3(φ)+Vstab.(φ) for the D7 em-
bedding (18) in the entire deformed conifold is finally given
by
VD3D3(φ)+Vstab.(φ) =2a2κ2|A0|2e−2aσ⋆(τ)
{U[τ,σ⋆(τ)]}2
|g(τ)|2/n
(
U[τ,σ⋆(τ)]
6
+ 1
a
1 −|W0|
|A0|
eaσ⋆(τ)
[g(τ)]1/n
!
+ F(τ)
)
+
D(φ)
{U[τ,σ⋆(τ)]}2 , (34)
D(φ) =D0
1 −27D0
64π2φ4
+ Dothers .
(35)
Here, we should regard the radial coordinate τ to be an im-
plicit function of the canonical inflaton φ given by (27). In
addition, as shown in Ref. [24] the residual angular isome-
try directions becomes degenerate along the trajectory (21).
Therefore D3-D3 separation |y −¯y| is purely radial and pro-
portional to the canonical inflaton φ(τ) for the entire deformed
conifold. In Fig. 1, we show the plot of the potential (34) with
the parameters given by Case 1 of Table I.
0.0
0.2
0.4
0.6
0.8
1.0
7.70
7.75
7.80
φ/φµ
1017 × V/M 4
Pl
0.00
0.01
0.02
0.03
0.04
0.05
7.660
7.665
7.670
7.675
7.680
7.685
7.690
FIG. 1: The plot of the potential (34) as a function of φ/φµ, with
φ2µ ≡3T3/2(2µ2)2/3. The parameters are set the same as Case 1
of Table I. We show two extreme cases of D(φ): either it is com-
pletely dominated by the Coulombic interaction (solid line) or by
the distant sources (dotted line). Note that the difference becomes
only noticeable at the region very close to the tip, as shown in the
inset, which magnifies the potential in this region. This implies in-
flation only ends when φ approaches close to the tip, even if the po-
tential is highly curved by the Coulombic term: in the case shown
here, φe ≈0.0105φµ, meanwhile the “potential” slow-roll parameter
with higher order corrections [33] gives φe ≈0.0700φµ. Note that
|η| ≡|M2
PlV′′/V| = 1 well before this point, at φ ≈0.324φµ.
Let us conclude this section by revisiting the η problem
discussed in Ref. [18], now with the potential (34) valid in the
region near the tip with the small field canonical inflaton given
in (28) and the stabilized volume (31). After some expansions,
we can obtain
VD3D3(φ)+Vstab.(φ)
VD3D3(0)+Vstab.(0) ≈1 + φ2
3M2
Pl
1 +O
1
σ0
+O(φ4).
(36)
Notice that the dependence of the gaugino condensate on the
mobile D3 brane position does give corrections to the infla-
ton mass in the near tip region. However, such corrections are
suppressed by the large stabilized volume σ0, and are insuf-
ficient to give small inflaton mass. Thus, η remains of order
one10. This is in fact consistent with the analysis in Ref. [18]
using the singular conifold approximation, that the inflection
point η = 0 only appears at some intermediate radius11.
Parametrization of the bulk effects
To account for the ultraviolet physics arising from at-
taching the warped throat to a compact Calabi-Yau, a use-
ful parametrization of the leading corrections was given in
Ref. [15].
The authors employed gauge/string correspon-
dence for the warped deformed conifold (see, for example,
Refs. [29, 31]), where the position of the mobile D3 is iden-
tified with the Coulomb branch vacuum expectation value of
the dual field theory. In such a holographic formulation, the
symmetry breaking bulk effects can be encoded by coupling
a field theory operator O∆of scaling dimension ∆to its dual
bulk mode and a perturbation to inflaton φ potential is gener-
10 Notice that on the other hand |ε| ≪1, as we do not have trans-Planckian
field displacement ∆φ/MPl ≪1.
11 Notice that the analysis here is accurate for the near tip region. The required
cancelation term ∝φ3/2 for obtaining inflection point, only appears when
the large radius canonical inflaton (28) and the associated stabilized volume
expression are substituted in the derivation.
7
ated as
∆V = −c∆a4
0T3
φ
φUV
∆
.
(37)
Here φUV =
p
3T3/2rUV and rUV is the radius at which the
deformed conifold throat joins the compact Calabi-Yau. The
positive constant c∆depends on the specific distant fluxes
or brane configurations12.
Varying its value allows us to
parametrize our ignorance about this information. The nor-
malization of a4
0T3 in (37) comes from the estimated energy
required to move the mobile D3 from its supersymmetric min-
imum to the four cycle moduli stabilizing D7 wraps. This is
proportional to the height of the anti de Sitter potential barrier
which in our more detailed setup should be identified explic-
itly with |VF(0,σF)|.
There can in fact be whole series of perturbations of the
form given in (37). However the two leading contributions
come from the lowest chiral multiplet of dimension 3/2, O3/2,
and the lowest non-chiral multiplet of dimension 2, O2. For
our case, the bulk potential as denoted in (2) is then given by
Vbulk(φ) = −|VF(0,σF)|
"
c3/2
φ
φUV
3/2
+ c2
φ
φUV
2#
.
(38)
One can of course include other higher dimensional operators
in Vbulk(φ). The terms above are merely to illustrate the im-
portance of bulk physics in our later sample parameter scan-
nings13. However, one should note that when more than one
(φ/φUV)∆is turned on in (38), there are generally additional
angular perturbations. This comes from the fact that individ-
ual coefficient c∆is obtained from integrating out complicated
angular dependences. When more than one c∆are involved, it
is generally not possible to perform such integrating out14.
CONSTRAINING PARAMETER SPACE: MICROSCOPIC
AND OBSERVATIONAL
Microscopic constraints
Let us first list out the explicit parameters specifying the
total single field inflaton potential V(φ) = VD3D3[φ,σ⋆(φ)] +
Vstab.[φ,σ⋆(φ)]+Vbulk[φ,σ⋆(φ)]; they are
n,|A0|,|W0|,s,ε,µ,c3/2 ,c2
.
(39)
12 To be specific, c∆used here only incorporate strictly bulk effects. This is
in contrast with Ref. [15], where the c∆coefficients there can receive both
local and bulk contributions.
13 In Ref. [34], an earlier attempt to perform parameter scanning using (38) is
given.
14 We are grateful to Daniel Baumann for pointing this out to us.
Here, we have used the F-term flatness condition DσW|σF =
0,
eaσF = |A0|
|W0|
1 + 2
3aσF
(1 + α)1/n ,
(40)
to exchange |W0| for σF. From the perspective of Kähler mod-
uli stabilization, σF should be regarded as a derived parame-
ter, which is obtained as soon as the hierarchy between |A0|
and |W0| is specified15. Before comparing with the observa-
tional data, there are additional microscopic requirements that
need to be satisfied a priori. Here, we list them below.
• The string coupling gs should be small, i.e. gs ≪1 to
ignore the string loop corrections to the supergravity
action. The physical radius of the three sphere at the
tip of deformed conifold is gsMα′; thus, we also need
gsM ≫1 [29].
• The ultraviolet cutoff rUV should be large such that
rUV/ls ≫1 for valid supergravity solution. This sets
the upper bound on the displacement for φ, hence the
total number of e-folds. Moreover, the unit of five form
flux N = KM controlling the size of conifold needs to
be large for the supergravity approximation to be valid.
These geometric requirements combine to give a strong
bound on the tensor-to-scalar ratio r [35],
4
N ≳
φUV
MPl
2
≳100 × r,
(41)
where the ≳sign is to indicate that bulk volume can
also give significant contribution to V w
6 . This can be ob-
tained from the relation between the four dimensional
reduced Planck mass MPl and the warped volume V w
6 of
the compact six manifold. Given N ≫1, the inequality
(41) implies that warped brane inflation yields negligi-
ble tensor-to-scalar ratio.
• The stabilized volume modulus σF should also be at
large values for the α′ corrections to be suppressed.
This can be ensured by tuning the bulk flux to generate a
large hierarchy between |A0| and |W0|, i.e. |A0|/|W0| ≫
1, since a = 2π/n is typically smaller than 1 so that
large σF can be readily produced. To avoid the back-
reaction of D7 branes on the deformed conifold, how-
ever, n should also be such that n/M ≪1. This ensures
that the resultant geometry is smooth at the end of dual-
ity cascade, rather than cascading into singular conifold
throat.
15 Furthermore, in this paper, we shall consider the configuration where mod-
uli stabilizing D7 brane is sufficiently far away from the tip of the deformed
conifold. Therefore the the term 1+α with α = ε/µ ≪1 in (40) only gives
insignificant shift in σF.
8
• Finally, the uplifting ratio s is bounded within the range
1 ≤s ≤O(3) to ensure a small positive cosmologi-
cal constant at the end of inflation. The upper bound
here arises from preventing runaway decompactifica-
tion. Such requirement effectively couples the scale of
|VF| and the scale of the uplifting term(s) Vuplift(φ).
Comparison with observations
In this section we shall first consider some generic features
of the inflaton potential (2) with c3/2 = c2 = 0, i.e. involv-
ing only VD3D3(φ) +Vstab.(φ) given by (34). In particular, we
discuss which parameters listed in (39) have more impact on
the overall scale or the detailed shape of the inflaton poten-
tial, as this is useful for an efficient full parameter space scan-
ning. Next, we shall present some sample parameter sets to
demonstrate that VD3D3(φ)+Vstab.(φ) can indeed yield obser-
vationally consistent results. Such scanning for our complete
potential is in line with the existing literature [16, 17]. Fi-
nally, we shall scan the perturbations due to Vbulk(φ) on these
observationally consistent local potentialVD3D3(φ)+Vstab.(φ),
and demonstrate that bulk contributions generically need to be
highly fine-tuned to preserve such results.
Let us first consider the amplitude of the power spectrum of
the curvature perturbation PR and the corresponding spectral
index nR , which are tightly constrained by recent cosmolog-
ical observations [2, 3]. On the largest observable scales the
slow-roll approximation holds at a good enough accuracy (see
later discussion), we can express them as
PR =
V
24π2εM4
Pl
= (2.41 ± 0.22)× 10−9,
(42)
nR =1 −6ε+ 2η = 0.963 ± 0.028,
(43)
at 95% confidence level. Here, (42) and (43) are evaluated at
φCMB, the value of the canonical inflaton at the CMB scale,
and should be determined by integrating backwards 60 e-
folds16 from the end of inflation. The inflationary scale is
expected to be approximately constant around the CMB scale,
and, in particular, for our model, it is expected to occur near
the “inflection point” where the majority of e-folds is gener-
ated. Explicitly, the combination
(s−1)|VF(0,σF)| ≈(s−1)a2|A0|2e−2aσF
3M2p(2σF)
≈V(φCMB) (44)
largely sets the overall scale of inflation in our model. The
deviation from (44) due to the motion of mobile D3 is essen-
tially a small fluctuation around it. If the energy associated
16 There exists some level of uncertainty on exactly when the perturbation on
the largest observable scales is generated. Depending on the detail of the
model, the corresponding e-fold is supposed to lie between 50 and 60 [36].
But provided that the curvature power spectrum is nearly scale invariant it
does not cause too significant differences. Thus in the remaining text we
evaluate PR and nR at 60 e-folds before the end of inflation.
with the inflaton is too large, this would in fact lead to run-
away decompactification [37]. The slow-roll parameter ε is
also small around the CMB scale, but it varies more rapidly
than V(φ). We therefore conclude to obtain an observation-
ally consistent value of (42), it is easier to fix the combination
(44) which sets the overall scale, then vary other parameters
such as ε and µ, which affect the shape of V(φ) around φCMB.
It is also worth noting that while the uplifting ratio s or
Vuplift(φ) is fixed, one can still vary the ratio between the dis-
tant uplifting (∝Dothers) and contribution from D3 at the tip
of (∝D0). This also varies the D3-D3 Coulombic attraction
in (1). However, as such highly warped attraction only be-
comes significant near the tip region, it is important to use
the full scalar potential (34) to study any change in the trajec-
tory. Furthermore, at the relatively large distance where the
CMB scale lies, the Coulombic attraction is effectively ab-
sent. The variation of D0/Dothers therefore should not affect
significantly the observational predictions17. This is indeed
the case as illustrated in Fig. 1 and Table I.
Now we would like to present some sample parameter scan-
nings for VD3D3(φ) +Vstab.(φ). The strategy is that we shall
further systematically fix the parameters n,|A0|,|W0| and s to
some appropriate fiducial values by hand. This allows us to
roughly fix the overall scale of the inflaton potential following
(44). We then generate a range of observationally consistent
parameter sets by scanning in ε-µ or equivalently α-β plane.
Let us briefly describe how the fiducial values for these
other parameters are chosen. The number of probe D7s n can
first be fixed to be sufficiently small. This is because n ap-
pears mostly with σ⋆or in [g(τ)]1/n. With αcosh(τ/2) < 1
and σF ≫1, the dependence of the inflaton potential on n is
insignificant comparing with other parameters. To fix the val-
ues of |A0|,|W0| and s, as mentioned earlier that their relative
sizes are fixed by compactification constraints (40), we need
to set the ratio |A0|/|W0| large to ensure the volume modulus
is fixed at large value σF. For the actual value of |A0|, we note
that as |A0| is related to the dynamical scale Λ at which gaug-
ino condensation takes place [17], therefore it is necessary to
have |A0|1/3 ∼Λ ≤MPl. To fix the uplifting ratio s, the resul-
tant cosmological constant should be small and positive at the
end of inflation, but not necessarily at our current value as, for
example, there can be further dynamical processes, e.g. topo-
logical changes after inflation, which can change its value.
The specific numerical values for {n,|A0|,|W0|,s} used in our
scanning are given in Table I.
With the full inflaton potential given by (34) and (38), we
can exactly solve the system and subsequently identify where
inflation ends, i.e. ε = 1. This is most easily done by solv-
ing, instead of the Friedmann equation, the Hamilton-Jacobi
17 It is in principle possible to finely tune the CMB scale to small radius [17],
but there one should again use the full potential valid for that region (34)
to study the effects of varying D0/Dothers on the trajectory.
9
equation
2M4
Pl[H′(φ)]2 −3M2
Pl[H(φ)]2 + V(φ) = 0.
(45)
We can thus calculate the exact number of e-folds Ne given by
Ne(φ) = M−1
Pl
Z φ
φe
dφ
√
2ε
,
(46)
with ε defined as (3) and φe given by solving (45), and sub-
sequently identify φCMB where Ne(φCMB) = 60. Note that for
φe, we explicitly consider two limiting cases, where the up-
lifting is exclusively by the distant sources or by the warped
D3. As mentioned earlier and checked in our scannings that
ε ≪1 until the tip of deformed conifold for distant uplifting,
thus φe = 0 in this case. Whereas for warped D3, φe can also
be determined at a small radius by solving (3). Essentially we
expect that the end point φe will vary continuously as we dial
between the two limit cases. Furthermore, at any viable φCMB
the potential is very flat so that |η| ≪1, we therefore make use
of the simplified slow-roll formulae (42) and (43) to estimate
PR and nR respectively, instead of solving the perturbation
equations mode by mode.
In Table I, we present three sets of α and β, which give sim-
ilar predictions on PR and nR for VD3D3(φ) +Vstab.(φ). The
values of W0 and A0 are the same in both Cases 1 and 2. The
scanned results suggest that locally there exists a region of
degeneracies in α-β plane with the other parameters fixed, as
explicitly demonstrated in the lower panels of Fig. 2. Fur-
thermore, if the other parameters are allowed to vary, we can
produce similar prediction in an even wider range of parame-
ter sets. A sample parameter set with different |A0| and |W0| is
presented as Case 3 in Table I. Note that from Fig. 2, a frac-
tional change of O(1)% in either α or β can easily move the
values of PR and nR to observationally inconsistent regimes.
Here, we have scanned only the vicinity of a given {α,β},
and it is not entirely clear (although suggestive) such O(1)%
tuning in α-β plane holds for a wider range. It would be in-
teresting to return to this issue in a more complete scanning in
the future.
Bulk effects scanning
Having presented a range of the observationally consistent
parameter sets for VD3D3(φ)+Vstab.(φ), we shall now consider
the perturbations on them, due to the unknown bulk physics
parametrized by Vbulk(φ) (38). In particular, for the local in-
flection point based inflationary trajectories, we shall perform
a sample scanning in the c3/2-c2 plane to demonstrate that they
typically need to be of order 10−8-10−9 to preserve consistent
observational predictions.
|W0|
|A0|
α
β
PR ×109
nR
Case 1 2.92485×10−6 0.0085 1/200 1/508 2.66644 0.933109
2.49420 0.932009
Case 2 2.92485×10−6 0.0085 1/100 1/320 2.59208 0.934267
2.42615 0.933175
Case 3
3.3×10−6
0.066 1/100 1/350 2.36186 0.934743
2.19847 0.933838
TABLE I: Three sets of parameters that give the viable values of PR
and nR . We have fixed n = 8 and s = 1.07535 for all the cases.
The values of the tensor-to-scalar ratio r = 16ε and the non-linear
parameter fNL = O(ε,η) are unobservably small and hence we do
not present them here. The first line of each case corresponds to the
complete domination of the distant sources, while the second to that
of the Coulombic interaction. Note that as shown in the first two
cases, with a given set of n, s, |W0| and |A0|, a different combination
of α and β yields similar values of PR and nR . Also, in the last case
with another set of |W0|, |A0|, α and β we can find observationally
consistent values of PR and nR .
For numerical purpose, we slightly recast (38) as
Vbulk(φ) =−|VF(0,σF)|
×
(
c′
3/2α
Z
τ
dτ′
K(τ′)
3/2
+ c′
2α4/3
Z
τ
dτ′
K(τ′)
2)
.
(47)
In the above we have used
φ
φUV
= φµ/φUV
21/3 ·3 α2/3
Z
τ
dτ′
K(τ′) ,
(48)
with φµ/φUV ≲1 being a number (This is denoted as Q−1
µ
in
Refs. [18, 24]) and various order one numerical factors are ab-
sorbed into a newly defined constant c′
∆. Hereafter, we shall
drop this prime notation. In general, as VD3D3(φ) yields a deli-
cate inflection point based inflation, we expect the value of c∆
needs to be finely tuned. In Table II we show a summary of
the effects of Vbulk.
Specifically, from Table II, we can see that a very slight dis-
turbance of the bulk effects of magnitude 10−9-10−8 for both
c3/2 and c2 can push the otherwise viable predictions into the
regions beyond 2-σ errors. As Vbulk is negative definite, it
pushes down the inflaton potential further so that the previ-
ously flat region becomes flatter or is even changed into a lo-
cal minimum. Naturally the amplitude of PR increases, while
nR deviates further from 1 as the value of the coefficients c3/2
and c2 get larger. These tendencies are clearly shown in Ta-
ble II. Occasionally Vbulk can improve the relevant predictions
to be closer to the current observations. For example, in Case
3, the bulk terms move the value of PR to the central value
of the observationally allowed region and leave nR more or
less the same with small c3/2 and c2. One may thus hope that
by adding Vbulk(φ) to an unviable VD3D3(φ)+Vstab.(φ), obser-
vationally consistent results can be obtained. However, we
expect in general c3/2 or c2 need to be of order 10−8-10−9 to
achieve such objective.
10
0
10
20
30
40
50
60
-14
-13
-12
-11
-10
-9
-8
log10 PR
Ne
0
10
20
30
40
50
60
0.70
0.75
0.80
0.85
0.90
0.95
1.00
nR
Ne
0.00498
0.00500
0.00502
0.00504
0.00506
0.001966
0.001968
0.001970
0.001972
0.001974
0.001976
0.001978
α
β
0.00498
0.00500
0.00502
0.00504
0.00506
0.001966
0.001968
0.001970
0.001972
0.001974
0.001976
0.001978
α
β
FIG. 2: (Upper panels) the plots of (left panel) log10 PR and (right panel) nR as functions of Ne, and (lower panels) the contour plots of (left
panel) PR and (right panel) nR in the α-β plane for Case 1 given in Table I. In the upper panels we show the two extreme cases where D(φ)
given by (35) is completely dominated by either the Coulombic interaction (solid line) or the distant sources (dotted line). Meanwhile, in the
lower panels we only present the case with the distant sources completely dominating. In the contour plot of PR , the contours denote 2.5,
2.6, 2.7, 2.8, 2.9, 3.0, 3.5, 4.0×10−9 from the innermost line. Likewise we have set 0.9325, 0.9300, 0.9275, 0.9250, 0.9225, 0.9200, 0.9175,
0.9150 for the contour plot of nR . The dots in the contour plots are numerical glitches. We have obtained qualitatively the same contour plots
when the Coulombic interaction is dominating instead, with the deep colored region a bit enlarged (PR ) and shrunk (nR ).
DISCUSSIONS
In this paper, we have discussed in detail the inflaton poten-
tial governing the motion of a mobile D3 in the entire warped
deformed conifold. In particular, we have included both the
effects of moduli stabilization and other bulk physics. We then
have performed some sample scannings to demonstrate that
without the bulk perturbations, there can be significant degen-
eracies in the conifold deformation parameter ε and the D7
embedding parameter µ for producing observationally consis-
tent predictions. However, as the bulk perturbations are in-
cluded, we have explicitly shown that their magnitudes need
to be 10−8-10−9 to preserve the observationally consistent pa-
rameter sets18. The results presented here provide the begin-
ning systematic steps towards a complete brane scanning in
18 An obviously interesting question would be whether the smallness of bulk
perturbation coefficients c∆really constitutes a significant fine-tuning, or
they are just tied to the choice of having inflection point inflation in the
throat. To answer this question fully, we believe it requires better than our
current understanding of UV physics and beyond the scope of investiga-
tions here.
the warped throat, and, in particular, highlight the importance
of the bulk effects.
It would be very interesting to follow the steps outlined
here and perform a full scanning over the parameters listed in
(39). This clearly requires intensive computational undertak-
ings. However given the rich parameter space and the degen-
eracies we have shown in the sample scannings, barring the
observation of the primordial gravitational waves, it is likely
that there remain significant regions in the parameter space for
the warped brane inflation to match the future data. Moreover,
a variant of the inflation model presented here is recently pro-
posed in Ref. [38]. In such a construction the gravitino mass
m3/2 can be made smaller than the Hubble scale H, hence cir-
cumventing the phenomenological bound given in Ref. [37].
It would clearly be interesting to generalize the analysis here
and scan the parameter space for such variant, and search for
an explicit example of a parameter set that gives TeV scale
gravitino mass and observationally consistent cosmological
predictions.
11
c3/2
c2
10−9
10−8
10−7
10−6
10−9
10−8
10−7
10−6
Case 1
Distant
PR ×109 2.71386
3.17635
13.7483
26670.1
2.74701
3.58118
37.8218
0.0559217∗
sources
nR
0.932540 0.927506 0.883792 0.657621 0.932149 0.923750 0.856258
0.552413
Coulomb PR ×109 2.53682
2.95080
11.9644
11348.2
2.56657
3.31098
31.2301
0.0138777∗
interaction
nR
0.931448 0.926480 0.883233 0.657668 0.931062 0.922766 0.855908
0.552458
Case 2
Distant
PR ×109 2.63754
3.08041
13.0733
22761.8
2.66847
3.45614
34.6474
0.0399836∗
sources
nR
0.933704 0.928724 0.885438 0.661093 0.933327 0.925100 0.858794
0.559159
Coulomb PR ×109 2.46750
2.86903
11.5801
10664.9
2.49564
3.20800
29.4348 0.01153290∗
interaction
nR
0.932613 0.927646 0.884425 0.659278 0.932238 0.924026 0.857777
0.556756
Case 3
Distant
PR ×109 2.40944
2.87842
14.8411
70552.3
2.44157
3.27915
43.8420
0.186355∗
sources
nR
0.934097 0.928393 0.879688 0.636830 0.933668 0.924284 0.850560
0.528232
Coulomb PR ×109 2.24107
2.65907
12.7450
27115.1
2.26982
3.01367
35.5661
0.0400323∗
interaction
nR
0.933199 0.927564 0.879325 0.636940 0.932776 0.923498 0.850383
0.528320
TABLE II: The effects of the bulk terms for each case of Table I. For definiteness, we have turned on either c3/2 or c2, not both of them at the
same time. This was also needed to ensure that we can avoid additional angular perturbations mentioned earlier in the main text. Also note
that the values of PR when c2 = 10−6, denoted by a superscript ∗in the last column, are bare ones and the factor of 109 is not multiplied.
Acknowledgement
We thank Gary Shiu for collaboration and discussions at
the early stage of this project. We are also grateful to Ana
Achúcarro, Daniel Baumann, James Cline, Shamit Kachru,
Gonzalo Palma, Fernando Quevedo, Koenraad Schalm and
Bret Underwood for comments and suggestions. HYC ap-
preciates the hospitality of Stanford Institute for Theoretical
Physics, where part of this work was conducted. The work of
HYC is supported in part by NSF CAREER Award No. PHY-
0348093, DOE grant DE-FG-02-95ER40896, a Research In-
novation Award and a Cottrell Scholar Award from Research
Corporation, and a Vilas Associate Award from the Univer-
sity of Wisconsin. JG is partly supported by the Korea Re-
search Foundation Grant KRF-2007-357-C00014 funded by
the Korean Government at the early stage of this work, and is
currently supported in part by a VIDI and a VICI Innovative
Research Incentive Grant from the Netherlands Organisation
for Scientific Research (NWO).
∗Electronic address: hchen46@wisc.edu
† Electronic address: jgong@lorentz.leidenuniv.nl
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arxiv | 0812.5113v1 | Ultraviolet Spectra of Local Galaxies and their Link with the High-z Population | The new generation of 8 to 10m class telescope is providing us with high-quality spectral information on the rest-frame ultraviolet region of star-forming galaxies at cosmological distances. The data can be used to address questions such as, e.g., the star-formation histories, the stellar initial mass function, the dust properties, and the energetics and chemistry of the interstellar medium. We can tackle these issues from a different angle by comparing the spectral properties of high-redshift galaxies to those of their counterparts in the local universe. I give a review of recent developments related to observations and empirical modeling of the ultraviolet spectra of local galaxies with recent star formation. The emphasis is on the youngest stellar populations with ages less than 100 Myr. Current uncertainties will be discussed, and areas where progress is needed in the future are highlighted. | Claus Leitherer | astro-ph | https://arxiv.org/abs/0812.5113v1 | Ultraviolet Spectra of Local Galaxies and their
Link with the High-z Population
Claus Leitherer
Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA
Abstract. The new generation of 8 to 10m class telescope is providing us with high-quality
spectral information on the rest-frame ultraviolet region of star-forming galaxies at cosmological
distances. The data can be used to address questions such as, e.g., the star-formation histories,
the stellar initial mass function, the dust properties, and the energetics and chemistry of the
interstellar medium. We can tackle these issues from a different angle by comparing the spectral
properties of high-redshift galaxies to those of their counterparts in the local universe. I give a
review of recent developments related to observations and empirical modeling of the ultraviolet
spectra of local galaxies with recent star formation. The emphasis is on the youngest stellar
populations with ages less than 100 Myr. Current uncertainties will be discussed, and areas
where progress is needed in the future are highlighted.
Keywords: ultraviolet; H II regions; star clusters; stellar populations; starburst galaxies; galactic
winds; intergalactic medium
PACS: 95.85.Mt; 98.58.Hf; 98.62.Lv; 98.62.Nx; 98.62.Ra.
INTRODUCTION
The first scientifically useful ultraviolet (UV) spectra of astronomical objects
outside the solar system were obtained in the 1960s when the 3-axis star-pointing
stabilization system of the Aerobee sounding rockets permitted the acquisition of
sufficiently deep spectrograms [1]. However, significant numbers of UV spectra of
normal galaxies could not be accumulated until the advent of the IUE satellite, which
had the capability of collecting multi-hour exposures necessary for extragalactic
studies [2]. The first- (FOS), second- (GHRS), and third-generation (STIS)
spectrographs of HST, together with HUT, each led to order-of-magnitude
improvements of spectral resolutions and progressively higher signal-to-noise. In
parallel with the progress in satellite-UV astronomy, a new generation of 8 to 10-m
class ground-based telescopes went on-line during the past decade. The telescopes
have produced restframe-UV spectra of star-forming galaxies at cosmological
distances whose quality rivals and often exceeds that of their local counterparts [3]. A
comparison of the average spectra of 16 local star-forming galaxies and of those of
811 Lyman-break galaxies (LBG) suggests striking similarity [4].
In this review, I will highlight the overall spectral similarity between local and
distant star-forming galaxies but at the same time point out some subtle but significant
differences. After a brief summary of the basic galaxy properties, I will cover the
stellar populations, the neutral and ionized interstellar medium (ISM), Lyman-Į, and
the escape of Lyman continuum radiation.
GALAXY PROPERTIES
Local star-forming galaxies targeted for UV spectroscopy are necessarily UV-
bright, a bias imposed by the low quantum efficiency of UV detectors and the
relatively small telescope sizes. Morphologically, these galaxies tend to be of late
Hubble types, and they include blue compact galaxies, H II galaxies, and nuclear
starbursts [4]. Stellar masses are of order 109 Mڒ, and absolute magnitudes are in the
range 16 < MB < 19. Oxygen abundances are as low as 1/20th Zڒ and as high as Zڒ,
with typical values similar to those of the Magellanic Clouds. Overall, local UV-bright
galaxies cover a parameter space that is similar to that occupied Lyman-Į emitters at
high redshift [5], but with the important difference of generally weak Lyman-Į
emission. The local sample is often quoted as the counterpart of LBGs. While the two
samples are similar in many respects, it is important to realize that the average
luminosities and masses of LBGs are 1 to 2 orders of magnitude higher than for the
local sample.
Since these galaxies were selected based on their UV brightness, they tend to have
low dust reddening. As a result, their morphologies are often quite similar at different
wavelengths, in particular when going from the UV to the optical. A comparison of
GALEX far- and near-UV and SDSS optical imagery supports this view [6].
Nevertheless, examples of UV-bright star-forming galaxies with strong local dust
attenuation exist. The UV may very well provide a rather biased view of the actual
star-formation activity. A striking example is the interacting galaxy pair VV 114 [7],
whose two members show a strong color contrast. One component is dominated by a
blue, high surface brightness complex of regions with a relatively weak near-infrared
(IR) nucleus. The other component is much redder and brighter in the near-IR but
inconspicuous in the UV. If this system were observed at high redshift in the absence
of spatial information, the apparently coincident UV and IR would arise in spatially
disjoint regions, and correlating them would be meaningless.
STELLAR POPULATION
The satellite-UV traces the most recently formed stars with masses of ~5 Mڒ and
above. The continuum below the Balmer break comes from late-O and early-B stars.
Superimposed on the continuum are strong, broad, blueshifted absorption lines,
sometimes with emission components, from O stars of all spectral type. On average,
these O stars have masses of order 50 Mڒ. The most prominent spectral lines are O VI
Ȝ1035, N V Ȝ1240, Si IV Ȝ1400, and C IV Ȝ1550 [8].
The UV lines originate in powerful stellar winds with stellar-mass-dependent
properties and cover a wide range of ionization potentials from a few eV to 114 eV
(O VI). This makes them suitable for studying the mass distribution and eventually the
initial mass function (IMF) of the most massive stars in the mass range between 10
and 100 Mڒ. A major outcome of numerous spectroscopic UV studies of local star-
forming galaxies is the ubiquity of a single Salpeter-like IMF in this mass range. This
result has independently been confirmed by other methods, such as the photo-
ionization modeling of optical nebular emission lines [9].
FIGURE 1. Composite cluster and field spectra for four dwarf galaxies obtained with STIS. The field
spectra have weaker N V, Si IV, and CIV lines indicating a deficit of the most massive stars. From [10].
However, one should be aware that essentially all these data were taken through
narrow slits extending over length scales of order 10 pc in each galaxy. The apertures
typically encompass one or more bright star clusters, which constitute the local peak
of the UV light in each galaxy. Fig. 1 illustrates this point. In this case, long-slit
spectroscopy with HST’s STIS allows separate studies of both the clusters and the
intercluster light, which is the diffuse, unresolved stellar emission. Comparison of the
cluster and intercluster light suggests weaker stellar NV, Si IV, and C IV lines in the
field. This translates into a deficit of very massive field stars. One interpretation could
be a steeper field IMF, which has fewer O stars. Alternatively, an age effect could be
responsible: field stars are older on average because field stars are the relics of
dissolved star clusters whose lifetimes of order 10 Myr are longer than O-star
lifetimes. Therefore, massive O stars may disappear before cluster dissolution and
never contribute to the field population [11].
The lesson learned for the interpretation of the restframe UV spectra of high-z
galaxies is to be aware of the bias that is inherent in local spectra. The latter usually
refer to only a few bright star clusters whose light contribution to the total is only a
few percent and whose stellar population may not be representative for the galaxy as a
whole. In contrast, spectra of distant galaxies encompass a much larger volume, and
using local template spectra may introduce a significant bias.
NEUTRAL AND IONIZED ISM
In addition to the already discussed stellar-wind lines, numerous strong interstellar
absorption lines are located in the wavelength region below 3000 Å. The lines can
easily be distinguished from the stellar lines by their line widths, whose values of a
few 102 km s1 are smaller by almost an order of magnitude.
FIGURE 2. UV absorption lines observed with FUSE. Left: Haro 11; right: NGC 3310. Note: O I
Ȝ989 is blended with N III Ȝ990. From [12].
The FUSE satellite was specifically optimized for observations of the Galactic and
extragalactic ISM. Its wavelength coverage was from 1175 Å down to the Lyman limit
at 912 Å, at a resolving power of 20,000. The ISM lines in a sample of 16 star-forming
galaxies observed with FUSE by [12] are all blueshifted and asymmetric. In Fig. 2 I
have reproduced the data for two of their program galaxies. Velocity displacements of
hundreds of km s1 can be seen. These displacements are indicative of galaxy-wide
outflows, also known as galactic superwinds [13]. The existence of such outflows in
essentially all local star-forming galaxies has been demonstrated from observations of
the cool (Na I), warm (HĮ), and hot (O VI and X-rays) gas. Stellar winds and
supernovae support a pressure-driven outflow, which expands along the direction of
the maximum density gradient.
The FUSE data of [12] suggest a trend of larger outflow velocities in galaxies with
larger specific star-formation rate SFR/M, where SFR is measured from the combined
UV and IR luminosities, and the K-band luminosity is used as a proxy for stellar mass.
Since the specific star-formation rate increases with redshift [14], one may speculate
that galactic superwinds are even more pronounced and prevalent in, e.g., LBGs. If so,
the importance of the escape of processed matter from galaxies into the surrounding
intergalactic medium (IGM) and leakage of ionizing radiation will increase from low
to high redshift.
FIGURE 3. Haro 11 as seen by HST. Field sizes are 20 × 20 arcsec2, corresponding to 8.1 × 8.1 kpc2.
The individual images show Haro 11 in the 1500 Å continuum, the B-band, HĮ, Lyman-Į, the Lyman-Į
equivalent width, and the Lyman-Į/HĮ ratio. The bottom panels show spatial cuts along rows and
columns for the Lyman-Į flux (left) and equivalent width (right). From [18].
LYMAN-ALPHA
If the nebular Lyman-Į line behaved like an ideal recombination line, its predicted
equivalent width (EW) for a standard young stellar population is of order 102 Å [15].
Such large EW values are never observed in local star-forming galaxies, which often
display Lyman-Į as a damped absorption profile [16]. This is surprising, as Lyman-Į
is typically observed as a strong emission line in high-redshift star-forming galaxies,
whose properties are otherwise quite similar to their low-redshift counterparts [17].
A major limitation of current UV spectrographs is their lack of spatial resolution
and/or small spatial coverage. Narrow-band Lyman-Į imaging can provide invaluable
complementary information. Fig. 3 summarizes the results of HST/ACS imagery of
Haro 11, whose luminosity (MB = 20.5) and oxygen abundance (log O/H +12 = 7.9)
make it an excellent analog of an LBG [18]. Most Lyman-Į photons are emitted in the
nucleus (bottom left) but since the stellar UV continuum is even more peaked towards
the nucleus (top left), the Lyman-Į equivalent width is very small in the center
(bottom right). The equivalent width is in fact quite high outside the nucleus where the
neutral hydrogen column is low but the absolute number of ionizing photons produced
there is too small to be of importance. Only 3% of the Lyman-Į photons expected to
be observed based on the HĮ recombination flux escape.
The reason for the higher Lyman-Į escape fraction in galaxies at high redshift may
lie with the ISM porosity and dynamics. Galactic winds are more powerful at high
redshift, leading to increased stirring of the ISM and creating an effective Lyman-Į
escape mechanism.
LYMAN CONTINUUM
A standard star-forming population emits approximately 10% of its luminosity as
ionizing radiation below 912 Å [19]. Most of this radiation is absorbed by the ambient
neutral hydrogen and by dust, as suggested by the observed recombination lines. Yet
the possibility exists, and even seems likely, that some fraction of the ionizing photons
will escape from both the H II regions and the diffuse ISM. If so, star-forming galaxies
could be an important source for the cosmological ionizing background radiation.
One can measure the escape fraction either in local galaxies using a far-UV detector
or in galaxies at cosmological redshift, whose restframe UV then becomes accessible
from the ground with 8-m class telescopes. Either technique has its advantages and
disadvantages. The “local” approach faces the obvious challenge of extreme UV
observations, whereas the “cosmological” measurement must account for the radiative
transfer in the IGM.
The FUSE survey of [12] can shed additional light on this issue. Nine program
galaxies have sufficiently high velocity to shift the intrinsic Lyman continuum out of
the Galactic foreground H I absorption. The spectra of five of these galaxies are
plotted in Fig. 4. No significant Lyman continuum emission is detected in any of the
target galaxies. The upper limits on the fluxes, when combined with simple models for
the geometry of the ISM, permit relatively stringent constraints on the Lyman
continuum escape fractions. For a picket-fence model of the ISM, average escape
fractions of less than about 1% are found.
FIGURE 4. Lyman continuum regions of five galaxies in order of increasing redshift observed with
FUSE. The wavelengths are in the observed frame. The intrinsic Lyman continuum is to the left of each
vertical dashed line. There is no convincing evidence of Lyman continuum emission in these spectra.
From [12].
Alternatively, the strength of the interstellar absorption lines can be used to infer
the neutral hydrogen opacity, and therefore the escape probability of hydrogen
ionizing photons. The FUSE spectral range includes species of several abundant
elements with ionization stages close to dominant, thus minimizing uncertain model
assumptions. The interstellar lines imply hydrogen column densities which limit the
Lyman photon escape to less than a few percent, consistent with the direct
measurement below 912 Å.
The FUSE result mirrors previous studies of the Lyman continuum in local star-
forming galaxies, which uniformly failed to detect significant Lyman continuum
radiation [20]. The evidence at high redshift is less clear. [21] reported a detection in
the composite spectrum of 29 LBGs with average redshift z = 3.40 ± 0.09. On the
other hand, a different LBG sample studied by [22] implied an upper limit 4.5 times
lower than inferred from the composite spectrum of [21]. Strikingly, two out of the 14
sample galaxies observed by [21] show a clear detection of Lyman continuum
radiation whereas the remaining twelve are non-detections. Averaging over the whole
sample of [22] leads to a mean escape fraction of 14%, with a large variation from
galaxy to galaxy.
The empirical result of a higher Lyman continuum escape fraction from lower to
higher redshift may again be understood in terms of more violent star formation in the
early universe. More powerful galactic superwinds that are initiated and supported by
stellar winds and supernovae increase the ISM porosity and create escape paths for the
stellar ionizing radiation. Better observational statistics and quantitative modeling are
required for determining how the escaping radiation compares to the contribution of
optically selected quasars at the same redshift and whether star-forming galaxies are
ultimately responsible for the reionization of the early universe.
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arxiv | 0812.4582v1 | Correlated variability in the blazar 3C 454.3 | The blazar 3C 454.3 was revealed by the Fermi Gamma-ray Space Telescope to be in an exceptionally high flux state in July 2008. Accordingly, we performed a multi-wavelength monitoring campaign on this blazar using IR and optical observations from the SMARTS telescopes, optical, UV and X-ray data from the Swift satellite, and public-release gamma-ray data from Fermi. We find an excellent correlation between the IR, optical, UV and gamma-ray light curves, with a time lag of less than one day. The amplitude of the infrared variability is comparable to that in gamma-rays, and larger than at optical or UV wavelengths. The X-ray flux is not strongly correlated with either the gamma-rays or longer wavelength data. These variability characteristics find a natural explanation in the external Compton model, in which electrons with Lorentz factor gamma~10^(3-4) radiate synchrotron emission in the infrared-optical and also scatter accretion disk or emission line photons to gamma-ray energies, while much cooler electrons (gamma~10^(1-2)) produce X-rays by scattering synchrotron or other ambient photons. | E. W. Bonning, C. Bailyn, C. M. Urry, M. Buxton, G. Fossati, L. Maraschi, P. Coppi, R. Scalzo, J. Isler, A. Kaptur | astro-ph | https://arxiv.org/abs/0812.4582v1 | arXiv:0812.4582v1 [astro-ph] 30 Dec 2008
Submitted to ApJ Letters
Preprint typeset using LATEX style emulateapj v. 08/22/09
CORRELATED VARIABILITY IN THE BLAZAR 3C 454.3
E. W. Bonning1, C. Bailyn2, C. M. Urry1, M. Buxton2, G. Fossati4, L. Maraschi3, P. Coppi2, R. Scalzo1, J.
Isler1, A. Kaptur2
Submitted to ApJ Letters
ABSTRACT
The blazar 3C 454.3 was revealed by the Fermi Gamma-ray Space Telescope to be in an exceptionally
high flux state in July 2008. Accordingly, we performed a multi-wavelength monitoring campaign on
this blazar using IR and optical observations from the SMARTS telescopes, optical, UV and X-ray
data from the Swift satellite, and public-release gamma-ray data from Fermi. We find an excellent
correlation between the IR, optical, UV and gamma-ray light curves, with a time lag of less than one
day. The amplitude of the infrared variability is comparable to that in gamma-rays, and larger than
at optical or UV wavelengths. The X-ray flux is not strongly correlated with either the gamma-rays
or longer wavelength data. These variability characteristics find a natural explanation in the external
Compton model, in which electrons with Lorentz factor γ ∼103−4 radiate synchrotron emission in
the infrared-optical and also scatter accretion disk or emission line photons to gamma-ray energies,
while much cooler electrons (γ ∼101−2) produce X-rays by scattering synchrotron or other ambient
photons.
Subject headings: galaxies: active — quasars: general — black hole physics — BL Lacertae objects:
individual (3C 454.3)
1. INTRODUCTION
Blazars are understood to be active galactic nuclei
(AGN) with aligned relativistic jets (Urry & Padovani
1995), so they offer a unique laboratory for study-
ing the physics of astrophysical jets. The spectral en-
ergy distributions (SEDs) of blazars have a characteris-
tic double-humped shape with a low-energy component
peaking anywhere from radio to X-rays, and a high-
energy component peaking at MeV to GeV energies (Fos-
sati et al. 1998).
Flat Spectrum Radio Quasars (FS-
RQs) like 3C 454.3 have SED peaks at radio-IR wave-
lengths and ∼1 GeV (Urry & Padovani 1995; Sam-
bruna et al. 1996). The low-energy component is well
modeled as synchrotron emission from relativistic elec-
trons in the jet (Konigl 1981; Urry & Mushotzky 1982),
while the origin of the second SED peak at high ener-
gies is not fully understood. Current explanations for the
gamma-ray emission fall into two categories, leptonic and
hadronic. Leptonic models produce high-energy flux by
inverse-Compton scattering of low-energy seed photons,
either the synchrotron photons themselves (Synchrotron
Self-Compton,
Jones et al. 1974) or photons from an
external source, such as thermal accretion disk emis-
sion or broad-line emission (Sikora et al. 1994; Dermer
& Schlickeiser 1993; Ghisellini & Madau 1996; Celotti
& Ghisellini 2008).
In hadronic models, protons that
are accelerated to very high energies in the jet produce
gamma-rays from neutral pion decay, proton synchrotron
1 Department of Physics and Yale Center for Astronomy and
Astrophysics, Yale University, PO Box 208121, New Haven, CT
06520-8121; erin.bonning@yale.edu
2 Department of Astronomy and Yale Center for Astronomy and
Astrophysics, Yale University, PO Box 208101, New Haven, CT
06520-8101
3 INAF - Osservatorio Astronomico di Brera, V. Brera 28, I-
20100 Milano, Italy
4 Department of Physics and Astronomy, Rice University, Hous-
ton, TX 77005
emission, and synchrotron emission from pair production
(M¨ucke & Protheroe 2001; M¨ucke et al. 2003; B¨ottcher
2007).
Both leptonic and hadronic models can ade-
quately fit single-epoch blazar SEDs, but variability of-
fers a test of either model.
3C 454.3 was among the more intense and vari-
able FSRQs detected with CGRO EGRET (Hart-
man
et
al.
1999),
varying
over
several
years
by
factors
of
up
to
five,
with
a
flare-state
flux
of
F>100 MeV ∼0.5×10−6 photons/s/cm2 (Hartman et al.
1993, 1999; Aller et al. 1997). Long-term optical vari-
ability has also been reported, with up to ∼3 mag
changes over several years Djorgovski et al. (2008). Dur-
ing a 2005 optical flare to R=12 (Villata et al. 2006),
3C 454.3 was detected with INTEGRAL at a flux of
F3−200 keV ∼3×10−2 photons/s/cm2 (Pian et al. 2006);
a radio flare followed about a year later (Villata et al.
2007).
3C 454.3 has been detected with the AGILE
gamma-ray satellite (Tavani et al. 2008), flaring in July
2007 and again in July 2008 (Vercellone et al. 2008; Gas-
parrini et al. 2008) with associated flaring at optical and
longer wavelengths (Ghisellini et al. 2007; Villata et al.
2008). On 24 July 2008, Tosti et al. (2008) confirmed the
high gamma-ray flux state of the source with a detection
by the Fermi Large Area Telescope (LAT) while still in
its post-launch commissioning phase. In the Fermi/LAT
first light image released on 26 August 2008, 3C 454.3
was among the brightest sources in the gamma-ray sky,
at the high end of its recorded gamma-ray intensity,
F0.1−300 GeV ∼4.4×10−6 photons/s/cm2.
Here we present data from our multi-wavelength op-
tical and infrared monitoring program of 3C 454.3 from
June to December 2008 with the Small and Moderate
Aperture Research Telescope System (SMARTS). We
correlate these data with Target of Opportunity observa-
tions carried out with the Swift X-ray Telescope (XRT)
and Ultraviolet and Optical Telescope (UVOT), as well
2
Bonning et al.
as with 0.1–300 GeV fluxes made public by the Fermi
Science Support Center. The observations are described
in Section 2. The light curves, correlation functions, and
SED are discussed in Section 3.
2. OBSERVATIONS
2.1. SMARTS
Photometric monitoring of 3C 454.3 was carried out on
the 1.3m telescope located at Cerro Tololo Interamerican
Observatory (CTIO) with the ANDICAM instrument.
ANDICAM is a dual-channel imager with a dichroic that
feeds an optical CCD and an IR imager, which can ob-
tain simultaneous data from 0.4 to 2.2 µ. Our campaign
began with observations in B, V, R and J-bands with a
cadence of one observation every 2 nights. After it be-
came clear that 3C 454.3 was exhibiting interesting and
varied behavior, we added K-band observations and in-
creased the cadence to one observation every night. The
SMARTS photometric data and light curves for 3C 454.3
as well as all other Fermi/LAT monitored blazars visi-
ble from CTIO are made publicly available on a 1-2 day
timescale on the web. 5
Optical
data
were
bias-subtracted,
overscan-
subtracted, and flat fielded using ccdproc in IRAF.
The optical photometry was calibrated using published
magnitudes of a secondary standard star6 in the field
of 3C 454.3 (Craine 1977; Angione 1971; Fiorucci et al.
1998).
Infrared data were sky-subtracted, flat fielded,
and dithered images combined using in-house IRAF
scripts. The infrared photometry was calibrated using
2MASS magnitudes of a secondary standard star (the
same star used in optical photometry calibration) in the
field of 3C 454.3. We estimated photometric errors by
calculating the 1-σ variation in magnitude of comparison
stars with comparable magnitude to 3C 454.3.
These
are as follows:
Berr = 0.02 mag, Verr = 0.02 mag,
Rerr = 0.02, Jerr = 0.04 mag, and Kerr = 0.04 mag.
Figure 3 shows the B-band light curve normalized to
its flux at JD 2454700. Figure 3 shows two SEDs for
3C 454.3: one averaged over the actively flaring period
up to JD 2454750, and a second averaged over the rel-
atively quiescent period after that day. To compute the
fluxes, magnitudes were dereddened using the extinction
relations in Cardelli et al. (1989) together with the value
for AB given by Schlegel et al. (1998) and converted into
flux densities using the zero-point fluxes given by Bessell
et al. (1998) and Beckwith et al. (1976)
2.2. Fermi
The Fermi Space Telescope (formerly GLAST) was
launched on 11 June 2008. The Fermi observatory Large
Area Telescope (LAT) is designed to measure the cos-
mic gamma-ray flux up to ∼300 GeV. The LAT is an
imaging, wide field-of-view high-energy pair conversion
telescope with energy range from ∼20 MeV to
>
∼300
GeV, and surveys the sky every three hours (Michelson
2007). As a service to the community and in order to sup-
port correlated multiwavelength observations, the LAT
Instrument Science Operations Center provides daily and
weekly averaged fluxes for a number of blazars, of which
5 http://astro.yale.edu/glast/index.html
6 Shown as star H in the finding chart at http://www.lsw.uni-
heidelberg.de/projects/extragalactic/charts/2251+158.html
3C 454.3 is one. Fluxes and 1σ uncertainties for three
bands, 0.1–300 GeV, 0.3–1 GeV, and 1–300 GeV, using
preliminary instrument response functions and calibra-
tions, are made available online roughly once per week,
with the caveat that the early flux estimates are not ab-
solutely calibrated, and may have variations of up to 10%
due to uncorrected systematic effects. Because the ob-
served variations are well correlated with independently
measured IR, optical, and UV variations, we conclude
the gamma-ray variations will not change significantly
even if they are eventually recalibrated, and in any case,
our key results are robust against 10% fluctuations in
gamma-ray intensity. We show the 3C 454.3 light curve
in the 0.1–300 GeV band in Figure 3 normalized to its
photon flux at JD 2454700. Fluxes shown in Figure 3
are computed from the publicly released data in the 0.3–
1 GeV and 1–300 GeV bands by assuming a power-law
spectrum of photon index Γ=2.
2.3. Swift
Since being identified in June 2008 as an extraordi-
narily bright gamma-ray source (Vittorini et al. 2008;
Gasparrini et al. 2008), 3C 454.3 has been the subject of
numerous Swift target of opportunity observations, in-
cluding one by PI Bonning covering 22 September - 02
October, 2008. The Swift satellite (Gehrels et al. 2004)
has three instruments: a coded-mask Burst Alert Tele-
scope (BAT, Barthelmy et al. 2005), an X-ray Telescope
covering the energy range 0.2–20 keV (XRT, Burrows
et al. 2005), and an Ultraviolet/Optical Telescope cover-
ing 170–600 nm (UVOT, Roming et al. 2005). Swift data
are made public to the community within a few days of
the observations; therefore we were able to collect all
available data within the period of our SMARTS obser-
vations. We reduced the data from the X-ray telescope
(XRT) and the Ultraviolet Optical Telescope (UVOT)
according to the standard recipes given by the Swift data
analysis manuals.
For each obsid, the UVOT data for each exposure were
co-added with the task uvotimsum. The source magni-
tudes were then computed from a source region of 5.5 arc-
sec using the task uvotsource, which performs aperture
photometry on the source and returns the count rate,
flux density, and magnitude in the Swift/UVOT photo-
metric system (Poole et al. 2008). We correct these for
interstellar extinction as described in Section 2.1. Light
curves from the UVOT B and W1 bands are shown in
Figure 3, and average fluxes before and after JD 2454750
in Figure 3.
For each obsid, the XRT level-2 event list was gener-
ated via xrtpipeline v. 0.11.5 with the default filtering
and screening criteria, selecting photon counting (PC)
data with XRT event grades 0-12.
We extracted the
source spectrum from a region centered at the source
with a radius of 60 arcsec and subtracted the background
from a nearby source-free region. Spectra were rebinned
to 25 cts/bin, fit with an absorbed power law, and the
flux was computed in 0.5 −2.0 keV and 2.0 −10.0 keV
bands. The X-ray light curve is shown in the bottom
panel of Figure 3.
3. RESULTS AND DISCUSSION
The correlated variability across all observed wave-
bands except the X-ray is readily apparent in Figure 3.
Correlated Variability in 3C 454.3
3
All observed bands save the X-ray show two promi-
nent peaks around JD 2454715 and a short flare near
JD 2454740. The amplitude is largest in the gamma-
rays and J band. Figure 2 shows the discrete correlation
function (DCF, Edelson & Krolik 1988; White & Peter-
son 1994) calculated for the gamma-ray (0.1–300 GeV)
flux versus light curves in the optical B band7, which has
the best temporal coverage, and infrared J-band, which
shows the strongest variations. The DCF shows a peak
correlation amplitude ∼0.7 at τ = 0, indicating no de-
tectable lag between IR/optical and gamma-ray fluxes.
Given the sampling, this means any lag is less than or
about 1 day. Similar results were reported by Vercellone
et al. (2009) for the earlier flare observed with AGILE,
though with much lower significance. The optical ver-
sus IR DCF shows even stronger correlation (amplitude
∼0.8), also with 0 ± 1 day lag.
Table 1 shows the fractional root mean square (rms)
variability amplitude (Vaughan et al. 2003) for each
band. The IR, optical, and UV variability amplitudes
decrease toward shorter wavelengths, suggesting the pos-
sible presence of steady thermal emission (UV accretion
disk emission plus Balmer continuum, Fe ii, and Mg ii
in the V and B bands) added to the steeper-spectrum
jet. Evidence for ‘big’ and ‘little’ blue bumps was found
previously in the SED of 3C 454.3 during periods of low
emission (Raiteri et al. 2007). The colors of 3C 454.3 are
redder at brighter levels, historically (Villata et al. 2006)
and in the present data, also supporting the presence of
thermal emission beneath the much brighter non-thermal
jet.
The closely correlated IR/gamma-ray variability of
3C 454.3 supports a model in which relativistic elec-
trons in the jet radiate IR/optical synchrotron photons
and inverse Compton scatter thermal photons to X- and
gamma-ray energies.
The observed gamma-ray flares
must be caused by changes in the injection luminosity
of the higher energy electrons, rather than variability
of the ambient thermal photons, since in that case there
would be higher amplitude variations in the UV than the
infrared. The implication of the short lag time (Fig. 2) is
that electrons of similar energy produce IR and gamma-
ray emission.
Figure 3 shows the SED in the high state (JD 2454680–
2454750) and at the lower final intensity (JD 2454750–
2454820).
The SED of the high flux state prior to
JD 2454750 shows an optical/IR flux level similar to that
of the May 2007 flare (Raiteri et al. 2008), intermedi-
ate between the high and low states reported by Rai-
teri et al. (2007) (and references therein), so not surpris-
ingly, the basic model parameters are similar. The op-
tical/UV emission is due to the highest energy electrons
(Lorentz factors ∼103−4) radiating via synchrotron in
a field of ∼10 Gauss, while the gamma-rays come from
inverse Compton scattering on the broad-line photons.
The bulk Lorentz factor is Γ ∼δ ∼10 −15 (where
δ = [γ(1 −β cos θ)]−1 is the Doppler beaming factor).
The lack of correlation seen in the DCF for 2–10 keV X-
rays with respect to the other wavebands finds a natural
explanation in the external Compton scenario, with the
7 For the B band, we include optical fluxes from both the Swift
and SMARTS telescopes in order to have complete coverage over
gaps in the individual light curves.
X-rays coming from low-energy electrons (γ ∼10–100)
inverse-Compton scattering external UV photons, rather
than higher energy electrons (∼103−4) scattering syn-
chrotron photons. An SSC component in X-rays would
introduce correlation between X-rays and gamma-rays,
which is not seen. The highest energy electrons (produc-
ing the IR/optical and gamma-ray emission) vary more
rapidly (the radiative timescales are shorter) while the
low energy electrons act as a reservoir and vary more
slowly.
More precise SED modeling is needed to determine de-
tailed model parameters, such as the energy density and
location of the thermal photons, the location and size of
the scattering region, the electron distribution, the bulk
Lorentz factor and jet orientation, etc.
This detailed
analysis will be deferred to a later paper. Still, some ad-
ditional conclusions can be made. The overall stability
of source parameters and the correlation imply that the
emission region is stable on time scales of ∼1 month. If
the electrons are localized in a fast moving knot (which
might become visible in VLBI maps in a few months), it
moves a distance γ2c∆t, roughly 1-10 pc, i.e., the jet pa-
rameters cannot change dramatically on this scale. How-
ever, the Sikora et al. (1994) model for 3C 454.3 can be
ruled out as the source of the rapid variations discussed
here, since their assumed source size of 1019 cm implies
∆t ≳1 year. Instead, their model might explain a slowly
changing, much larger region of the jet.
In conclusion, 3C 454.3 shows very strong, correlated
variability between the peak of the synchrotron compo-
nent (at infrared, optical and UV wavelengths) and the
peak of the gamma-ray component. No such correlation
is seen between X-rays and any other band. These results
suggest that the variability arises from changes in the
electron luminosity at a compact location in the jet. The
highly variable infrared through UV emission, particu-
larly in the brightest state, is dominated by synchrotron
emission from a compact region of high-energy electrons
in the jet, with a smaller contribution from a relatively
steady accretion disk.
The slowly varying low-energy
part of the electron spectrum gives rise to relatively sta-
ble X-ray emission via scattering. The gamma-rays vary
in a correlated way because they result from the same
high-energy electrons up-scattering ambient UV photons.
SMARTS observations of LAT-monitored blazars are
supported by Fermi GI grant 011283. CDB, MMB and
the SMARTS 1.3m observing queue also receive support
from NSF grant AST-0707627. This research has made
use of the NASA/IPAC Infrared Science Archive, which
is operated by the Jet Propulsion Laboratory, Californina
Institute of Technology, under contract with the National
Aeronautics and Space Administration.
4
Bonning et al.
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TABLE 1
Fractional Variability Amplitude
Band
Fvar
K
0.510 ± 0.0004
J
0.603 ± 0.0001
R
0.472 ± 0.001
V
0.385 ± 0.001
B
0.362 ± 0.001
U
0.193 ± 0.002
W1
0.165 ± 0.003
M2
0.142 ± 0.004
W2
0.140 ± 0.004
2-10 keV
0a
0.1-300 GeV
0.455 ± 0.015
aThe X-ray sample variance was equivalent to the mean square error, leading to a value of Fvar consistent with zero.
Correlated Variability in 3C 454.3
5
Fig. 1.— Multi-wavelength light curves of 3C 454.3 at (top panel) gamma-ray (0.1–300 GeV), UV (W1), optical (B), and IR (J) wavelengths
from Fermi LAT, Swift UVOT, and SMARTS. Fluxes have been normalized to JD 2454700. Light curves are offset for clarity; minor tick
spacing corresponds to 50% change. Fluxes at JD 2454700 are 2.83×10−6 cts s−1 at 0.1–300 GeV, 1.64×10−11 erg s−1 cm−2 in W1,
2.21×10−11 erg s−1 cm−2 in B, and 3.62×10−11 erg s−1 cm−2 in J. (Bottom panel) Swift XRT 2-10 keV light curve, normalized to flux at
JD 2454700 (2.90×10−11 erg s−1 cm−2). The IR/optical/UV variations are well correlated with the gamma-ray variations, with a lag of
≲1 day, while the (minimal) X-ray variability is uncorrelated. The variability has much higher amplitude in the J-band than in B, which
can be explained if there is an relatively constant blue component, as expected for an accretion disk. At z=0.859, Balmer continuum from
an accretion disk, as well as Fe ii and Mg ii emission lines would be redshifted into the B and V bands; Hα is shifted into the J band.
6
Bonning et al.
Fig. 2.— Discrete correlation function for optical B-band (black squares) and infrared J-band (red triangles) versus gamma-ray (0.1–300
GeV) light curves. Error bars on the γ - J-band DCF are comparable to the γ - B-band DCF and are omitted for clarity. The DCF
peaks at zero lag, supporting external Compton models in which the variability is due to changes in the spectrum of relativistic electrons
that both radiate the optical/IR synchrotron emission and up-scatter soft photons to GeV energies. The DCF has a hint of a shoulder at
∼3 −5 days (negative lags correspond to the gamma-rays leading the other band). This may result from slightly higher electron energies
for the gamma-rays, which would give them shorter radiative timescales.
Correlated Variability in 3C 454.3
7
Fig. 3.— Spectral energy distributions for high (JD 2454680–2454750) and low (JD 2454750–2454820) states of 3C 454.3. Error bars not
shown are smaller than the plotted points. Fermi/LAT fluxes are derived from the average count rate in the 0.3-1 GeV and 1-300 GeV
bands assuming a power-law spectrum with photon index Γ = 2. The long-wavelength component in the low state is very flat, not unlike
an accretion disk spectrum, while the variable component is clearly an infrared-bright jet. |
arxiv | 0812.5111v1 | Biases and Uncertainties in Physical Parameter Estimates of Lyman Break Galaxies from Broad-band Photometry | We investigate the biases and uncertainties in estimates of physical parameters of high-redshift Lyman break galaxies (LBGs), such as stellar mass, mean stellar population age, and star formation rate (SFR), obtained from broad-band photometry. By combining LCDM hierarchical structure formation theory, semi-analytic treatments of baryonic physics, and stellar population synthesis models, we construct model galaxy catalogs from which we select LBGs at redshifts z ~ 3.4, 4.0, and 5.0. The broad-band spectral energy distributions (SEDs) of these model LBGs are then analysed by fitting galaxy template SEDs derived from stellar population synthesis models with smoothly declining SFRs. We compare the statistical properties of LBGs' physical parameters -- such as stellar mass, SFR, and stellar population age -- as derived from the best-fit galaxy templates with the intrinsic values from the semi-analytic model. We find some trends in these distributions: first, when the redshift is known, SED-fitting methods reproduce the input distributions of LBGs' stellar masses relatively well, with a minor tendency to underestimate the masses overall, but with substantial scatter. Second, there are large systematic biases in the distributions of best-fit SFRs and mean ages, in the sense that single-component SED-fitting methods underestimate SFRs and overestimate ages. We attribute these trends to the different star formation histories predicted by the semi-analytic models and assumed in the galaxy templates used in SED-fitting procedure, and to the fact that light from the current generation of star-formation can hide older generations of stars. These biases, which arise from the SED-fitting procedure, can significantly affect inferences about galaxy evolution from broadband photometry. | Seong-Kook Lee, Rafal Idzi, Henry C. Ferguson, Rachel S. Somerville, Tommy Wiklind, Mauro Giavalisco | astro-ph | https://arxiv.org/abs/0812.5111v1 | arXiv:0812.5111v1 [astro-ph] 30 Dec 2008
Biases and Uncertainties in Physical Parameter Estimates of
Lyman Break Galaxies from Broad-band Photometry
Seong-Kook Lee
Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles
Street, Baltimore, MD 21218-2686
joshua@pha.jhu.edu
Rafal Idzi
Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles
Street, Baltimore, MD 21218-2686
Henry C. Ferguson
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218
Rachel S. Somerville
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218
Tommy Wiklind
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218
and
Mauro Giavalisco
Astronomy Department, University of Massachusetts, Amherst, MA 01003
ABSTRACT
We investigate the biases and uncertainties in estimates of physical parame-
ters of high-redshift Lyman break galaxies (LBGs), such as stellar mass, mean
stellar population age, and star formation rate (SFR), obtained from broad-band
photometry. These biases arise from the simplifying assumptions often used in
fitting the spectral energy distributions (SEDs). By combining ΛCDM hierar-
chical structure formation theory, semi-analytic treatments of baryonic physics,
and stellar population synthesis models, we construct model galaxy catalogs from
– 2 –
which we select LBGs at redshifts z ∼3.4, 4.0, and 5.0. The broad-band photo-
metric SEDs of these model LBGs are then analysed by fitting galaxy template
SEDs derived from stellar population synthesis models with smoothly declining
SFRs. We compare the statistical properties of LBGs’ physical parameters – such
as stellar mass, SFR, and stellar population age – as derived from the best-fit
galaxy templates with the intrinsic values from the semi-analytic model. We find
some trends in these distributions: first, when the redshift is known, SED-fitting
methods reproduce the input distributions of LBGs’ stellar masses relatively well,
with a minor tendency to underestimate the masses overall, but with substantial
scatter. Second, there are large systematic biases in the distributions of best-fit
SFRs and mean ages, in the sense that single-component SED-fitting methods
underestimate SFRs and overestimate ages. We attribute these trends to the
different star formation histories predicted by the semi-analytic models and as-
sumed in the galaxy templates used in SED-fitting procedure, and to the fact that
light from the current generation of star-formation can hide older generations of
stars. These biases, which arise from the SED-fitting procedure, can significantly
affect inferences about galaxy evolution from broadband photometry.
Subject headings: galaxies: evolution — galaxies: fundamental parameters —
galaxies: high-redshift — galaxies: statistics — galaxies: stellar content — meth-
ods: statistical
1.
Introduction
Over the last decade, astronomy has experienced a boom of panchromatic surveys of
high redshift galaxies – such as the Great Observatories Origins Deep Survey (GOODS,
Giavalisco et al. 2004a) and the Cosmic Evolution Survey (COSMOS, Scoville et al. 2007)
– thanks to the developments of space telescopes, like Hubble Space Telescope (HST) and
Spitzer Space Telescope (Spitzer), and also of ground-based facilities, including Keck obser-
vatory, Very Large Telescope (VLT), and Subaru telescope. Spectral energy distributions
(SEDs) constructed from the photometric data of wide wavelength coverage have been used
to constrain galaxy parameters, such as stellar masses, SFRs, and stellar population ages,
of high redshift galaxies via comparison with simple stellar population synthesis models
(e.g. Papovich et al. 2001; Shapley et al. 2001, 2005; Shim et al. 2007; Stark et al. 2007;
Verma et al. 2007). Accompanied byselection techniques to separate high-redshift galaxy
samples efficiently via their broad-band colors and magnitudes, these surveys have provided
increasing information about the nature of high-redshift galaxies.
– 3 –
However, despite recent advances in our knowledge of the properties of high redshift
galaxies, more accurate estimation of physical parameters is necessary for addressing sev-
eral important issues in galaxy evolution and cosmology.
For example, it is as yet un-
certain whether the Lyman break galaxies (LBGs, Steidel et al. 1996; Giavalisco 2002) –
high-redshift, star-forming galaxies selected according to their rest-frame ultraviolet (UV)
colors – will evolve into large ellipticals (Adelberger et al. 1998; Steidel et al. 1998), or into
smaller spheroids, such as small ellipticals/spiral bulges or subgalactic structures, which will
later be merged to form larger galaxies at z ∼0 (Lowenthal et al. 1997; Sawicki & Yee 1998;
Somerville et al. 2001). Accurate knowledge of physical parameters, such as stellar masses
and SFRs, serves as one of the important elements in discriminating among the possible evo-
lutionary descendants of LBGs. Also, accurately constraining LBG’s stellar-population ages
and stellar masses is necessary for determining how much the LBG populations contributed
to cosmic reionization. Better estimation of these parameters is also crucial in comparing
different galaxy populations at different redshifts.
The stellar populations of LBGs as well as other populations of high-redshift galaxies
have been studied by several authors using SED-fitting methods to compare the photometric
SEDs of observed galaxies with various galaxy spectra from stellar population synthesis
models.
Papovich et al. (2001) investigated 33 spectroscopically-confirmed LBGs with a
redshift range, 2.0 ≤z ≤3.5.
They found that the mean lower limit of stellar masses
of these LBGs is ∼6 × 109 M⊙with upper limits of ∼3-8 times larger and that the
mean age is ∼120 Myr (assuming solar metallicity and Salpeter (1955) IMF) with broad
range between 30 Myr and 1 Gyr. More importantly, they concluded that the most robust
parameter constrained through SED-fitting is stellar mass, while stellar population age and
especially star formation rate (SFR) are only poorly constrained. Also, they speculated that
SED-fitting methods (single-component fitting) can only give lower limits to galaxies’ stellar
masses because the results from the SED-fitting methods are largely driven by the light from
the most massive, most recently formed stars. Shapley et al. (2001) studied the physical
parameters of 74, z ∼3 LBGs with spectroscopic redshifts. They confirmed that among
various physical parameters, stellar mass is the most tightly constrained, and that constraints
on other parameters such as dust extinction and age are weak. With the assumption of solar
metallicity and a Salpeter IMF, the median age (defined as time since the onset of current
star formation) was found to be tsf ∼320 Myr with a large spread from several Myr to
more than 1 Gyr. The median stellar mass was 1.2×1010 h−2 M⊙, which is higher than
the Papovich et al. (2001) value. However, the Shapley et al. (2001) LBGs were generally
brighter than LBGs in the Papovich et al. (2001) sample. The two studies found similar
stellar masses for LBGs with similar rest-frame UV luminosities. Shapley et al. (2001) also
found that about 20 % of the LBGs have best-fit ages older than 1 Gyr, stellar masses larger
– 4 –
than 1010 M⊙, and SFR ∼30 M⊙yr−1, and they interpreted these as galaxies which have
formed their stars over a relatively long period in a quiescent manner. Shapley et al. (2001)
also noted that more luminous galaxies are dustier than less luminous ones and that younger
galaxies are dustier and have higher SFR.
More recently, Verma et al. (2007) studied 21 z ∼5 LBGs, six of which have confirmed
spectroscopic redshifts and the remaining 15 of which have photometric redshifts. These
LBGs were found to be moderately massive with median stellar mass ∼2 × 109 M⊙, and
to have high SFRs with a median of ∼40 M⊙yr−1. They also found that the stellar mass
estimates are the most robust of all derived properties. Best-fit ages have a large spread with
a median value ∼25 Myr, assuming a metallicity of one-fifth solar (0.2 Z⊙) and a Salpeter
IMF. Verma et al. (2007) also compared their z ∼5 LBGs with Shapley et al. (2001)’s z ∼3
LBGs, assuming the same IMF and metallicity (i.e. solar metallicity and Salpeter IMF), and
concluded that these two samples of LBGs with similar rest-frame UV luminosity are clearly
different. Specifically, z ∼5 LBGs are much younger (≲100 Myr) and have lower stellar
masses (∼109 M⊙) than z ∼3 LBGs. The fraction of young (age < 100 Myr) galaxies is
∼70% for z ∼5 LBGs while it is ∼30% for z ∼3 LBGs. They also concluded that these
z ∼5 LBGs are the likely progenitors of the spheroidal components of present-day massive
galaxies, based on their high stellar mass surface densities. Stark et al. (2007) analysed 72
z ∼5 galaxies (not just LBGs) with photometric redshifts. They performed an SED-fitting
analysis on a spectroscopically confirmed subset of 14 z ∼5 galaxies to derive best-fit stellar
masses ranging between 3×108 and 2×1011 M⊙, and ages from 1 Myr to 1.1 Gyr. Three out
of these 14 galaxies have stellar masses in excess of 1011 M⊙. Using stellar mass estimates
from both spectroscopically confirmed and candidate galaxies, they calculated a stellar mass
density at z ∼5 of 6 × 106 M⊙Mpc−3, which is much larger than the integration of the
star formation rate density (SFRD) from z ∼10 to 5. They attributed this discrepancy
either to significant dust extinction or to undetected low-luminosity star-forming galaxies at
z ≳5. Shim et al. (2007) studied the properties of 1088 massive LBGs whose best-fit stellar
masses are larger than 1011 M⊙at z ∼3. They derived stellar masses of these massive LBGs
through SED-fitting with the assumption of a Salpeter IMF, and noted that LBGs which
are detected in mid-infrared wavelength are the ones with large stellar mass and severe dust
extinction among LBG population.
The SED-fitting method is also used to constrain physical parameters for high-z Lyman-
α emitting galaxies (LAEs, Finkelstein et al. 2007) and for submillimetre galaxies (Dye et al.
2008). Finkelstein et al. (2007) analysed z ∼4.5 LAEs through SED fitting and found their
ages range from 1 Myr to 200 Myr (assuming a constant star formation history), and stellar
masses from 2 × 107 −2 × 109 M⊙. Dye et al. (2008) used SED fitting to study physical
parameters of SCUBA (Submillimetre Common-User Bolometer Array) sources, finding an
– 5 –
average stellar mass of ∼1011.8±0.1M⊙.
However, the estimated parameter values and implications derived from these values
are prone to several sources of error including errors inherent in SED-fitting methods. It
is unclear whether or not SED-fitting methods deliver biased estimates of parameters such
as stellar mass, age, and SFR. It is also unclear how much worse the parameter estimates
become if spectroscopic redshifts are unavailable and the photometric data must be used to
constrain not only the star formation history, but also the redshift. The critical question is
how far we can trust the physical parameters derived using mainly photometric data under
the assumption of simple star-formation histories (SFHs), when real galaxies are expected
to have more complex formation/evolution histories with several possible episodes of star-
formation.
In this paper, we address this issue by comparing the statistical distributions of intrin-
sic physical parameters of model Lyman break galaxies (LBGs) from semi-analytic models
(SAMs) of galaxy formation, with the estimates of those same parameters derived from
stellar population synthesis models through the commonly used SED-fitting method.
There have been several studies which tried to constrain the stellar populations of LBGs
or of high-redshift galaxies by comparing them with predictions from theoretical models, such
as semi-analytic models (e.g. Somerville et al. 2001; Idzi et al. 2004) or cosmological hy-
drodynamic simulations (e.g. Nagamine et al. 2005; Night et al. 2006; Finlator et al. 2007).
Somerville et al. (2001) compared z ∼3 and z ∼4 observed LBGs with semi-analytic mod-
els to investigate possible scenarios for LBGs, and Idzi et al. (2004) put constraints on the
properties of z ∼4 LBGs through comparison of colors of observed LBGs and SAMs model
LBGs. Nagamine et al. (2005) and Night et al. (2006) used cosmological simulations to place
constraints on properties of UV-selected, z ∼2 galaxies and z ∼4 −6 LBGs. Finlator et al.
(2007) constructed a set of galaxy templates from hydrodynamical simulations and used
them to place constraints on properties of 6 z ≤5.5 observed galaxies. Our approach is
distinct from these studies. Here, instead of comparing model galaxies with observed ones,
we perform SED fitting on model galaxies, derive best-fit parameters, and compare them
with the intrinsic values given in the model, trying to understand quantitatively as well as
qualitatively the biases and uncertainties of SED-fitting methods in constraining the physical
parameters of LBGs.
To accomplish this, we construct galaxy formation histories by combining ΛCDM hier-
archical structure formation theory with semi-analytic treatments of gas cooling, star for-
mation, supernova feedback, and galaxy mergers. Then, using the stellar population synthe-
sis models of Bruzual & Charlot (Bruzual & Charlot 2003, hereafter BC03), and a simple
model for dust extinction, we derive the photometric properties of these model galaxies.
– 6 –
After selecting LBGs through appropriate color criteria, we compare the photometric spec-
tral energy distributions (SEDs) in observed-frame optical (HST/ACS) through mid-infrared
(Spitzer/IRAC) passbands of these model LBGs with those of SED templates from stellar
population synthesis model of BC03 using a χ2-minimization method. In this SED-fitting
procedure, a large multi-parameter space is explored to minimize any prior. Each parameter
– such as star-formation time scale, age, dust-extinction, and metallicity – in this parame-
ter space spans a broad as well as physically realistic range. The distributions of physical
parameters of these model LBGs derived from this SED-fitting method are compared with
the input distributions of these parameters from the semi-analytic model.
Various parameters in the semi-analytic model are calibrated to reproduce the col-
ors of observed LBGs in the southern field of the GOODS (GOODS-S). The GOODS
(Giavalisco et al. 2004a) is a deep, multiwavelength survey which covers a combined area
of ∼320 arcmin2 in two fields – GOODS-N centered on the Hubble Deep Field-North
(HDF-N), and GOODS-S centered on the Chandra Deep Field-South (CDF-S). The exten-
sive wavelength coverage of the photometric data in the GOODS fields including HST/ACS
(Advanced Camera for Surveys) and Spitzer/IRAC (Infrared Array Camera) is beneficial
for the derivation of various physical parameters of galaxies including stellar mass, SFR,
and age.
Deep observations (reaching z850 ∼26.7) in the GOODS fields can probe the
high redshift universe in a comprehensive and statistically meaningful manner. This has en-
abled many authors to investigate high-redshift LBG populations in the GOODS fields (e.g.
Giavalisco et al. 2004b; Idzi et al. 2004; Papovich et al. 2004; Lee et al. 2006; Ravindranath et al.
2006; Yan et al. 2006; Stark et al. 2007; Verma et al. 2007).
In this work, we use band-
passes for the HST/ACS filters F435W (B435), F606W (V606), F775W (i775), F850LP (z850),
VLT/ISAAC (Infrared Spectrometer And Array Camera) J, H, Ks bands, and Spitzer/IRAC
3.6, 4.5, 5.8, 8.0 µm channels for SED-fitting. The CTIO (Cerro Tololo Inter-American Ob-
servatory) MOSAIC U band is used for selecting U-dropouts.
A description of the semi-analytic model of galaxy formation used in this work is given in
§ 2, with the Lyman break galaxy sample selection and the SED-fitting procedures explained
in § 3. The statistical properties of the derived physical parameters for model LBGs are
shown in § 4. We analyse, in detail, the effects of various factors on the biases in SED-fitting
in § 5, and the bias in estimating SFR from rest-frame UV magnitude in § 6. We discuss
the effects of these biases on the galaxy evolution studies in § 7 and a summary and our
conclusions are given in § 8. Throughout the paper, we adopt a flat ΛCDM cosmology, with
(Ωm, ΩΛ) = (0.3,0.7), and H0 = 100h km s−1 Mpc−1, where h=0.7, and all magnitudes are
expressed in the AB magnitude system (Oke 1974).
– 7 –
2.
Semi-Analytic Models of Galaxy Formation
Semi-analytic models (SAMs) of galaxy formation are embedded within the framework
of a ΛCDM initial power spectrum and the theory of the growth and collapse of fluctuations
through gravitational instability. The models include simplified physical treatments of gas
cooling, star formation, supernova feedback, dust extinction, and galaxy merging. In this
work, we use the SAMs model run which was constructed and used in Idzi (2007). The
Idzi (2007) work was based on the Somerville & Primack (1999) model, which is also used
by Somerville et al. (2001) and Idzi et al. (2004). Specifically, among various model runs in
Idzi (2007), we use here the model run which showed the closest match to the rest-frame
UV-continuum and UV-optical colors1 of observed U- & B-dropouts in the GOODS-S field.
The details of the model can be found in the above references, and here we briefly highlight
the most important aspects of the model (including some changes from the models described
in the original papers).
2.1.
Description of Semi-Analytic Models
The semi-analytic model used in this work implements the method of Somerville & Kolatt
(1999) to build merging histories of dark matter halos. After dark matter merging histo-
ries are constructed, the semi-analytic treatments for various physical processes, such as gas
cooling, mergers, star formation (both in a merger-induced burst mode and in a quiescent
mode), supernovae feedback, chemical evolution, and dust extinction, are applied to realize
predictions of the formation history of a statistical ensemble of galaxies.
A newly formed dark matter halo (residing at top of the tree) contains pristine shock-
heated hot gas at the virial temperature. When a halo collapses or undergoes a merger with
a larger halo, the associated gas is assumed to be shock-heated to the virial temperature
of the halo. This gas then radiates energy and consequently cools. For small halos at high
redshift, cooling is limited by the accretion rate, since the amount of gas that can cool at any
given time cannot exceed the amount of hot gas contained within the halo’s virial radius.
Once halos merge, the galaxies within them remain distinct for some time. The central
galaxy of the largest progenitor halo is set as the central galaxy of the merged dark matter
halo. All the other galaxies become satellites, which then fall in towards the central galaxy
due to dynamical friction.
Unlike in Somerville et al. (2001), collisions between satellite
1V606 - i775 and V606 - IRAC 3.6 µm colors are used for U-dropouts, and i775 - z850 and z850 - IRAC 4.5
µm colors are used for B-dropouts as diagnostics to determine best-fit model run.
– 8 –
galaxies are neglected here.
Both quiescent and merger-driven modes of star formation are included in the model.
We assume that every “major” galaxy-galaxy merger above a certain mass-ratio threshold
(1:4) triggers a starburst, which converts 100 % of the available gas into stars in a tenth of
a halo dynamical time. Quiescent star formation is modeled with a Kennicutt-like law, such
that the star formation rate is proportional to the mass of cold gas in the disk divided by
the dynamical time. We also scale the SFR by a power-law function of the galaxy circular
velocity, such that star formation is less efficient in low-mass galaxies. This mimics the effect
of a SF threshold, as implemented in more recent semi-analytic models (e.g. Somerville et al.
2008). The total star formation rate of a galaxy is the sum of the burst and quiescent modes.
Supernova feedback is also modeled via a simple recipe, where the rate of reheating of
cool gas by supernovae is proportional to the SFR times a power-law function of circular
velocity, such that the reheating is more efficient in smaller mass galaxies. If the halo’s virial
velocity is less than a preset ejection threshold (set to 100 km s−1 in our model), then all of
the reheated gas is ejected from the halo; otherwise the reheated gas is placed in the hot gas
reservoir within the halo. The gas and metals that are ejected from the halo are distributed
outside of the halo with a continuation of the isothermal r−2 profile that we assumed inside
the halo. This material falls in gradually as the virial radius of the halo increases due to the
falling background density of the Universe.
Chemical evolution is treated by assuming a constant mean mass of metals produced
per mass of stars. The metals produced in and ejected from stars are first deposited into the
surrounding cold gas, at which point they may be ejected from the disk and mixed with the
hot halo gas. The metallicity of any newly formed stars is set to equal the metallicity of the
ambient cold gas at the time of formation.
For each galaxy, the SAM predicts the two-dimensional distribution of the mass in
stars of a given age and metallicity. We convolve this distribution with the SSP models of
BC03 (using the Padova 1994 isochrones) to create a synthetic SED. We adopted a Chabrier
(2003) IMF with lower and upper mass cutoffs of mL
=
0.1 M⊙and mU
=
100 M⊙.
We then use the model of Madau (1995) to account for the opacity of intervening HI in the
intergalactic medium as a function of redshift, and convolve the SEDs with the response
functions appropriate to the ACS, ISAAC, IRAC, and MOSAIC photometric bands.
Dust extinction is treated assuming that the face-on optical depth in the V-band is
τV,0 = τdust,0×( ˙m∗)βdust where τdust,0 and βdust are free parameters set as 1.2 and 0.3, to match
the observations in the GOODS-S. This choice is motivated by the observational results of
Hopkins et al. (2001). The dependence of the extinction on wavelength (the attenuation
– 9 –
curve) is calculated using a Calzetti attenuation curve (Calzetti et al. 2000).
In summary, we use the semi-analytic models to build galaxies that have rich and varied
star formation histories that are motivated by the hierarchical galaxy formation picture. Star
formation is bursty and episodic. Stars have a distribution of metals consistent with the
assumed star formation history, and the dust content is plausible. In contrast, the simple
SED templates which we will use to analyze these model galaxies have uniform metallicity
and monotonically declining star formation rates.
3.
Lyman Break Galaxy Samples and Spectral Energy Distribution Fitting
3.1.
Model Galaxy Catalog and Lyman Break Color Selection
We created a suite of model runs by varying the uncertain model parameters controlling
the burstiness of star formation and dust extinction. As stated in § 2, the rest-frame UV-
continuum and UV-optical colors of model LBGs from each semi-analytic model run were
compared with the colors of LBGs observed in the GOODS-S field, and the best-fit model
run was selected based on these comparisons. From this best-fit model run, a model galaxy
catalog, which contains a total of 44281 galaxies within a redshift range 2.3 ≤z ≤5.7, has
been constructed. This catalog carries the various physical parameters such as stellar mass
and SFR, along with the (dust-extinguished) broad band photometry for several bandpasses
used in the GOODS observations.
The photometric data in this catalog are derived by
combining the SAM’s star formation and enrichment histories with BC03 stellar population
synthesis models, and including models for dust extinction and absorption by the IGM, as
described in § 2.
These photometric data are used to select high-redshift, star-forming galaxies through
the Lyman-break color selection technique. The Lyman-break color selection technique –
which uses the ‘Lyman-break’ feature at λ ∼912 ˚A in galaxy spectra and the Lyman-α
forest flux deficit between 912 ˚A and 1216 ˚A together with blue colors at longer rest-frame
UV wavelengths to identify star-forming galaxies located at high redshift – has been shown
to be an effective way to construct large samples of high-redshift, star-forming galaxies from
optical photometric data sets (Madau et al. 1996; Steidel et al. 2003). Spectroscopic follow-
ups for the LBG samples have verified the robustness and efficiency of this technique in
building up high-z galaxy samples (Steidel et al. 1996, 1999; Vanzella et al. 2006)
The color criteria used in this work to select U-, B-, & V-dropout model LBGs are as
follows:
– 10 –
(U −B435) ≥0.62 + 0.68 × (B435 −z850) ∧
(1)
(U −B435) ≥1.25 ∧
(2)
(B435 −z850) ≤1.93
(3)
for U-dropouts (z∼3.4),
(B435 −V606) > 1.1 + (V606 −z850) ∧
(4)
(B435 −V606) > 1.1 ∧
(5)
(V606 −z850) < 1.6
(6)
for B-dropouts (z∼4), and
((V606 −i775) > 1.4667 + 0.8889 × (i775 −z850)) ∨((V606 −i775) > 2.0) ∧
(7)
(V606 −i775) > 1.2 ∧
(8)
(i775 −z850) < 1.3
(9)
for V-dropouts (z∼5).
Here, ∧means logical ‘AND’, and ∨is logical ‘OR’.
The criteria for B- & V-dropouts used here are the same ones which have been tested
and used in previous works on GOODS survey fields (e.g., Giavalisco et al. 2004b; Idzi et al.
2004; Papovich et al. 2004; Idzi 2007). The criteria for U-dropouts are set (1) to select z ∼
3 star-forming galaxies, (2) to effectively avoid contamination from foreground stars, and
(3) to minimize the overlap with B-dropouts. These color criteria and additional magnitude
cuts for the ACS z-band (zAB ≤26.6) and IRAC channel 1 or channel 2 (m3.6µm ≤26.1 or
m4.5µm ≤25.6), which correspond to detection magnitude limits for each bands in GOODS-
S, give samples of LBGs containing 2729 objects for U-dropouts with redshift range 3.1 ≤
z ≤3.6, 2638 objects for B435-dropouts with 3.6 ≤z ≤4.6, and 808 for V606-dropouts with
4.6 ≤z ≤5.6.
Figure 1 shows the redshift distributions of model U-, B-, and V-dropout galaxies se-
lected with the above criteria. The color selection criteria effectively isolate three distinct
redshift intervals. Because the models include a realistic distribution of galaxy luminosities,
the galaxies preferentially lie toward the lower redshift boundary of each redshift interval,
as they do for real surveys.
– 11 –
3.2.
Spectal Energy Distribution Fitting
The goal of this work is to compare the statistical distributions of various physical
parameters of model Lyman break galaxies with the ones derived from the SED-fitting
method. To do this, we performed χ2-minimization to find the best-fit stellar population
synthesis model templates and best-fit parameters derived from them. Here, we use as the
stellar population synthesis model the BC03 model with Chabrier (2003) IMF and Padova
1994 evolutionary tracks. This is the same set of SSP models used to construct the SEDs
for the SAM galaxies. Only few discrete values of metallicities are available in the BC03
model library; among these, two sub-solar (0.2 Z⊙and 0.4 Z⊙) and solar metallicities were
used. Studies of high redshift LBG spectra have shown that their metallicities are subsolar
or solar (Z ∼0.2 −1.0 Z⊙) (Pettini et al. 2001; Shapley et al. 2003; Ando et al. 2004). For
internal dust extinction, the Calzetti extinction law (Calzetti et al. 2000) is used with 0.000
≤E(B −V ) ≤0.950 with a step size 0.025, and intergalactic absorption by neutral hydrogen
according to Madau (1995) is included. The star formation histories are parameterized as
∝e−t/τ with the e-folding time, τ ranging between 0.2 and 15.0 Gyr, and the time since onset
of star formation, t spanning from 10 Myr to 2.3 Gyr, with limiting t being smaller than
the age of the universe at each corresponding redshift. The parameter values used in our
SED-fitting are summarized in table. 1. From the observed LBGs in GOODS-S, mean errors
for different magnitude bins at each passband of ACS, ISAAC and IRAC are calculated,
and assigned to each SAM galaxy photometric value according to their magnitudes in the
calculation of χ2. Fluxes of the SAM galaxies in the ACS B435, V606, i775, z850 bands (B435
band photometry are not used in fitting for V606-dropouts), ISAAC J, H, Ks bands, and
IRAC 3.6, 4.5, 5.8, 8.0 µm channels are compared with those of BC03 model templates2,
and based on the calculated χ2 values, the best-fit galaxy template is determined for each
model LBG. The best-fit stellar mass is calculated by multipling the mass-to-light ratio and
bolometric luminosity of the best-fit template.
The best-fit stellar mass equals to the integration of the SFR over time, correcting for
stellar mass recycling.
M∗= A × [
Z t0
0
Ψ(t′, τ)dt′ −Γ(t0, τ)]
(10)
Here, t0 and τ are the best-fit t and τ respectively, M∗is the best-fit stellar mass, and
A is a normalization factor.
Ψ(t, τ) and Γ(t, τ) are unnormalized SFR and stellar mass
2U-band data is used only for selecting U-dropout samples, and is not used in the SED-fitting procedure.
– 12 –
recycling fraction respectively, and can be known from the best-fit t and τ. The best-fit
instantaneous SFR is then,
SFR(t0, τ) = A × Ψ(t0, τ),
(11)
where Ψ(t0, τ) = e−t0/τ.
Not correcting for recycling would result in a ∼29 % underestimation of the correct SFR
and 8 ∼9 % overestimation of the mean age3. The statistical behaviors of various physical
parameters, such as stellar mass, SFR, and mean age, derived from these best-fit templates
are compared with the intrinsic distributions from the SAMs in the following sections.
4.
Constraining Physical Parameters using the SED-Fitting Method
4.1.
Physical Parameters of Model Lyman Break Galaxies
In the ideal case, in which the SED-fitting methods can determine various physical
parameters accurately, the one-to-one comparison between intrinsic SAM parameters and
best-fit parameters from SED-fitting should show no deviation from a straight line with a
slope of unity. However, there are several factors that introduce errors in the parameters
derived from SED-fitting, which include the complex star formation/merging histories of
individual galaxies and photometric errors. So, the interesting question is how well the SED-
fitting methods can recover the intrinsic or true statistical distributions of various physical
parameters – such as stellar mass, star formation rate (SFR), mean stellar population age,
and so on – which are used to constrain galaxies’ formation histories.
The star formation rate of a galaxy is an ill-defined quantity. Strictly speaking, it is
always zero except at those instants in time when a collapsing ball of gas begins to generate
energy through nuclear fusion. It is common in the literature to implicitly average over some
time interval ∆t without specifying ∆t. In our view, it is important to be specific about the
timespan for this averaging. Henceforth in this paper, we define the SFR to be the mass in
stars formed in the past 100 Myr. Our reason for this choice is as follows: unlike the BC03
model galaxy templates whose SFRs are assumed to decrease exponentially with time, the
star formation activity in semi-analytic model galaxies – and probably in real LBGs also
3In some studies, SFR is derived first from the best-fit t and τ, then stellar mass is calculated by integrating
the SFR over time. In these cases, failure to correct for stellar mass recycling results in an overestimation
of stellar mass.
– 13 –
(e.g. Papovich et al. 2005) – is more episodic and complex. So, instantaneous SFRs or SFRs
averaged over very small ∆t are less meaningful for SAM LBGs.
In this work, we repeat our SED-fitting experiment with several different choices of
conditions: (1) holding redshift fixed and fitting redshift as an additional free parameter, (2)
using various combinations of passbands of ACS, ISAAC, and IRAC, and (3) with limited
τ ranges. We also try two-component templates as well as single-component templates.
In this section, we show the SED-fitting results in cases where we use all the passbands
from ACS, ISAAC, and IRAC: first, assuming the redshift is known, and second, fitting
redshift as an additional free parameter.
Figures 2, 3, and 4 compare the intrinsic distributions to the fitted values of stellar
mass, SFR (averaged over the past 100 Myr), and (mass-weighted) mean stellar population
age for z ∼3.4 (U-dropouts), 4.0 (B-dropouts), and 5.0 (V-dropouts) model LBGs, with the
redshift fixed to the actual value in the SAM model catalog.
Figure 2 shows that the SED-fitting method recovers relatively well the input distribu-
tions of stellar masses in spite of the very different star formation histories (SFHs) in the
semi-analytic model and in the templates from the BC03 stellar population synthesis models.
The mean values of the SED-derived stellar masses of U-, B-, and V-dropouts differ from the
intrinsic mean values by ∼19 % (U-dropouts), 25 % (B-dropouts), and 25 % (V-dropouts) of
the intrinsic mean values, in the sense that the SED-fitting method systematically underesti-
mates the mean values of stellar masses for all of three dropout samples. This trend may be
attributed to the fact that light from a recent generation of star formation can easily mask
(some portion of) the presence of an older stellar population. Our results are consistent with
the earlier arguments that the stellar masses of LBGs are underestimated when derived from
(single-component) SED-fitting methods (e.g. Papovich et al. 2001; Shapley et al. 2005).
However, figures 3 and 4 show that there are larger biases in constraining SFRs and
mean ages of galaxies using the SED-fitting method. As can be seen in these figures, the
mean SFRs derived from the SED-fitting method systematically underestimate the intrinsic
mean values by ∼65 % for U-dropouts, ∼58 % for B-dropouts and ∼62 % for V-dropouts,
while the stellar population mean ages are overestimated by about factors of two for all three
sets of dropouts.
The bottom rows of figure 5, 6, and 7 show the results when the redshift is allowed to float
as an additional free parameter in the fit – which is analogous to the case of color-selected
LBG samples without spectroscopic (or pre-calculated photometric) redshift information.
The bottom rows of the first columns of figure 5, 6, and 7 show that when we allow
– 14 –
redshift to vary as an additional free parameter, the differences between the distributions of
SAM intrinsic and SED-derived stellar masses increase especially for B- and V-dropouts. The
mean values of the SED-derived stellar masses are underestimated by 25 % (U-dropouts),
51 % (B-dropouts), and 43 % (V-dropouts) when redshifts were allowed to float.
More interestingly, bimodalities in the distributions of SED-derived SFRs and mean
ages become more significant, as can be seen in the second and third columns of the bottom
row of figures 5, 6, and 7. The change is more significant for B- and V-dropouts. When we
fix the redshift, we can only see hints of the existence of this bimodality in the SFR and
age distributions of B-dropouts. The existence of these bimodalities, which are absent in
the intrinsic distributions, indicates that there are sub-populations of LBGs whose behavior
in the SED-fitting procedure is distinct from others, and the different behaviors of these
subpopulations are exaggerated when we do not fix the redshifts. The directions of these
bimodal distributions show that for this sub-population of LBGs, the SED-fitting method
does not underestimate (or even overestimates) SFRs and underestimates mean ages. The
characteristic SFHs of this sub-population of galaxies are discussed in § 4.4
SED-fitting generally underestimates redshifts slightly and the ranges of redshift dis-
crepancies, (zSED −zSAM)/(1+zSAM) are ∼-0.137 −0.043 for U-dropouts, ∼-0.115 −0.044
for B-dropouts, and ∼-0.052 −0.010 for V-dropouts.
For galaxies whose redshifts are
severely underestimated, stellar masses and ages are severely underestimated and SFRs are
overestimated. The means of |zSED −zSAMs|/(1+zSAMs) are 0.022, 0.032, and 0.014 for U-,
B-, and V-dropouts.
Table. 2 summarizes the main results of the SED-fitting in the case where we fix the
redshifts to the values in the SAM model galaxy catalog (analogous to observations with
spectroscopic redshifts) as well as in the case where we allow redshift to float as an additional
free parameter (analogous to observations without spectroscopic redshifts). The contents of
table. 2 are the mean values of SAM intrinsic and SED-derived stellar masses, SFRs, and
mean ages as well as redshifts for each set of dropouts.
4.2.
Biases in Constraining Physical Parameters
In section 4.1, we showed the statistical behavior of various physical parameters derived
from the SED-fitting compared with the intrinsic distributions. In this and the next section,
we investigate more thoroughly the biases in the statistical properties of physical parameters
derived from SED-fitting, focusing in particular on the dependencies of the offsets on various
galaxy parameters.
– 15 –
First, in this section, we show how the discrepancies in LBGs’ stellar mass, SFR, and
mean age depend on the magnitudes and colors of LBGs. Specifically, we investigate the
behavior of biases as functions of rest-frame UV and optical magnitudes (i.e. ACS V606 and
IRAC m4.5µm for U-dropouts, ACS i775 and IRAC m5.8µm for B-dropouts, and ACS z850 and
IRAC m5.8µm for V-dropouts), and rest-frame UV and UV-optical colors (B435 −V606 and
i775 −m3.6µm for U-dropouts, V606 −i775 and i775 −m3.6µm for B-dropouts, and i775 −z850
and z850 −m4.5µm for V-dropouts) in figures 8-10 for the case of fixed redshifts and in
figures 11-13 for the case of allowing redshift as an additional free parameter.
In these
figures, relative errors of stellar mass, SFR, and mean age are defined as ∆rM∗= (M∗,SED −
M∗,SAM)/M∗,SAM, ∆rSFR = (SFRSED −SFRSAM)/SFRSAM, and ∆rAge = (AgeSED −
AgeSAM)/AgeSAM, respectively. Here, valueSED is stellar mass, SFR, or mean age derived
from SED-fitting, and valueSAM is the intrinsic stellar mass, SFR, or mean age of each
galaxy. These figures clearly show that the SFRs of most galaxies are underestimated and
mean ages are almost always overestimated.
For both the case of fixed redshifts and where we vary redshift as a free parameter, the
magnitude- and color-dependent behaviors are similar; however, the plots for B-dropouts
when redshifts are fixed (figure 9) and for U-, B-, and V-dropouts when redshifts are allowed
to vary freely (figures 11-13) reveal subpopulation(s) of LBGs whose behaviors in SED-fitting
are distinguished from the majority of LBGs in each dropout sample even in stellar mass
estimation.
The bimodality which is clearly seen in figure 9 is not evident in the mass
distributions shown in section 4.1. According to figure 9 and figures 11-13, galaxies which
are belong to this subpopulation show distinguished pattern of biases from the majority of
galaxies: (1) Their stellar masses are more severely underestimated than the majority of
galaxies. (2) SFRs are less severely underestimated or even overestimated. (3) Mean ages
are underestimated for these galaxies in the subpopulation while they are overestimated for
the majority of galaxies. Also, this subpopulation of galaxies is more likely to reside on the
fainter side of the rest-frame UV magnitude distribution. A more detailed investigation of
this subpopulation is given in section 4.4. It should also be noted that figure 12 indicates that
there are actually (at least) two subpopulations whose behavior in the SED-fitting procedure
are distinct from the majority of LBGs.
The offsets in stellar mass, SFR, and mean age show relatively clear dependencies on
rest-frame UV-optical color. (1) The stellar mass is more likely underestimated for redder
LBGs, (2) the bluer the LBG is in rest-frame UV-optical color, the more the SFR is un-
derestimated and the more the mean age is overestimated. Rest-frame UV-optical color is
considered to be a crude indicator of a galaxy’s stellar population age, so dependencies of
offsets on rest-frame UV-optical color may indicate that (one of) the main cause(s) of the
biases in SED-fitting is the mean age (or SFH), in the sense that systematic offsets between
– 16 –
intrinsic and best-fit SFRs and mean ages increase for younger galaxies.
Interestingly, stellar masses tend to be overestimated for galaxies that are blue in rest-
frame UV-optical color. These are also the galaxies whose biases in SFR and mean age are
largest, which suggests that the mass-overestimation and age-overestimation share the same
origin.
The parameter that shows the clearest dependence on rest-frame optical magnitude is
stellar mass. Stellar mass is more likely to be underestimated for LBGs which are brighter
in rest-frame optical magnitudes, while stellar masses of fainter LBGs are more likely to
be overestimated. At both faint and bright rest-frame optical magnitude, SFRs are clearly
underestimated, while relative errors are distributed widely.
This indicates that several
different factors may contribute to the biases in SFR estimates.
The offsets show almost no dependence on rest-frame UV-color, and relatively weak
dependence on rest-frame UV magnitude. The dispersions of offsets are larger for galaxies
whose rest-frame UV magnitudes are fainter, and SFRs are more likely underestimated for
galaxies with bright rest-frame UV magnitudes.
These color/magnitude dependencies become more complex when we do not fix the
redshifts. More specifically, for some galaxies with red rest-frame UV-optical color, stellar
masses are greatly underestimated, SFRs are severely overestimated, and ages are severely
underestimated.
These trends are similar for the subpopulation of B-dropouts shown in
figure 9.
Also, there are also some hints of galaxies with blue rest-frame UV-optical color, whose
stellar masses are underestimated, SFRs are roughly correct, and ages are underestimated.
Table. 3 lists the means and standard deviations of relative offsets for physical param-
eters such as stellar masses, SFRs, mean ages, and redshifts for each set of dropouts. Here,
the relative offset for each parameter is defined as (V alueSED −V alueSAM)/(V alueSAM).
For redshift, the relative offset is defined slightly differently as (zSED −zSAM)/(1 + zSAM),
following the convention used in the literature.
4.3.
Origins of the Biases of Galaxy Population Parameters
In this section, we investigate the dependencies of the fitting discrepancies on intrinsic
properties of the SAM model galaxies, such as stellar mass, SFR, age, and specific SFR
(SSFR; defined as SFR per unit stellar mass) to investigate the causes of these biases in
SED fitting.
– 17 –
Figures 14, 15, and 16 show how relative errors (as defined in previous section) in stellar
mass, SFR, and mean age are correlated with intrinsic stellar mass, SFR, mean age and
SSFR for U-, B-, and V-dropouts, with redshift fixed. These correlations shed light on the
origins of the biases found in sections 4.1 and 4.2.
4.3.1.
Origin of Bias in Age Estimation
The clearest correlations are seen in ‘∆rM∗vs. M∗,SAM’, ‘∆rAge vs. AgeSAM’, and
‘∆rAge vs. SSFRSAM’. The tight correlation of relative mean age errors with intrinsic ages
is not unexpected from the correlation between relative mean age errors and rest-frame UV-
optical colors. This correlation and the one between relative mean age errors and intrinsic
SSFRs are a strong indication that mean ages and/or SSFRs are the main cause of bias. The
sense of bias is that the stellar population mean age overestimates are worse for galaxies with
the youngest intrinsic ages and/or largest SSFRs (i.e. galaxies whose current SF activity
is strong compared with the past SF activity), as can be seen in figure 17. In this figure,
which shows the intrinsic mean ages (y-axis) and SSFRs (x-axis) of B-dropout galaxies, blue
dots represent galaxies whose relative age errors are 0.0 ≤∆rAge ≤0.75, i.e.
galaxies
with the smallest ∆rAge. Green dots are for galaxies with 0.75 < ∆rAge ≤2.0, and large
red dots are for galaxies with the largest ∆rAge (> 2.0). This figure clearly shows that
galaxies with very large age overestimates are relatively young galaxies with high SSFRs.
The Young mean ages and large SSFRs of these galaxies indicate that they have experienced
a relatively high level of SF activity recently. The mismatch between the SFHs predicted
by the semi-analytic galaxy formation model and the simple SED templates from the BC03
stellar population model is expected to be largest for galaxies with this type of SFH. In the
BC03 templates, SFRs are assumed to decrease exponentially, so the strongest SF activity
always occurs at early times, which is nearly the opposite of the SFHs of these galaxies.
This difference makes ages overestimated severely for these galaxies. Figures 18 and 19,
which show SF activity as a function of lookback time for individual galaxies, support this
speculation. Figure 18 shows the typical SFHs (we will refer this type of SFH as ‘type-2’
from now on) of galaxies whose age-overestimation is largest. The star formation histories
of these galaxies are clearly distinct from the SFHs of galaxies whose age-overestimation is
smallest. The typical SFHs of galaxies with the smallest age-overestimation are shown in
figure 19 (‘type-1’ SFHs from now on). In figure 17, there is a dearth of galaxies with old
age and high SSFR and galaxies with young age and small SSFR. Very young galaxies with
low SSFR would not pass our color selection criteria and/or magnitude limits or would not
be detected in real galaxy samples. Old galaxies with high SSFR, in contrast, would be
probably detected. To have a high SSFR and an old mass-weighted stellar population age,
– 18 –
galaxies would need to have had a very extreme SFH – for example, two strong, very short,
widely separated bursts.
4.3.2.
Origins of Bias in Stellar Mass Estimation
How does this age overestimation affect other derived physical parameters, such as stellar
mass and SFR? When the mean stellar age is overestimated, some portion of the galaxy’s
luminosity will be attributed to older stars, with consequently higher mass-to-light ratios,
than would be the case for the true SFH. This leads to an overestimation of the stellar mass.
In previous studies, it has been suspected that the stellar masses of galaxies are under-
estimated through single-component SED-fitting, since light from the recent star formation
can easily mask some portion of the older stellar population. This is confirmed through the
experiments done with the composite BC03 templates in section 5.4. As stated in section
4.1, the mean values of stellar masses are underestimated by ∼19-25 %, which is in qual-
itative agreement with the arguments of previous studies. However, figures 8-10 and 14-16
show that the stellar mass is not always underestimated. For some galaxies with very small
stellar masses and/or relatively young ages, the stellar mass can be overestimated. These
galaxies are the ones for which the age overestimation is large. We speculate that the stellar
mass estimation of LBGs is affected by two different factors. One factor is the fact that
the recent generation of star formation can dominate the broadband SED, leading to the
underestimation of stellar mass. The other is the SFH difference between SAM galaxies and
BC03 templates. If the age is overestimated due to the SFH difference, the mass-to-light
ratio is overestimated, which results in the stellar-mass overestimation.
In figures 14-16, it can be seen that the stellar masses are most likely underestimated for
the galaxies with the oldest ages and/or smallest SSFRs, for which the age-underestimation
is minimal. More clearly, (a)-(c) in figure 20 show that the stellar masses are underestimated
when the age estimation is nearly correct. This confirms that the stellar mass is underes-
timated in single-component SED-fitting when the effect of the other origin of bias – SFH
difference between the SAM galaxies and the BC03 templates – is minimal
In figure 20 (a)-(c), we can also see that the age discrepancies and the stellar-mass
discrepancies are correlated. Stellar masses tend to be underestimated for galaxies with the
smallest ∆rAge, while they are overestimated for most galaxies with the largest ∆rAge.
This supports the speculation that the age-overestimation (due to the SFH mismatch be-
tween the SAM galaxies and the BC03 templates) causes the mass-overestimation through
mass-to-light ratio overestimation. Evidently, in the stellar mass estimation, two sources of
– 19 –
bias are compensating with each other. While SFH differences between the SAM galaxies
and the BC03 templates tend to make stellar masses to be overestimated in SED-fitting,
recent SF activity can easily mask older stellar population causing the stellar masses to be
underestimated. The compensating effects of these two biases in the estimation of stellar
mass explains why the stellar mass distributions are recovered better than other parameters
through the SED-fitting, and also provides a clue as to why the stellar mass has turned out to
be the most robust parameter in earlier studies based on the SED-fitting (e.g. Papovich et al.
2001; Shapley et al. 2001, 2005).
4.3.3.
Origins of Bias in Star Formation Rate Estimation
Let us now consider the sources of bias in the SFR estimates. First, when ages are
overestimated, the SED fitting erroneously assigns some portion of the luminosity to older
stars instead of stars that are just forming. Figures (d)-(f) of figure 20 show that the SFRs
are most severely underestimated for galaxies with the largest ∆rAge.
However, the SFR still tends to be underestimated even when the age estimates are
nearly correct. Even some of the most extreme underestimates can be found for galaxies
whose SED-derived age is correct to within a factor of two. This indicates that there is
another source of bias in the SFR estimation.
Interestingly, the correlation between the relative SFR error and the relative stellar mass
error shown in (g)-(i) of figure 20 reveal distinct behaviors of the upper envelopes in this
correlation between galaxies with the positive relative stellar mass errors and galaxies with
the negative errors, providing another indication that there are two sources of bias in the
SFR estimation. For galaxies whose stellar masses are underestimated – i.e. galaxies whose
age-overestimation is small due to the little SFH difference or galaxies with type-1 SFH –
the SFR discrepancy is proportional to the stellar mass discrepancy. As stated in § 3.2, the
SFR is calculated from the estimated stellar mass accumulated over 100 Myr. Thus, for
galaxies with the smallest age discrepancy (with the type-1 SFH), the underestimation of
stellar mass results in the SFR underestimation.
However, unlike in the stellar mass estimation, both of these two origins of bias –
the age-overestimation due to the SFH mismatch and the hidden old stellar population by
the recent SF activity – cause the SFRs to be underestimated. This results in the overall
underestimation of the SFR distributions and the large offsets in the mean SFRs.
In addition to these two origins, the well-known ‘age-extinction degeneracy’ leads to
more significant SFR underestimation. For galaxies with large (positive) ∆rAge, dust ex-
– 20 –
tinctions tend to be underestimated due to the ‘age-extinction degeneracy’. This further
deflates estimated SFRs for these galaxies.
In summary, the main origins of biases in estimating physical parameters, such as stellar
mass, SFR, and mean age, are: (1) the differences in the assumed SFHs in the SAM galaxies
and in the BC03 stellar population templates, (2) the effects of the recent SF activity hiding
some portion of old generations of stellar population, and (3) the age-extinction degeneracy.
In the stellar mass estimation, issues (1) and (2) compete with each other, resulting in the
best-fit stellar mass distributions that resemble the intrinsic distributions. For the SFRs, all
of these issues work in the same direction, leading to the large offsets in the distributions
and in the mean values. The mass-weighted mean ages are mostly affected by issue (1).
Figures 21-23, and figure 24 are similar plots with figures 14-16, and 20, but when red-
shift is an additional free parameter in the SED-fitting procedure. These figures show trends
similar to the redshift-fixed case except that subpopulations are more evident, especially for
B- and V-dropouts. The increased ambiguity due to the lack of redshift information evi-
dently enlarges the subsets of galaxies that behave distinctly from the majority of galaxies in
the SED-fitting. However, for the majority of galaxies, the lack of redshift information does
not significantly affect the SED-fitting results, which is not surprising given the relatively
small mean redshift errors in SED-fitting.
There is a subset of galaxies whose stellar masses are severely underestimated, SFRs
are severely overestimated, and ages are severely underestimated. These are similar trends
with those shown by a subpopulation of B-dropouts when redshift is fixed. However, the
number of galaxies which belong to this subpopulation substantially increases when the
redshift is allowed to float as an additional free parameter. For these galaxies, redshift is
underestimated in the SED fitting. For U-dropouts, this sub-population is not as significant
as for B- or V-dropouts.
Also, there are galaxies with high SSFRs/young ages which act differently if we perform
the SED-fitting without fixing redshift. For these galaxies, ages are greatly overestimated,
stellar masses are overestimated, and SFRs are greatly underestimated when we fix redshift
in the SED-fitting (i.e. these are the galaxies with type-2 SFHs). When we vary redshift
freely as a free parameter, redshifts are slightly underestimated and ages/stellar masses are
underestimated. The SFRs are similar to or slightly higher than the intrinsic values.
– 21 –
4.4.
Characteristics of Sub-populations in the Fitted Distributions
Figures 9, 15, and (b), (e), and (h) of figure 20 reveal the presence of a subpopulation
of B-dropout galaxies whose behavior in the SED-fitting is distinct from the majority of
galaxies. For these galaxies, the ages are underestimated, the SFRs are overestimated, and
the stellar masses are more severely underestimated than other galaxies. What makes the
behavior of the galaxies in this subpopulation different from the majority of galaxies?
Because one of the main origins for biases in SED fitting is the SFH difference between
the SAM galaxies and the BC03 templates, it is plausible that the SFHs of these galaxies are
distinct from others. Figure 25 shows SFHs of typical model galaxies in this subpopulation
(’type-3’ SFHs from now on). Generally, they have small SSFR values like those shown in
figure 19. However, the SF activity in figure 25 shows a slower increase and more rapid
decrease with a peak at later time compared with the galaxies shown in figure 19. The
gradual decreases of the SFRs shown in figure 19 are not significantly different from the
exponentially decreasing SFRs assumed in the BC03 model, which makes the relative age
errors small for these galaxies.
On the other hand, for galaxies shown in figure 25, the
strong SF activity, which occurred relatively recently, dominates SEDs.
Combined with
the age-extinction degeneracy, this causes the mean ages to be severely underestimated,
distinguishing behaviors of these galaxies in the SED-fitting. The purple crosses in figure
17 represent galaxies in this sub-population. They are not clearly distinguished from other
galaxies with the type-1 SFH in this age-SSFR domain, but have, on average, slightly younger
ages than the type-1 SFH galaxies with similar SSFR.
Of course, the SFHs of all galaxies are not clearly divided into typical examples shown
in figures 18, 19, or 25. For example, when only ACS and IRAC fluxes are used (i.e. if
smaller number of passbands are used; see § 5.2.2), more galaxies behave similarly to galaxies
with SFH type-3, making the bimodal distributions of SFRs and ages more prominent.
5.
Effects of Parameter Changes on the Results of the SED-fitting
In the following sections, we investigate how the bias in the SED fitting behaves as we
change some of the conditions in the SED-fitting procedure, such as the range of e-folding
time of star formation history, τ, combinations of broad passbands used, or the assumed
SFH.
– 22 –
5.1.
Two-Component Fitting
Here, we try to allow more complex star formation histories in the SED-fitting, by
using the two-component templates instead of single component ones. Some studies tried
this method to constrain the hidden mass in old stellar population or for better estima-
tion of total stellar mass. To construct the two-component stellar population templates,
various combinations of simple SFHs have been tried in the literature, including: (1) com-
bining a maximally old, instantaneous burst with a more moderately decreasing SFH (e.g.
Papovich et al. 2001; Shapley et al. 2005), or (2) adding a secondary young bursty SFH
component to an old, slowly decreasing SFH (e.g. Kauffmann et al. 2003; Drory et al. 2005;
Pozzetti et al. 2007). Also, different authors have used different ways of fitting two com-
ponents: (1) fitting with a young SF component first, then fitting the residual SED with
an old component (Shapley et al. 2005), or (2) constructing the combined templates with
various ratios between a young and an old simple SFH templates (Kauffmann et al. 2003;
Pozzetti et al. 2007).
5.1.1.
Slowly Decreasing Star Formation Histories with a Secondary Burst
Here, we constructed the two-component SFH templates by adding (maximally) old,
very slowly varying SFH templates (τ = 15 Gyr), and younger, more bursty templates (τ
= 0.2 Gyr). Old components are assumed to start forming at zf = 10. The star formation
activity of the secondary burst is constrained to be initiated at least 200 Myr later than
that of the old component, and at most 500 Myr earlier than the observed time, with the
percentage of young templates varying between 5 % to 95 % (with 5 % step size). This
construction is expected to reflect better the SFHs of some galaxies in the semi-analytic
models used in this study (for example, galaxies with type-2 SFHs shown in figure 18),
and using this type of composite templates ought to reduce the systematic bias for galaxies
with type-2 SFHs. The SFHs constructed in this way can mimic the SFHs of galaxies that
experienced the secondary star formation due to merger/interaction.
The trends in the SED-derived distributions of various parameters in this type of two-
component fitting are: (1) the stellar masses are underestimated more severely than in
the single-component fitting (middle row of figure 26), (2) the SFR/age distributions show
reduced offsets compared with the single-component fitting (middle rows in figures 27 and
28), and (3) the bimodalities that existed in the age distributions have disappeared, while the
bimodalities in the SFR distributions are enhanced for B- and V-dropouts (for U-dropouts,
the SFR distribution becomes much broader). These bimodalities in the SFR distributions
are, however, not driven by the subpopulation of galaxies with type-3 SFHs, as can be seen
– 23 –
below.
The first two trends are not unexpected since we are adding a young stellar population
with a low mass-to-light ratio to the SED templates. This young component makes the
best-fit ages younger, thereby reducing bias in the mean ages, while its lower mass-to-light
ratio makes the best-fit stellar masses smaller, thereby increasing bias. Younger ages, and
thus increased dust extinctions, lead to higher SFRs, and decrease bias in the SFRs.
Figure 29 – showing the correlation between the relative errors arising in the two-
component fitting and the relative errors in the single-component fitting – reveals behaviors
of galaxies with different SFH types in the two-component fitting performed in this section.
Galaxies whose ages are greatly overestimated in the single-component fitting (with type-
2 SFHs) show significantly reduced age-overestimation in the two-component fitting. This
improvement in the age estimation is expected. In the single-component fitting, the ages are
greatly overestimated for the SFH type-2 galaxies because the young component is ignored.
By adding a young component in the templates, ages are better fitted for this type of
galaxies improving the age estimation. For these galaxies, the stellar masses are generally
overestimated in the single-component fitting. Improving the age estimates also improves
the stellar-mass estimates
However, when the age is relatively well constrained in the single-component fitting
(i.e. for galaxies with type-1 SFHs), an added young component causes the underestimation
of ages, and therefore more severe underestimation of the stellar mass which is already
underestimated in the single-component fitting. The age underestimation of these galaxies
couples with the age-extinction degeneracy, leading to the overestimation of the SFRs shown
in (g), (h), and (i) in figure 29. This causes the bimodalities in the SFR distributions shown
in middle row of figure 27.
Interestingly, for a small number of galaxies, the ages and the stellar masses derived in
the two-component fitting are higher than the ones derived in the single-component fitting.
These galaxies are the ones with SFH type-3 and are manifested as a subpopulation in
figures 9 and 15. Through the single-component fitting, ages and stellar masses of these
galaxies are greatly underestimated, and the SFRs are severely overestimated due to the
large age underestimation. The higher values of the ages and stellar masses derived through
the two-component fitting reduce the errors in the age and stellar mass for these galaxies.
For these galaxies with the type-3 SFHs, the best-fit ts are small in the single-component
fitting. So, the actual effect of the two-component fitting performed in this section is to
add an old component for these galaxies, while for the other galaxies (with the type-1 or
type-2 SFHs), the the effect of the two-component fitting is to add a young component. This
makes the best-fit ages from the two-component fitting older than the ones derived from the
– 24 –
single-component fitting, the best-fit masses higher (due to higher mass-to-light ratios), and
the best-fit SFRs lower for galaxies with type-3 SFHs. Older best-fit ages of the type-3
SFH galaxies derived in the two-component fitting removes the bimodalities shown in the
SED-derived age distributions in the single-component SED-fitting.
The age distributions derived in the two-component fitting are narrower than the in-
trinsic distributions (middle row of figure 28). The ages tend to be underestimated for older
galaxies (possibly with SFHs of type-1 or type-2), while ages are more likely overestimated
for younger galaxies (probably SFH type-3 galaxies). The SFR distributions derived in the
two-component fitting are more extended toward the high SFRs than in the single-component
fitting for U-dropouts (bringing the SFR distributions closer to the intrinsic distributions).
For B- and V-dropouts, the bimodality in the SFR distribution is enhanced, i.e. the SFRs
are overestimated for more galaxies.
In summary, the two-component fitting performed in this section reduces bias in the SFR
and age distributions, but increases the offsets in the stellar mass distributions. However,
the detailed investigation reveals that the changes of behavior in the two-component fitting
compared with the case of the single-component fitting are different for galaxies with different
types of SFH. Errors in the estimation of ages and stellar masses are reduced for galaxies
with SFHs type-2 or type-3, while the stellar mass errors increase for galaxies with type-1
SFHs.
5.1.2.
Maximally Old Burst Combined with Slowly Varying Younger Components
In the previous section, we experimented with two-component fitting by adding a young,
burst-like (τ = 0.2 Gyr) component to a more continuously varying (τ = 15 Gyr) old
component. Such two-component templates are expected to match better the type-2 SFHs,
and turned out to give better age estimates for the galaxies with type-2 as well as type-3
SFHs.
In this section, we perform another type of two-component fitting in an attempt to
give a better constraint on the hidden old stellar mass. To achieve this we construct the
two-component SFH templates in a similar way done as in § 5.1.1, but exchange the roles
of a τ = 0.2 Gyr component and a τ = 15 Gyr component. We add an old, τ = 0.2 Gyr
component formed at zf = 10 and younger τ = 15 Gyr components with various ages. The
star formation activity of the young components is constrained to start at least 200 Myr
later than an old burst, to make the two-component templates clearly distinguished from
the single-component template. By adding an old, bursty component to a more continuously
– 25 –
varying SFH component, we can expect that this method will give us higher mass than the
single-component fitting.
The bottom row of figure 26 shows that the stellar mass distributions are moved toward
higher values than the ones derived in § 5.1.1 (middle row of figure 26) and also than the ones
from the single-component fitting. Compared with the intrinsic distributions, the derived
stellar mass distributions from the two-component fitting with an additional old, burst-like
component are slightly crowded at high stellar mass for U- and B-dropouts. The bimodalities
both in the SFRs and ages have disappeared (bottom rows of figures 27 and 28).
The mean values of stellar masses are higher by ∼19 %, 54 %, and 1 % than the values
from the single-component fitting for U-, B-, and V-dropouts, respectively. For individual
galaxies, the stellar mass from the two-component fitting with an old, burst-like component
can be as large as several times of the stellar mass from the single-component fitting. For
a few galaxies (mostly with type-3 SFHs), the stellar mass from the two-component fitting
with an old burst can reach ∼4 −9 times of the stellar masses from the single-component
fitting. However, the relatively small increase in the mean values of the best-fit stellar mass
(especially for V-dropouts) indicates that the young component dominates the SEDs even
in the two-component fitting performed in this section.
The dominance of the young component can be seen by the fact that the mean values of
the best-fit ages derived in the two-component fitting are much younger (by ∼39 %, 35 %,
and 42 % for U-, B-, and V-dropouts) than the mean values derived in the single-component
fitting. (The star-formation time scale of the young component in the two-component fitting
performed in this section is fixed as τ = 15 Gyr, and this leads to younger best-fit ages – see
§ 5.3.2, below.)
Even though the two-component fitting performed in this section gives higher values
of the mean stellar mass, it is not always true that stellar masses derived from the single-
component fitting and this kind of two-component fitting bracket the true, intrinsic stellar
mass. For some galaxies with small stellar mass (with log (M∗/M⊙) ≲9.5), even the single-
component fitting overestimates the stellar mass. On the other hand, the stellar mass derived
through the two-component fitting with an old, burst-like component often remain smaller
than the intrinsic value for some massive galaxies.
If we were to limit the fractional contribution of the young component to smaller values
than allowed here – as done, for example, in Kauffmann et al. (2003) – the stellar masses
derived in the two-component fitting would become higher than the ones derived in this
section. Also, we can derive higher stellar masses from the two-component fitting: (1) by
fitting the old component first, then fitting the younger component to the residual fluxes (i.e.
– 26 –
forcing the contribution from the old component to increase), or (2) by setting the formation
redshift (zf) of the old component higher (i.e. increasing the mass-to-light ratio of the old
stellar component). However, even with these more extreme settings, it is still possible that
the derived stellar masses for some very massive galaxies will be smaller than the intrinsic
ones.
In summary, it is not universally true that the single-component fitting and the two-
component fitting (with an old, burst component added on more continuous SFH compo-
nents) bracket the true stellar mass.
5.2.
Effects of Wavelength Coverage
The main results presented in § 4 are based on the analysis using broadband photometric
information from observed-frame optical through MIR range – i.e. ACS B435- to z850-bands,
ISAAC J- to Ks-bands, & IRAC 3.6 µm through 8.0 µm. However, not all the observed
LBGs have photometric data with this wavelength coverage. Before the Spitzer era, the
majority of the observed photometric data only covers up to the observed-frame NIR range.
Thus, it is interesting and important to examine how the results vary as we use different
combinations of passbands in the SED-fitting.
5.2.1.
SED-fitting without IRAC Data
Several authors investigated the effects of inclusion of IRAC photometry (of wavelength
coverage of ∼3-10 µm) in constraining the properties of high-redshift galaxies (Labb´e et al.
2005; Shapley et al. 2005; Wuyts et al. 2007; Elsner et al. 2008). Shapley et al. (2005) anal-
ysed z ∼2 star-forming galaxies with and without IRAC photometric data. They reported
that the SED-derived stellar mass distribution shows little change with the inclusion of IRAC
data, while including IRAC data can reduce errors in the stellar mass estimation for indi-
vidual galaxy. Investigating 13 z ∼2 −3, red (Js −Ks > 2.3) galaxies, Labb´e et al. (2005)
showed that the best-fit ages are younger without IRAC data for dusty star forming galaxies,
while there is little change for old, dead galaxies. Wuyts et al. (2007) studied 2 < z < 3.5,
K-selected galaxies, and showed that inclusion of IRAC data does not change the overall
distributions of stellar masses and ages. Analyzing 0 < z < 5 observed galaxies, Elsner et al.
(2008) showed that the mean stellar masses increase when derived omitting Spitzer data.
The discrepancy is maximum at z ∼3.5 with log(MU−4/MU−K) ∼−0.5, and decreases with
– 27 –
redshift at z > 3.5. (At z ≤3.0 or z ≥4.0, log(MU−4/MU−K) ≤−0.3.)4 In addition, they
reported that despite this overall trend, the stellar masses and mean ages decrease without
IRAC data for some very young (faint) galaxies. Thus, there is no clear consensus in the
literature on the benefits of including IRAC.
The effects of omitting IRAC data are shown in the third rows of figures 5, 6, and 7).
Without IRAC data, the stellar mass estimates shift to lower values for U- and B-dropouts,
but are virtually unchanged for V-dropouts. Conversely, the SFR estimates are virtually
unchanged for U- and B-dropouts, but shift to lower values (increasing the discrepancy with
the SAMs) for V-dropouts. The age distributions shift to younger ages for the U- and B-
dropouts (which actually brings them into better agreement with the intrinsic distributions)
while for V-dropouts, the age distribution is only slightly changed.
Evidently, removing the IRAC photometry for the U- and B-dropouts increases the
dominance of the younger stellar populations due to the shorter wavelengths. This drives
the best-fit ages to lower values, resulting in lower stellar masses for a given amount of
stellar light. The stellar masses are reduced by ∼57 % and ∼48 % for U- and B-dropouts,
respectively, relative to the results from fits, in which the IRAC data are included.
The age and stellar mass decreases are largest for the galaxies with the high SSFRs
and young ages, i.e. galaxies with type-2 SFHs (similar with the ones shown in figure 18).
These galaxies have roughly two components of stellar populations – an ‘old’, slowly varying
component and a ‘young’, burst-like component. In the SED-fitting with IRAC photometry
included, the ‘old’ component dominates the SED, resulting in much older best-fit ages than
the intrinsic ages. However, without IRAC photometry, the SEDs cover only up to the rest-
frame ∼4000 −5000 ˚A. Due to the resulting shortage of information at long wavelengths,
the ‘young’ component comes to dominate the SED. This moves the best-fit ages younger,
even younger than the intrinsic ages in extreme cases.
The changes of the best-fit age and stellar mass for V-dropouts are much smaller (∼13
% and ∼4.8 % of decreases, respectively). This is presumably due to the generally younger
ages of V-dropouts. As can be seen in figures 18 and 19, V-dropouts, on average, started
forming stars more recently than U- and B-dropouts. Therefore, the proportion of the old
stellar population hidden without IRAC data is much smaller for V-dropouts than U- and
B-dropouts.
Interestingly, the bimodalities in the SFR- and age-distributions shown (for B-dropouts)
in figures 3 and 4 disappear when we exclude IRAC photometry. For the type-3 SFHs shown
4MU−4 and MU−K are stellar masses that are derived with and without IRAC data, respectively.
– 28 –
in figure 25, the SFRs have lower values and the ages (and stellar masses) have higher values
than the ones derived including IRAC bands.
The behaviors of these galaxies are thus
opposite to the majority of B-dropouts.
What makes this sub-population of galaxies (with type-3 SFHs) behave differently from
other galaxies? The difference of the SFHs between the ones shown in figure 25 (i.e. type-3
SFHs) and the ones shown in figure 19 (i.e. type-1 SFHs) becomes significant when the
lookback time is larger than ∼400 −500 Myr. With only ACS and ISAAC photometry,
which covers only up to the rest-frame ∼4000 −5000 ˚A, the SFHs at early time are hard to
constrain. Therefore, the SED-fitting without IRAC data cannot discriminate between SFH
type-1 (figure 19) and type-3 (figure 25), resulting in the disappearance of the bimodalities
in the SFR and age distributions.
Figure 30 shows the ratios of the best-fit stellar masses, SFRs, and ages with and without
IRAC photometry for U-, B-, and V-dropouts. Without IRAC photometry, the stellar masses
are underestimated (compared with when IRAC data are included) for most U-dropouts
(figure 30-(a)), and for the majority of B-dropouts (figure 30-(b)). For the subpopulation
of galaxies in B-dropouts, the best-fit stellar masses without IRAC data are larger than the
ones with IRAC data. For V-dropouts (figure 30-(c)), the stellar masses are underestimated
for some galaxies, and overestimated for other galaxies, making the distributions with and
without IRAC data similar.
In summary, the effect of removing the IRAC data depends on the redshift and/or
SFHs. This redshift and SFH dependence can explain the apparent disagreement between the
different previous investigations, since the samples included galaxies with different redshift
ranges and also different types of galaxies with possibly different SFHs.
5.2.2.
SED-fitting without ISAAC data
Next, we perform the SED-fitting with only ACS and IRAC photometry, excluding
the J, H, and Ks-band ISAAC photometry. The effect of omitting ISAAC photometry is
insignificant for U-dropouts and for the majority of B- and V-dropouts. As can be seen by
comparing the second and the fourth rows from the top in figure 5, the distributions of best-
fit stellar masses, SFRs, and ages show little change without ISAAC data for U-dropouts.
The mean stellar mass and mean age increase by 1.9% and 2.4% compared with the values
derived using ACS + ISAAC + IRAC photometry.
For B- and V-dropouts, the bimodalities in the SFR/age distributions become more
prominent as can be seen in the fourth rows of figures 6 and 7. For some galaxies whose age
– 29 –
offsets are small when the SED-fitting is done with full photometry, the best-fit ages become
much younger if we use only ACS and IRAC photometry. This age underestimation leads
to the stellar mass underestimation and SFR overestimation as explained in § 4.3, and the
extinction overestimation due to the well-known age-extinction degeneracy enhances the SFR
overestimation further. These galaxies skew the mean values of stellar mass and mean age
of total sample lower by about 10 % for B- and V-dropouts, even though the majority of
galaxies show little change.
5.3.
Effects of τ-range Used in SED-fitting
The changes of the allowed ranges of parameters, such as τ, t, or metallicity, in the
SED-fitting would affect the derived values of physical parameters, as well. Here, we focus
on the effects of the different range of τs used in the SED-fitting on the estimation of physical
parameters of LBGs. This investigation is beneficial for the comparison with previous works
done using the SED-fitting methods with various τ ranges as well as for better understanding
biases of the SED-fitting methods.
5.3.1.
SED-fitting with τ ≤1.0 Gyr Templates
First, we limit the τ range to ≤1.0 Gyr during the SED-fitting. Through this experi-
ment, we can look into the biases arising due to the usage of ‘not-long-enough’ τ values in
the SED-fitting.
As expected, the enforced smaller τ values cause smaller best-fit t, to match t/τ val-
ues, compared with the case when the full range of τ is allowed from 0.2 Gyr to 15 Gyr.
This bias systematically makes the best-fit ages to be younger. As explained in § 4.3, this
age-underestimation leads to the mass-underestimation, increasing differences between the
intrinsic- and SED-derived stellar masses. The effects of the limited τ values on the SFR
estimation is complicated due to the age-extinction degeneracy. The offsets due to the lim-
ited τ values as ≤1.0 Gyr are greatest for B-dropouts because of the severely enhanced
bimodalities. Relative changes of the mean stellar masses and ages when we limit τ range
to ≤1.0 Gyr, compared with the mean values derived with the full range of τs, are shown
in table. 4.
As can be seen in figure 31, the lowered best-fit values of ages/stellar masses are largely
driven by galaxies with preferentially young intrinsic ages (top row) and/or high intrinsic
SSFRs (middle row). These are mostly the galaxies with SFH type-2, whose ages are severely
– 30 –
overestimated when we fit the SEDs with the full range of τs (from 0.2 Gyr to 15 Gyr)
(bottom row of figure 31).
The type-2 star formation histories have two main components. One component is a
relatively low level of long-lasting SF activity, which corresponds to large τ values. The
other is a relatively young, strong SF activity, which is more likely represented by small
τ. When we fit SEDs with full range of τs, best-fit models tend to be determined by the
underlying, long-last SFH component giving the severely overestimated ages to these galaxies
(see § 4.3.1). In the case when only small values of τ (≤1.0 Gyr) are allowed in the SED-
fitting procedure, the best-fit models are more likely determined by the young SF component
with relatively small star formation time scale. The best-fit t values then are much smaller
than the ones derived utilizing the full range of τs, resulting in much younger best-fit mean
ages and much smaller best-fit stellar masses.
Another significant feature is the more prominent bimodalities in the distributions of
best-fit SFRs and ages. Galaxies in the smaller sub-population have very young best-fit
ages and very high best-fit SFRs compared with the remaining galaxies. Figure 31 shows
the increase in the number of galaxies which belong to this sub-population as an effect of
limitation on the allowed τ values. They are the galaxies with small intrinsic SSFRs (middle
row) as galaxies with type-1 or type-3 SFHs. The discrepancies between the intrinsic- and
best-fit ages are very small when the full range of τs is used in the SED-fitting (bottom row),
which means they behave like the galaxies with type-1 SFH. However, when we restrict τ
as ≤1.0 Gyr, their best-fit ages become much younger and join the sub-population. The
bottom row of figure 31 also shows that galaxies which are in this subpopulation when τ has
the full range remain in the subpopulation when τ is restricted to be ≤1.0 Gyr.
In summary, if we limit τ to be ≤1.0 Gyr in the SED-fitting, the overall trends are: (1)
the best-fit ages (and hence the best-fit stellar masses) become smaller for the galaxies with
type-2 SFHs. (2) a larger number of galaxies joins the subpopulation that has much smaller
best-fit ages and stellar masses than the intrinsic SAMs values. These trends result in (1)
an overall downward shift of the age/stellar mass distributions (increasing the discrepancies
between the intrinsic- and SED-derived stellar mass distributions) and (2) more prominent
bimodalities in the SFR/age distributions.
5.3.2.
SED-fitting with τ=15 Gyr Templates
Next, we hold τ fixed at 15 Gyr, which is equivalent to assigning a constant star forma-
tion rate, considering the age of the universe at redshifts ∼3 −5. Figure 32 reveals that the
– 31 –
best-fit values of tτ15 generally increase for galaxies with small best-fit tall derived with full
range of τs, but decrease if the best-fit talls are large. Here, tτ15 refers the best-fit t derived
if we fix τ at 15 Gyr, and tall is the best-fit t obtained when we allow the full range of τ.
This behavior is caused by the restriction on t to be younger than the age of the universe
at each redshift If the best-fit t is already large for the full range of τs, there is no room
to increase t to match the red color of these old galaxies. Instead, the fitted value of the
extinction increases and the fitted age generally decreases.
Also, with the larger value of τ, larger t does not always results in older mean ages while
smaller t always makes mean age younger, since mean age is a function of τ as well as t.
Therefore, the mean ages (and the stellar masses also as a result) are slightly underestimated
overall. Galaxies whose best-fit ts are very small (≤0.2 Gyr) show little differences in best-fit
t with or without the τ = 15 Gyr restriction.
Table. 4 shows the relative changes in the mean values of stellar masses and ages when
we set τ = 15 Gyr compared with the case when we allow the full range of τ.
5.4.
SED Models with Extreme SFHs from BC03 Model
We further test what would be the results of the SED-fitting for the galaxies with the
extreme star formation histories (SFHs). To do this, here, we construct three types of toy
models from the BC03 model, replacing SAM galaxies. The parameter settings used in these
toy models are summarized in table. 5. The aims of each toy model are to examine the biases
which arise: (1) when we use shorter τs than real in the SED fitting (toy model 1), (2) when
we use longer τs than real (toy model 2), and (3) when we try the single-component fitting
for the galaxies with clearly distinct, two generations of star formation.
5.4.1.
Effects of SED-fitting Using Too Small τs
First, in the case (1), the toy model SEDs have a very long SF time scale (τ = 15.0
Gyr). By restricting τ for the SED fitting not to exceed 1.0 Gyr, we can re-examine more
transparently (because we compare the same BC03 models) what would happen if we fit the
SEDs with τ values much shorter than the actual SF time scales of (model or real) galaxies.
As expected, the best-fit mean ages are underestimated.
To match colors (or SED
shapes) of ‘τ = 15 Gyr’ samples with much shorter τs, the SED templates with smaller ts
are found as the best-fit templates, which causes the mean ages are systematically underes-
timated. The amount of the age underestimation increases with age. The relative age error
– 32 –
reaches up to ∼30 % underestimation for the oldest galaxies (with t = 1.0 Gyr).
The systematic underestimation of ages leads to systematic mass underestimation, since
the given amount of light from galaxy is attributed to younger (more massive) stars with
lower mass-to-light ratios. The relative mass underestimation increases as the relative age
underestimation increases. For the oldest galaxies whose relative age underestimation reaches
up to ∼30 %, relative mass underestimations are ∼9 - 19 %. The ‘Younger-than-input’
best-fit ages result in higher average SFRs, whose relative overestimation spans from ∼0
through 9 %.
For the samples with t as small as 0.01 Gyr, all the physical parameters, such as stellar
masses, SFRs, and mean ages, are well recovered through the SED-fitting. This reflects the
fact that the effects of SFH difference are insignificant for the very young galaxies.
In summary, if one tries to fit galaxies’ SEDs with relatively short τs, the resultant
best-fit stellar masses and mean ages can be underestimations of the true values especially
for galaxies which have very extended SFHs for relatively long timescales. These trends
confirm the speculation of § 5.3.1. The SFRs are overestimated, but the relative errors are
not as large as those of stellar masses/mean ages.
5.4.2.
Effects of SED-fitting Using Too Large τs
In the case (2), the input toy models have shorter τ than the values allowed in the SED-
fitting procedure. This highlights the biases that can arise in the SED-fitting if galaxies
have much shorter SF time-scales (probably burst-like) than the τ values allowed in the SED
fitting.
The direction of bias in the best-fit mean ages is divided into two regimes depending on
the actual age (or t) of each galaxy. For the model galaxies with small enough t (i.e. t =
0.1, 0.2 Gyr) compared with the age of the universe at corresponding redshift (which is z ∼
4, in this case), the best-fit ts are larger than the input ts to match the SEDs with longer τs
than the input (τ = 0.2 Gyr), leading to the overestimation of the mean ages by amount of
∼30 % for the galaxies with the input t = 0.1 Gyr, and ∼50 % for the ones with the input
t = 0.2 Gyr.
For the similar reason as in the case (1) (but, in the opposite direction), the age over-
estimation results in the mass overestimation by attributing light to older stars with higher
mass-to-light ratios. The mass overestimation is about ∼11-13 %. The ‘older-than-input’
best-fit ages make the SFRs underestimated by ∼22-34 %.
– 33 –
However, for the model galaxies with relatively large values of t, which are not much
shorter than the age of the universe, there is not so much room for t to increase. So, for the
toy model galaxies with input t = 1.0 and 1.3 Gyr, the best-fit ts are only slightly larger
than the input ts, and the mean ages are younger than the input due to larger τs than
input. Here, the stellar masses are overestimated even for the underestimated mean ages,
mainly because of the more extended SFHs. The important difference between the models
with large input ts and small input ts is that the derived dust extinctions, parameterized as
E(B −V ), are greatly overestimated to compensate the ‘younger-than-input’ mean ages for
the SED models with large input ts.
During the SED-fitting procedure, the best-fit ts generally move in the direction which
‘correct’ the difference in τs – ‘smaller-than-input’ ts for the long input τ models, and ‘larger-
than-input’ ts for the short input τ models. However, for the toy model galaxies with small
τs and large ts, the SED-fitting cannot overcome the τ difference by adjusting the best-fit
ts due to the restriction that t be smaller than the age of the universe. Instead, the best-fit
E(B −V ) starts to be overestimated by the amount of ∆E(B −V ) ∼0.25-0.3 for the input-t
= 1.0 Gyr models and ∼0.45 for the input-t = 1.3 Gyr models. This large bias, in turn,
results in a large bias toward overestimated SFRs. The best-fit SFRs are about ∼13-17
times of the input SFRs for input-t = 1.0 Gyr models, and reach up to ∼70-110 times of the
input SFRs for the input-t = 1.3 Gyr models. The extinction overestimation also leads to
the larger mass overestimation, but the effects are not as dramatic as in the SFR estimation.
This example illustrates how the well-known ‘age-extinction’ degeneracy affects the results
of the SED-fittings, especially for the galaxies with extreme SFHs and/or for the case when
parameter (for example, τ) space allowed during the SED-fitting is not sufficiently large.
In summary, if one tries to fit galaxies’ SEDs with very long τs only, the resultant stellar
masses are generally overestimated. If galaxies have sufficiently young ages (compared with
the age of the universe at the redshifts where galaxies reside) the mean ages are overesti-
mated, as the best-fit ts to be much larger than the input ts to compensate the τ difference.
However, for old galaxies whose input ts are compatible to the age of the universe at cor-
responding redshift, the direction of bias in the age estimation is that the mean ages are
underestimated (even though the best-fit ts are slightly larger than the input ts). Instead,
the dust extinctions are greatly overestimated, which makes the best-fit SFRs erroneously
high (up to two orders of magnitude).
– 34 –
5.4.3.
Effects of Single Component SED-fitting for Galaxies with Two Generations of Star
Formation
Lastly, in the case (3), the model galaxies have two clearly distinguished generations
of star formation with t = 0.1 & 1.0 Gyr. Both of the components are set to have τ = 0.2
Gyr. When we try to derive the best-fit physical parameters of these two component model
galaxies via the single-component SED-fitting, both the mean ages and stellar masses are
underestimated, as the result of the older stellar population being at least partially ignored.
The age underestimation is minimal (∼10 %) for the toy model galaxies in which the
young-to-old ratio is smallest (i.e. young component fraction ∼0.1). The age underesti-
mation increases as the proportion of young component increases (up to ∼77-88 % for the
ones with young/old ∼1.0). This implies the best-fit parameters tend to be determined by
the stellar component whose fractional occupation is large. However, as the young compo-
nent fraction increases further to young/old = 2.6, the age underestimation decreases and
becomes similar to that for the galaxies with young/old = 0.52, because the mean ages of
these galaxies (with young/old = 2.6) are already small enough to be greatly underestimated.
As stated in previous sections, the age underestimation propagates to the mass under-
estimation. The mass underestimation is minimal (∼9 −11 %) for the model galaxies with
young/old = 0.1 due to the lowest age underestimation, and for the ones with young/old =
2.6 because they have the smallest portion of their mass in the old component. This means
the mass underestimation depends on two factors – the degree of age underestimation (i.e.
the amount of the mis-interpretation of the mass-to-light ratio) and the fraction of stellar
mass in the old component. The mass underestimation for galaxies with young/old = 0.26,
0.52, and 1.0, is in the range 21-40 %.
The SFRs are underestimated by ∼14-43 %, and the discrepancy is larger for the galax-
ies with larger fractional mass in the old component. This is because the best-fit parameters
are affected by the old component while the SFR contribution of the old component is very
small (∼0.2-4 % of the total SFR, depending on the mass fraction of the old component).
The best-fit τs are larger than the input (i.e. 0.2 Gyr), and the best-fit ts are larger than
the input young component (i.e. 0.1 Gyr) due to the effects of the old component, and this
leads to lower SFRs.
In summary, if one tries to fit galaxy’s SED with single-component SED-fitting when the
real SFH has two episodes, the mean ages and the stellar masses are generally underestimated
indicating the old components are to some extent masked. And, the SFRs are underestimated
since the old components with very little current SF activity pollute the SEDs.
– 35 –
6.
SFR Estimation from Rest Frame Ultra-violet Luminosity
For galaxies with a roughly constant star-formation rate, the extinction-corrected UV
luminosity is expected to provide a reasonably good estimate of the star-formation rate.
This is fortunate, because often the only data available for high-redshift galaxies are a few
photometric data points in the rest-frame UV.
The relation between SFR and UV luminosity can be calibrated using spectral synthesis
models, such as BC03.
Kennicutt (1998) notes that the calibrations differ over a range
of ∼0.3 dex, when converted to a common reference wavelength and IMF, with most of
the difference reflecting the use of different stellar libraries or different assumptions about
the star-formation timescale.
The calibrations usually assume constant or exponentially
declining star-formation rates.
It is interesting to see how well this technique works for the more varied star-formation
histories of the semi-analytic models. In this case, we are using BC03 and the same IMF for
both the calibration and the SAM galaxies, so the discrepancies in derived SFRs must be
primarily due to the different star-formation histories.
Here, we derive SFRs of SAM B-dropout galaxies with the assumption that we do not
know their redshifts, which is similar to the case when there is no spectroscopic redshift
information for a color-selected LBG sample.
The (dust-uncorrected) rest-frame UV (λ0 = 1500 ˚A) luminosity of each SAM B-dropout
galaxy is calculated from the i775 band flux as
Lν,1500 = 4πd2
L
(1 + z) × fν,i775,
(12)
assuming all B-dropouts are at z = 4.0.
Here, fν,i775 and Lν,1500 are specific flux at i775 band and specific luminosity at rest-frame
1500 ˚A, respectively, and dL is luminosity distance at redshift z, which is assumed to be 4.0.
There are two major possible sources of systematic biases, which can arise due to the
assumption that all galaxies are at z = 4.0: (1) ignoring the bolometric correction – which
is a decreasing function of redshift and is small – a factor of 1.0 ±0.2 in the redshift range of
B-dropouts (3.6 ≤z ≤4.6) for Chabrier (2003) IMF and solar metallicity, and (2) ignoring
error in luminosity distance (dL).
These two sources of bias act in opposite direction. Not including bolometric correction
causes the UV luminosity to be slightly overestimated at high redshift, since it is a decreasing
– 36 –
function of redshift. Underestimation of the luminosity distances for galaxies at high redshift
(z > 4.0) leads to underestimated UV luminosities. The correction factor due to the error
in dL estimation is slightly larger at high-redshift than the bolometric correction factor,
and ranges from 0.8 to 1.4 in the redshift range of B-dropouts. Together, this will cause
rest-frame UV luminosity to be slightly underestimated at the high-redshift end of the range.
For all B-dropout galaxies, dust-extinction is assumed to be E(B −V ) = 0.15, which is
the same value used for dust-correction in high-redshift LBG studies, such as Giavalisco et al.
(2004b) and Sawicki & Thompson (2006).
Star-formation rates are then calculated using the conversion of Kennicutt (1998) di-
vided by 2.0 to correct for the different assumed IMF, because Kennicutt (1998) uses a
Salpeter (1955) IMF with a mass range 0.1 to 100 M⊙.
Figure 33 shows the ratio of dust-corrected, UV-derived SFR to intrinsic SFR as a
function of redshift. In this figure, we can see the redshift dependent behavior of the star-
formation rate calibration from the rest-frame UV. This behavior is a combined effect of
luminosity-distance underestimation, which is small, though, as explained above, and the
redshift-dependent difference in average dust-extinction. The large scatter of SFR ratios at
a given redshift reflects the variation of galaxies’ intrinsic dust-extinction and SFH.
Differences in dust extinction can have a large effect. For example, small change in
assumed value of mean E(B −V ) will significantly change the derived SFR values – for a
range of E(B −V ) from 0.10 to 0.20, the dust-correction factor vary from 2.6 to 6.5.
With the assumed dust-extinction of E(B −V ) = 0.15, mean UV-derived SFR is 11.099
M⊙yr−1.
For comparison, mean values of intrinsic and SED-derived SFRs are 15.650
M⊙yr−1 and 6.638 M⊙yr−1, respectively5.
Figure 34 shows the distributions of intrinsic SFRs (left), and of SFRs derived by
assuming all galaxies are at z = 4.0 and assuming dust-extinction of E(B −V ) = 0.15
(right).
The distribution of SFRs derived from rest-frame UV luminosity, assuming all
galaxies are at z = 4.0, is much narrower than the intrinsic one.
5Since calibration between UV luminosity and SFR is derived assuming constant SFR over 100 Myr
(Kennicutt 1998), comparison between UV-derived SFR and intrinsic or SED-derived SFR, which is averaged
over last 100 Myr, is relevant.
– 37 –
7.
Discussion
We have shown in this paper that there are significant biases in the physical parameters
derived from the SED-fitting of standard τ-model to broadband photometry. Biases are
severe especially for the SFR and mean age, but even the SED-derived stellar masses are
biased. We now address in some detail how the biases in the derivation of these physical
parameters can affect the investigation of high-z galaxies and the inferred galaxy evolution
studies.
7.1.
Artificial Age bimodality
Figures 3 and 4 show that there are bimodalities in the SFRs and mean ages derived
through the SED-fitting (most clearly shown for B-dropout LBGs) while there is no such
bimodality in the intrinsic distributions. These bimodalities are enhanced when we try to fit
the SEDs omitting some available input information, such as NIR photometry from ISAAC or
the spectroscopic redshift (figures 5, 6, and 7). The main origin of these artificial bimodalities
is the mismatch of the SFHs between the SAM model galaxies and the templates from the
BC03 stellar population model as explained in § 4.4. Such bimodalities can lead to the false
interpretation that there are clearly distinguished populations among the similarly selected
star-forming galaxies.
Recently, Finkelstein et al. (2008) analysed 14 Lyman-α emitting galaxies (LAEs) and
found that there is a clear bimodality in their age distribution, which are derived through
SED-fitting, such that their ages either very young (< 15 Myr) or old (> 450 Myr). Based
on this bimodality, they concluded that there are two distinct populations of LAEs – dusty
starbursts and evolved galaxies. However, according to the results presented in our work, it
is possible that this age bimodality reported in Finkelstein et al. (2008) may not be real but
an artifact which arises in the SED-fitting procedure due to the difference of their SFHs. Of
course, caution should be applied in interpreting the SED-fitting results of the LAEs based
on our analysis performed for the LBGs. It is still controversial how similar (or how different)
the LBGs and LAEs are, despite some indications of similarities in their physical parameters
(Lai et al. 2007) and the possible overlap between LBGs and LAEs (Rhoads et al. 2008).
Shapley et al. (2005) also reported a subset of galaxies in their sample of z ∼2, star-
forming galaxies with extremely young ages (with t ≤10 Myr). The galaxies in this sub-
population are relatively less massive (<log M∗/M⊙> = 9.64 (calculated from column 7
in table 3 of Shapley et al. (2005)), while the mean value of total sample is 10.32. More
dramatic difference between galaxies in this subset and other remaining galaxies is shown
– 38 –
in the derived SFR distribution. Among 72 galaxies, there are 10 galaxies whose SFRs are
larger than 200 M⊙yr−1, and among these 10, nine galaxies belong to this subset of galaxies
with t ≤10 Myr according column 8 in their table 3 (one remaining galaxy with high SFR
has the best-fit t = 15 Myr). In contrast, almost half of their sample has very low SFRs
(≤10M⊙yr−1). Based on our analysis, it is plausible that the 10 galaxies with the very
high SFRs have similar SFHs as shown in figure 25 (i.e. type-3 SFHs), and therefore their
ages and stellar masses are underestimated while their SFRs are greatly overestimated. If
so, the ages and SFRs of their total sample would present more continuous distributions.
Bimodalities in the inferred ages are also seen among z ∼5 LBGs of Verma et al. (2007)
and among 14, z ∼5 LBGs with spectroscopy of Stark et al. (2007).
More interestingly, in the Shapley et al. (2005) sample, there are three galaxies whose
physical parameters derived through the SED-fitting methods do not agree with the indi-
cations from their rest-frame UV spectra. The SED-fitting results indicate that these are
old (t/τ >> 1) galaxies with very low level of SFRs (∼a few M⊙yr−1), while there are
indications of a young population of stars with active star formation in there spectra (see
§ 4.5 in Shapley et al. (2005) for detail). This apparent disagreement can be explained if
these galaxies have the star formation histories similar with the ones shown in figure 18 (i.e.
type-2 SFHs).
7.2.
Possible Descendants of High Redshift Star Forming Galaxies
There is great interest in connecting LBGs to their possible descendants – i.e. deter-
mining whether or not they are progenitors of local massive ellipticals (e.g. Lowenthal et al.
1997; Adelberger et al. 1998; Sawicki & Yee 1998; Steidel et al. 1998; Somerville et al. 2001).
Many properties of LBGs – including their sizes, morphologies, number densities, clustering
properties, and physical properties – are closely linked to this issue. Therefore, more accu-
rate estimation of LBGs’ physical parameters are clearly important for addressing this issue
properly.
According to our analysis, the stellar masses derived using the single-component SED-
fitting methods tend to underestimate the true values, on average, by 19-25 %. Moreover,
bias in the stellar mass estimation seems to strongly depend on the stellar mass itself, in
a sense that the stellar masses are more severely underestimated for more massive galaxies
(see figures 14, 15, and 16). For some of very massive galaxies (M∗> 1010M⊙), the best-fit
stellar masses can be less than half of the intrinsic stellar masses.
Accompanied by the
SFR underestimation, which is generally more severe than the stellar mass underestimation,
– 39 –
this discrepancy can significantly affect the discrimination among possible evolutionary de-
scendants of massive LBGs. Since the current (or recent) SFR is a measure of the possible
additional stellar mass which can be added to LBGs during their evolution to the lower
redshift, the underestimation of the stellar mass and SFR of massive LBGs can plausibly
lower their possible stellar masses at lower redshift or at z ∼0 significantly.
Interestingly, studies based on the clustering properties of LBGs speculated that mas-
sive, high-z LBGs could be the progenitors of local massive ellipticals (e.g. Adelberger et al.
1998; Steidel et al. 1998), while Sawicki & Yee (1998), based on the SED-fitting analysis,
suggested that LBGs would not become sufficiently massive at low-z to be massive ellipti-
cals unless experience significant number of mergers.
Biases and uncertainties in the estimation of LBGs’ ages – which are shown not only to
be large and but also to vary significantly as the fitting parameters change in our study –
can also affect understanding of LBG properties and the predictions regarding their possible
evolutionary paths. For example, biases in the age estimation would propagate into errors in
their estimation of the star-formation duty cycle, which would, in turn, affect the estimates
of the number of galaxies which have similar stellar masses/ages with detected LBGs, but
are undetected due to their reduced SFRs.
8.
Summary & Conclusions
In this paper, we examine how well the widely-used SED-fitting method can recover
the intrinsic distributions of physical parameters – stellar mass, SFR, and mean age – of
high-redshift, star-forming galaxies. To this end, we construct model high-redshift galaxies
from the semi-analytic models of galaxy formation, make a photometric catalog via the BC03
synthesis model, and select LBGs through the appropriate color selection criteria based on
their broadband colors. Then, we perform SED-fitting analysis, comparing the photometric
SEDs of these model galaxies with various galaxy spectral templates from the BC03 stellar
population model to derive the distributions of best-fit stellar masses, SFRs, and mean ages.
We use this test to explore (rather exhaustively) the errors and biases that arise in such SED
fitting and the underlying causes of these errors and biases.
Here are the summary of the main results of this work.
1. When we fix the redshift to the given value in the SAM catalog and use ACS/ISAAC/IRAC
passbands, the SED-fitting method reproduces relatively well the input distributions of stel-
lar masses with a minor tendency to underestimate the stellar masses and with substantial
scatter for individual galaxies. The mean stellar masses are underestimated by about 19∼25
– 40 –
%, which is due to the fact that the old generations of stars can be hidden by the current
generation of star formation. The distributions of SFRs and mean ages show larger offsets
than the stellar mass distributions. The SFRs are systematically underestimated and the
mean ages are systematically overestimated, and these trends mainly reflect the difference in
the SFHs predicted by the semi-analytic models and assumed in the simple galaxy templates
used in the SED fitting. The well-known ‘age-extinction degeneracy’ plays an important role
in biasing the derived SFRs.
2. When we use redshift as an additional free parameter, the discrepancy between the
intrinsic- and SED-derived stellar mass distributions increase (i.e., the overall stellar mass
underestimation becomes worse), while the bimodalities which appear in the SFR & mean
age distributions become more significant. The distributions of offsets of individual galaxies
indicate that there exist sub-population(s) of LBGs whose behaviors are distinct from the
majority of LBGs in the SED-fitting. The SED-fitting generally underestimates the redshift
slightly.
3. The age overestimates are clearly related to the intrinsic age and specific star for-
mation rate of each galaxy.
The overestimation of mean ages is worse for galaxies with
younger ages and higher SSFRs. Inspection of the SFHs of individual SAM galaxy confirms
that the main origin of the bias in the age estimation is the difference of assumed SFHs in
SAM galaxies and the simple galaxy templates used in the SED fitting. This bias in the
age-estimation propagates into the stellar mass and SFR estimations, in the sense that the
age-overestimation leads to the mass-overestimation and SFR-underestimation. The SFR-
underestimation is further enhanced by the ‘age-extinction degeneracy’. Another source of
biases is the dominance of the current generation of star formation over the old generation(s)
of star formation in the SED-fitting. This causes both of the stellar masses and SFRs to be
underestimated.
4. We perform two types of two-component SED-fitting: (1) adding a young, bursty
component to an old component with long-lasting SFH, and (2) combining an old, bursty
component with a younger, long-lasting component.
The changes of behaviors in these
two types of two-component fitting depend on galaxies’ SFHs. Generally, compared with the
best-fit values in the single-component fitting, the best-fit stellar masses are generally smaller
in the two-component fitting with a young, bursty component embedded in an older, long-
lasting component, while they become larger in the two-component fitting with an old burst
combined with a younger, long-lasting component. For the majority of galaxies (mainly,
with type-1/type-2 SFHs), the best-fit ages become younger and the best-fit SFRs become
higher in both types of the two-component fitting. The behavior of galaxies with type-3
SFH is opposite: the best-fit ages are older and the best-fit SFRs are smaller than the values
– 41 –
derived in the single-component fitting.
5. We perform the SED-fitting with different combinations of passbands – by omitting
IRAC data or ISAAC data. If we fit the galaxy SEDs with ACS and ISAAC data only
omitting IRAC data, the derived distributions of stellar masses, SFRs and mean ages are
significantly affected. The detailed behaviors of change in the SED-fittings with and without
IRAC data strongly depend on galaxy’s redshift and SFH. Alternatively, if we fit the SEDs
of LBGs with ACS and IRAC data only, the derived distributions of stellar masses, SFRs,
and means age do not show significant changes except for the enhanced bimodalities in the
SFR/age distributions for B-/V-dropouts. This indicates that the significant changes occur
only for a small fraction of galaxies, while the effects of ISAAC data in the SED-fitting
are insignificant for the majority of LBGs. These experiments demonstrate the usefulness of
the observed-frame MIR data from IRAC in constraining physical properties of the high-z,
star-forming galaxies.
6. When the allowed range of τ (star formation e-folding timescale) is limited to be
insufficiently short or long, biases in the SED-fitting increase, in general, compared with the
case when sufficiently broad range of τ is allowed (from 0.2 Gyr to 15.0 Gyr in this study).
The mean values of the best-fit stellar mass and age are smaller than the values derived
with the full range of τ used. The age shifts are larger when we limit τ to be small (≤1.0
Gyr) than when we use very large value of τ (= 15.0 Gyr). Detailed behaviors of change for
individual galaxy depend on the SFH.
7. The experiments with the SED templates constructed from the BC03 model (§ 5.4)
isolate the effect of limiting the allowed range of τ. If τ is restricted to be τ ≤1 Gyr, both
the stellar mass and mean age are underestimated while the SFR is overestimated. If only
very long values of τ are used, the stellar mass is overestimated for the majority of model
galaxies (from BC03 model) as a result of the mis-assignment of light to older stars (with
consequently higher mass-to-light ratio). The biases in the SFR and age estimation depend
on the age of galaxy. If the age (more exactly, value of t, i.e. time since the onset of star
formation) is long, compatible to the age of the universe, the mean age is underestimated and
E(B −V ) is greatly overestimated leading to the severe overestimation of SFR. Otherwise,
the mean age is overestimated and SFR is underestimated. In the case when one tries the
single-component SED-fitting for the galaxies with two clearly distinguished generations of
star formation (resembling repeated-burst models), the stellar mass, SFR, and mean age are
all underestimated.
8. Star-formation rates estimated from the UV luminosity alone may be less biased than
those estimated from SED-fitting, provided one has a reasonable estimate of E(B −V ). This
is mainly due to the fact that the results are less subject to the degeneracy between age and
– 42 –
dust-extinction when we use only rest-frame UV photometry. However, the bias depends on
galaxies’ redshift, and more significantly on the estimation of mean dust-extinction, which is
challenging. A relatively small change in E(B −V ) would result in large bias in UV-derived
SFRs.
9. We show that biases arising in the SED-fitting procedure can affect studies of the high-
redshift, star-forming galaxies. The different directions and amounts of biases depending on
galaxy’s SFH can produce the artificial bimodalities in the age or SFR distributions. This
can affect the interpretation of the properties and nature of these galaxies. Also, the stellar
mass underestimation for massive LBGs, combined with the SFR underestimation, can affect
the interpretation of possible evolutionary paths for these massive LBGs.
In conclusion, we show that single-component SED-fitting generally slightly underes-
timates the LBGs’ stellar mass distributions, while the SFR distributions are significantly
underestimated and the age distributions are significantly overestimated. The main causes
of these biases are: (1) the difference of assumed SFHs between in the SAM galaxies and
simple templates used in the SED-fitting, and (2) the effects of the current generation of star
formation masking the previous generation(s) of stellar population. The well-known ‘age-
extinction degeneracy’ (or ‘age-extinction-redshift degeneracy’ in the case when redshifts are
allowed to vary freely as an additional free parameter during the SED-fitting procedure)
plays an additional role, mainly in the estimation of SFR distributions. Consequently, the
directions and amounts of the biases in the SED-fitting strongly depend on galaxy’s star
formation history (SFH). If we change various inputs or fitting parameters in SED-fitting,
such as the range of τ – the e-folding timescale of star formation, combinations of pass-
bands used, or assumed SFHs, the derived distributions of best-fit stellar masses, SFRs and
ages can change dramatically. Due to the compensating causes of biases, the best-fit stellar
mass distributions are more stable against these changes than the SFR/age distributions.
Moreover, the behaviors of individual galaxy in various settings of the SED-fitting strongly
depend on galaxy’s SFH.
These biases arising in the SED-fitting can have significant effects in the context of
the galaxy formation/evolution studies as well as cosmological studies. Besides the effects
discussed in § 7, biases in the estimation of stellar mass and SFR (which are dependent on
the various input settings in the SED-fitting, on available passbands and more importantly
on individual galaxy’s SFH) can affect, for example, the estimation of the stellar mass
function as well as the global density of the stellar mass and SFR, which are important
probes in galaxy formation/evolution. The blind comparison among various works – which
were done with different input settings in the SED-fitting, with the different sets of available
photometric data (e.g. works done before or after Spitzer-era), or with differently-selected
– 43 –
galaxy samples – can also misleads us. Therefore, appropriate caution should be applied to
the estimates of physical parameters of high-redshift, star-forming galaxies through the SED-
fitting, and also to the interpretation in the context of galaxy formation/evolution based on
(the comparisons of) the derived results.
We acknowledge useful discussions of the issues in this paper with our many colleagues on
the GOODS, FIDEL, and COSMOS teams, including Casey Papovich. SL thanks Myungshin
Im for useful suggestions applied in this work. This work is supported in part by the Spitzer
Space Telescope Legacy Science Program, which was provided by NASA, contract 1224666
issued by the JPL, Caltech, under NASA contract 1407, and in part by HST program number
GO-9822, which was provided by NASA through a grant from the Space Telescope Science
Institute, which is operated by the Association of Universities for Research in Astronomy,
Incorporated, under NASA contract NAS5-26555.
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This preprint was prepared with the AAS LATEX macros v5.2.
– 47 –
3.0
3.5
4.0
4.5
5.0
5.5
Redshift
0
100
200
300
400
500
600
700
Number of Galaxies
B-dropouts
V-drop
U-drop
Fig. 1.— Redshift distributions of model U-, B-, & V-dropout galaxies selected through
Lyman break color criteria as explained in § 3.1.
– 48 –
Fig. 2.— Distribution of the logarithm of stellar masses for U-dropouts (left column), B-
dropouts (middle column), and V-dropouts (right column) when galaxy redshifts are fixed
and all of the ACS/ISAAC/IRAC passbands are used in the SED-fitting procedure. Figures
in the top row are the intrinsic distributions from the semi-analytic models and figures in
the bottom row are the distributions of best-fit values from SED-fitting. Stellar masses are
given in M⊙.
– 49 –
Fig. 3.— Distribution of the logarithm of SFRs averaged over last 100 Myr (in M⊙yr−1)
for U-dropouts (left column), B-dropouts (middle column), and V-dropouts (right column)
when galaxy redshifts are fixed and all of the ACS/ISAAC/IRAC passbands are used. The
top row shows the intrinsic distributions from the semi-analytic models and the bottom row
shows the distributions of best-fit values from SED-fitting.
– 50 –
Fig. 4.— Distribution of the stellar-mass weighted mean stellar-population ages (given in
Gyr) for U-dropouts (left column), B-dropouts (middle column), and V-dropouts (right
column) when galaxy redshifts are fixed and all of the ACS/ISAAC/IRAC passbands are
used for SED fitting. Figures in the top row are the intrinsic distributions from the semi-
analytic models and figures in the bottom row are the distributions of best-fit values from
SED-fitting.
– 51 –
Fig. 5.— Comparison of the statistical distributions of stellar masses (left column), SFRs
(middle column), and mean ages (right column) of U-dropouts when we use ACS, ISAAC,
& IRAC (the second row), when we use ACS & ISAAC only (i.e. without using IRAC
bands, the third row), and when we use ACS & IRAC only (i.e. without ISAAC, the
fourth row), all with the redshift held fixed. The bottom row shows the case when we vary
redshift as a free parameter. Intrinsic distributions are shown in the top row. Stellar masses
are given in M⊙, SFRs are given in M⊙yr−1, and mass-weighted mean ages are given in
Gyr.
– 52 –
Fig. 6.— Same as figure 5, but for B-dropouts.
– 53 –
Fig. 7.— Same as figures 5 and 6, but for V-dropouts.
– 54 –
Fig. 8.— Relative errors of stellar masses, SFRs, and mean ages vs. rest-frame UV (ACS
V606) and optical (IRAC 4.5µm) magnitudes, and vs.
rest-framge UV (B435 −V606) and
UV-optical (i775 −m3.6 µm) colors of U-dropout LBGs in case when ACS/ISAAC/IRAC
passbands are used and galaxy redshifts are fixed at the input values during the SED fitting.
One object with ∆SFR/SFRSAM larger than 3 is excluded in figures in the middle row for
visual clarity.
– 55 –
Fig. 9.— Relative errors of stellar masses, SFRs, and mean ages vs. rest-framge UV (ACS
i775) and optical (IRAC 5.8µm) magnitudes, and vs. rest-frame UV (V606 −i775) and UV-
optical (i775 −m3.6 µm) colors of B-dropout LBGs in case when ACS/ISAAC/IRAC pass-
bands are used and galaxy redshifts are fixed at the input values.
– 56 –
Fig. 10.— Relative errors of stellar masses, SFRs, and mean ages vs. rest-frame UV (ACS
z850) and optical (IRAC 5.8µm) magnitudes, and vs. rest-frame UV (i775 −z850) and UV-
optical (z850 −m5.8 µm) colors of V-dropout LBGs in case when ACS/ISAAC/IRAC pass-
bands are used and galaxy redshifts are fixed.
– 57 –
Fig. 11.— Relative errors of stellar masses, SFRs, and mean ages vs. rest-frame UV (ACS
V606) and optical (IRAC 4.5µm) magnitudes, and vs.
rest-framge UV (B435 −V606) and
UV-optical (i775 −m3.6 µm) colors of U-dropout LBGs in case when ACS/ISAAC/IRAC
passbands are used and galaxy redshifts are allowed to vary as an additional free parameter.
– 58 –
Fig. 12.— Relative errors of stellar masses, SFRs, and mean ages vs. rest-frame UV (ACS
i775) and optical (IRAC 5.8µm) magnitudes, and vs. rest-frame UV (V606 −i775) and UV-
optical (i775 −m3.6 µm) colors of B-dropout LBGs in case when ACS/ISAAC/IRAC pass-
bands are used and galaxy redshifts are allowed to vary as an additional free parameter.
– 59 –
Fig. 13.— Relative errors of stellar masses, SFRs, and mean ages vs. rest-frame UV (ACS
z850) and optical (IRAC 5.8µm) magnitudes, and vs. rest-frame UV (i775 −z850) and UV-
optical (z850 −m4.5 µm) colors of V-dropout LBGs in case when ACS/ISAAC/IRAC pass-
bands are used and galaxy redshifts are allowed to vary as an additional free parameter.
One object with ∆M∗/M∗,SAM larger than 2.5 is excluded from the figures in the top row
for visual clarity.
– 60 –
Fig. 14.— Relative errors of stellar masses, SFRs, and mean ages vs. intrinsic stellar masses,
SFRs, mean ages, and SSFRs for U-dropout LBGs. ACS/ISAAC/IRAC passbands are
used and redshifts are fixed. One object with ∆SFR/SFRSAM larger than 3 is excluded
from the figures in the middle row for visual clarity.
– 61 –
Fig. 15.— Relative errors of stellar masses, SFRs, and mean ages vs. intrinsic stellar masses,
SFRs, mean ages, and SSFRs for B-dropout LBGs. ACS/ISAAC/IRAC passbands are used
and redshifts are fixed.
– 62 –
Fig. 16.— Relative errors of stellar masses, SFRs, and mean ages vs. intrinsic stellar masses,
SFRs, mean ages, and SSFRs for V-dropout LBGs. ACS/ISAAC/IRAC passbands are
used and redshifts are fixed.
– 63 –
1
2
3
4
5
6
7
8
M
A
S
R
F
S
S
x1e-9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
M
A
S
e
g
A
Fig.
17.— Intrinsic age vs.
specific SFR (SSFR) for B-dropout galaxies.
Blue dots
represent galaxies with 0.0 ≤∆Age/AgeSAM ≤0.75.
Green dots are for galaxies with
0.75 < ∆Age/AgeSAM ≤2.0, and larger, red dots are for ones with ∆Age/AgeSAM > 2.0.
Purple crosses show age and SSFR for galaxies whose mean ages are underestimated, unlike
the majority of B-dropouts. Ages are given in Gyr, and SSFRs are given in yr−1.
– 64 –
0.0
0.5
1.0
1.5
2.0
0.0
0.5
1.0
1.5
SFR
x1e+8
z=3.190
Age (SAM) = 140 Myr
Age(best-fit) = 924 Myr
0.0
0.5
1.0
1.5
0
1
2
3
4
5
6
7
x1e+8
z=3.372
Age (SAM) = 176 Myr
Age(best-fit) = 894 Myr
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0
0.5
1.0
1.5
SFR
x1e+8
z=3.703
Age (SAM) = 120 Myr
Age(best-fit) = 671 Myr
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0
0.5
1.0
1.5
x1e+8
z=4.125
Age (SAM) = 123 Myr
Age(best-fit) = 665 Myr
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Lookback Time
0.0
0.5
1.0
1.5
2.0
2.5
SFR
x1e+8
z=4.619
Age (SAM) = 103 Myr
Age(best-fit) = 530 Myr
0.0
0.2
0.4
0.6
0.8
1.0
Lookback Time
0
1
2
3
4
5
x1e+8
z=5.243
Age (SAM) = 91 Myr
Age(best-fit) = 473 Myr
Fig. 18.— Intrinsic star formation histories of typical dropout galaxies among the galaxies
with the largest overestimation of mean stellar population ages (blue line), along with star
formation histories of the best-fit BC03 template (red line). These galaxies are the ones with
type-2 SFH (see text). The red, dotted vertical line shows the point where lookback time is
100 Myr. SFR are measured over past 100 Myr timespan in this work. Lookback time is
given in Gyr.
– 65 –
0.0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
SFR
x1e+8
z=3.216
Age (SAM) = 559 Myr
Age(best-fit) = 594 Myr
0.0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
x1e+9
z=3.449
Age (SAM) = 747 Myr
Age(best-fit) = 833 Myr
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
0.0
0.5
1.0
1.5
2.0
SFR
x1e+9
z=3.805
Age (SAM) = 592 Myr
Age(best-fit) = 634 Myr
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
0.0
0.2
0.4
0.6
0.8
1.0
x1e+9
z=3.943
Age (SAM) = 613 Myr
Age(best-fit) = 689 Myr
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Lookback Time
0.0
0.2
0.4
0.6
0.8
1.0
SFR
x1e+9
z=4.761
Age (SAM)
= 380 Myr
Age(best-fit)
= 442 Myr
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Lookback Time
0.0
0.2
0.4
0.6
0.8
x1e+9
z=4.765
Age (SAM)
= 375 Myr
Age(best-fit)
= 442 Myr
Fig. 19.— Intrinsic star formation histories of typical dropout galaxies among the galaxies
with the smallest overestimation of mean stellar population ages (blue line) along with star
formation histories of the best-fit BC03 template (red line). These galaxies are the ones with
type-1 SFH (see text). The red, dotted vertical line shows the point where lookback time is
100 Myr. SFR are measured over past 100 Myr timespan in this work. Lookback time is
given in Gyr
– 66 –
Fig. 20.— Correlations among relative errors of stellar masses, SFRs, and mean ages for
U-,B-,V-dropout galaxies. ACS/ISAAC/IRAC passbands are used and redshifts are fixed
in the SED-fitting.
– 67 –
Fig. 21.— Relative errors of stellar masses, SFRs, and mean ages vs. intrinsic stellar masses,
SFRs, mean ages, and SSFRs for U-dropout LBGs. ACS/ISAAC/IRAC passbands are
used and redshifts are allowed to vary as a free parameter.
– 68 –
Fig. 22.— Relative errors of stellar masses, SFRs, and mean ages vs. intrinsic stellar masses,
SFRs, mean ages, and SSFRs for B-dropout LBGs. ACS/ISAAC/IRAC passbands are used
and redshifts are allowed to vary.
– 69 –
Fig. 23.— Relative errors of stellar masses, SFRs, and mean ages vs. intrinsic stellar masses,
SFRs, mean ages, and SSFRs for V-dropout LBGs. ACS/ISAAC/IRAC passbands are
used and redshifts are allowed to vary. One object with ∆M∗/M∗,SAM larger than 2.5 is
excluded from the figures in the top row for visual clarity.
– 70 –
Fig. 24.— Correlations among relative errors of stellar masses, SFRs, and mean ages for U-
,B-,V-dropout galaxies. ACS/ISAAC/IRAC passbands are used and redshifts are allowed
to vary.
– 71 –
0.0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
2.0
SFR
x1e+9
z=3.724
Age(input)
= 417 Myr
Age(best-fit)
= 34 Myr
0.0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
x1e+9
z=3.838
Age(input)
= 442 Myr
Age(best-fit)
= 25 Myr
0.0
0.5
1.0
1.5
Lookback Time
0
1
2
3
4
5
6
7
SFR
x1e+8
z=3.966
Age(input)
= 367 Myr
Age(best-fit)
= 29 Myr
0.0
0.5
1.0
1.5
Lookback Time
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
x1e+9
z=4.361
Age(input)
= 308 Myr
Age(best-fit)
= 29 Myr
Fig. 25.— Star formation histories of typical B-dropout galaxies with underestimated mean
stellar population ages (type-3 SFH, see text). The dotted vertical line shows the point
where lookback time is 100 Myr. SFR are measured over past 100 Myr timespan in this
work. Lookback time is given in Gyr.
– 72 –
Fig. 26.— Stellar mass distributions of U-dropouts (left column), B-dropouts (middle col-
umn), and V-dropouts (right column). Figures in the top row show the distributions of
intrinsic stellar masses. The middle row is for stellar masses from the two-component fitting
with a secondary young burst added to a main component (as investigated in § 5.1.1), and
the bottom row is for stellar masses from the two-component fitting with a maximally old,
burst component plus a younger, long-lasting component (§ 5.1.2). Stellar masses are given
in M⊙.
– 73 –
Fig. 27.— SFR distributions of U-dropouts (left column), B-dropouts (middle column), and
V-dropouts (right column). Figures in the top row show intrinsic SFR distributions. The
middle row is for SFRs from the two-component fitting with a secondary young burst added to
a main component (§ 5.1.1), and the bottom row is for SFRs from the two-component fitting
with a maximally old burst component plus a younger, long-lasting component (§ 5.1.2).
SFRs are given in M⊙yr−1.
– 74 –
Fig. 28.— Mass-weighted stellar-population mean age distributions of U-dropouts (left col-
umn), B-dropouts (middle column), and V-dropouts (right column). Figures in the top row
show intrinsic age distributions. The middle row is for ages from the two-component fitting
with secondary a young burst added to a main component (§ 5.1.1), and the bottom row
is for ages from the two-component fitting with a maximally old burst component plus a
younger, long-lasting component(§ 5.1.2). Age is given in Gyr.
– 75 –
-1 0
1
2
3
4
5
M
A
S
e
g
A
/
e
g
A
b
∆
-1
0
1
2
3
4
5
V-dropouts
(c)
-1 0
1
2
3
4
5
M
A
S
e
g
A
/
e
g
A
b
∆
-1
0
1
2
3
4
5
B-dropouts
(b)
-1 0
1
2
3
4
5
M
A
S
e
g
A
/
e
g
A
b
∆
-1
0
1
2
3
4
5
M
A
S
e
g
A
/
e
g
A
a
∆
U-dropouts
(a)
-1.0 -0.5 0.0 0.5 1.0
∗
M
A
S
,∗
M
/
M
b
∆
-1.0
-0.5
0.0
0.5
1.0
V-dropouts
(f)
-1.0 -0.5 0.0 0.5 1.0
∗
M
A
S
,∗
M
/
M
b
∆
-1.0
-0.5
0.0
0.5
1.0
B-dropouts
(e)
-1.0 -0.5 0.0 0.5 1.0
∗
M
A
S
,∗
M
/
M
b
∆
-1.0
-0.5
0.0
0.5
1.0
∗
M
A
S
,∗
M
/
M
a
∆
U-dropouts
(d)
-1 0
1
2
3
4
5
M
A
S
e
g
A
/
e
g
A
b
∆
-1
0
1
2
3
4
V-dropouts
(i)
-1 0
1
2
3
4
5
M
A
S
e
g
A
/
e
g
A
b
∆
-1
0
1
2
3
4
B-dropouts
(h)
-1 0
1
2
3
4
5
M
A
S
e
g
A
/
e
g
A
b
∆
-1
0
1
2
3
4
M
A
S
R
F
S
/
R
F
S
a
∆
U-dropouts
(g)
Fig. 29.— Correlations between relative errors in two-component fitting with a secondary
young burst (§ 5.1.1) and relative errors in single-component fitting. Figures in the top
row show correlations between relative age errors in two-component fitting and relative age
errors in the single-component fitting. The middle row is for correlations between relative
stellar mass errors in the two-component fitting and relative stellar mass errors in the single-
component fitting. The bottom row is for correlations between relative SFR errors in the two-
component fitting and relative age errors in the single-component fitting. ∆avalue represents
‘(value from two-component fitting) - (intrinsic value)’, and ∆bvalue represents ‘(value from
single-component fitting) - (intrinsic value)’.
– 76 –
Fig. 30.— Ratio of the best-fit stellar masses ((a)-(c)), SFRs ((d)-(f)), and mean ages ((g)-
(i)) with and without IRAC photometry for U-dropouts ((a), (d), and (g)), B-dropouts ((b),
(e), and (h)), and V-dropouts ((c), (f), and (i)) dependent on the best-fit values with IRAC
photometry. ‘AS’ stands for values without IRAC photometry, and ‘ASR’ for values with
IRAC photometry. Objects with AgeAS/AgeASR > 4.5 are excluded from the figures in the
bottom row for visual clarity.
– 77 –
0.0 0.2 0.4 0.6 0.8
M
A
S
e
g
A
-1.0
-0.5
0.0
0.5
s
t
u
o
p
o
r
d
−
V
0.0 0.2 0.4 0.6 0.8
M
A
S
e
g
A
-1.0
-0.5
0.0
0.5
s
t
u
o
p
o
r
d
−
B
0.0 0.2 0.4 0.6 0.8
M
A
S
e
g
A
-1.0
-0.5
0.0
0.5
ll
a
−
1
τ
ll
ae
g
A
/
e
g
A
∆
s
t
u
o
p
o
r
d
−
U
-9.5
-9.0
-8.5
-8.0
M
A
S
)
R
F
S
S
(
g
ol
-1.0
-0.5
0.0
0.5
-9.5
-9.0
-8.5
-8.0
M
A
S
)
R
F
S
S
(
g
ol
-1.0
-0.5
0.0
0.5
-9.5
-9.0
-8.5
-8.0
M
A
S
)
R
F
S
S
(
g
ol
-1.0
-0.5
0.0
0.5
ll
a
−
1
τ
ll
ae
g
A
/
e
g
A
∆
-1 0
1
2
3
4
5
M
A
S
−
ll
a
M
A
S
e
g
A
/
e
g
A
∆
-1.0
-0.5
0.0
0.5
-1 0
1
2
3
4
5
M
A
S
−
ll
a
M
A
S
e
g
A
/
e
g
A
∆
-1.0
-0.5
0.0
0.5
-1 0
1
2
3
4
5
M
A
S
−
ll
a
M
A
S
e
g
A
/
e
g
A
∆
-1.0
-0.5
0.0
0.5
ll
a
−
1
τ
ll
ae
g
A
/
e
g
A
∆
Fig. 31.— (Ageτ1 −Ageall)/Ageall as a function of intrinsic age (top row), intrinsic specific
SFR (middle row), and (Ageall −AgeSAM)/AgeSAM (bottom row) for U-dropouts (left col-
umn), B-dropouts (middle column), and V-dropouts (right column). Here, Ageτ1 and Ageall
are the best-fit mean ages derived when we limit τ as ≤1.0 Gyr and when we allow τ to
vary from 0.2 Gyr to 15.0 Gyr, respectively. AgeSAM is an intrinsic age, and SSFRSAM is
an intrinsic SSFR.
– 78 –
Fig. 32.— Best-fit tall dependent discrepancies in best-fit t (tτ15 −tall). tall is the best-fit t
obtained when we allow the full range of τ, and tτ15 is the best-fit t derived if we use single
value of τ (= 15.0 Gyr) in the SED-fitting. tτ15 and tall are in Gyr.
– 79 –
Fig. 33.— Ratio of SFR derived from rest-frame UV luminosity to intrinsic SFR as a function
of redshift for B-dropouts. All B-dropouts are assumed to be at z = 4.0, and to be extincted
by dust with amount of E(B −V ) = 0.15.
– 80 –
0.5
1.0
1.5
2.0
)
R
F
S
(
g
ol
0
200
400
600
800
r
e
b
m
u
N
0.5
1.0
1.5
2.0
)
R
F
S
(
g
ol
0
200
400
600
800
Fig. 34.— Distributions of intrinsic SFRs (left), and SFRs derived from rest-frame UV
luminosity assuming all galaxies are at z = 4.0 and have mean dust-extinction of E(B−V ) =
0.15 (right) for B-dropout galaxies.
– 81 –
Table 1.
Fitting Parameters
IMF
τ (Gyr)
t (Gyr)
metallicity (Z⊙)
internal extinction
IGM extinction
Chabrier
0.2-15.0
0.01-2.3a
0.2, 0.4, 1.0
Calzettib
Madau
at is limited to be smaller than the age of the universe at each galaxy’s redshift.
b0.000 ≤E(B −V ) ≤0.950 with ∆E(B −V ) = 0.025
– 82 –
Table 2.
Mean Values of Stellar Population Parameter Distributions
Redshift
Stellar Mass (M⊙)
SFR (M⊙/yr)
Mean Age (Gyr)
U-dropouts
SAMsa
3.36
9.376×109
11.005
0.465
SEDz−fixb
3.36
7.594×109
3.887
0.915
SEDz−freec
3.27
7.038×109
4.682
0.870
B-dropouts
SAMsa
4.02
8.888×109
15.650
0.353
SEDz−fixb
4.02
6.680×109
6.638
0.667
SEDz−freec
3.88
4.327×109
19.891
0.420
V-dropouts
SAMsa
4.97
7.946×109
18.920
0.249
SEDz−fixb
4.97
5.983×109
7.287
0.508
SEDz−freec
4.88
4.496×109
16.905
0.344
aIntrinsic values from SAM catalogs
bValues derived in the SED-fitting with redshifts fixed as input values
cValues derived in the SED-fitting with redshifts allowed to vary freely
– 83 –
Table 3.
Means and Standard Deviations of Relative Offsetsa for various Stellar
Population Parameters
Redshift
Stellar Mass
SFR
Mean Age
U-dropouts
SEDz−fixb
· · ·
-0.039±0.232
-0.585±0.185
1.075±0.510
SEDz−freec -0.021±0.015
-0.108±0.230
-0.530±0.258
0.955±0.502
B-dropouts
SEDz−fixb
· · ·
-0.120±0.228
-0.558±0.288
1.009±0.586
SEDz−freec -0.030±0.031
-0.371±0.355
0.212±1.181
0.273±1.012
V-dropouts
SEDz−fixb
· · ·
-0.169±0.179
-0.597±0.133
1.162±0.589
SEDz−freec -0.014±0.012
-0.339±0.274
-0.094±0.964
0.487±0.875
aRelative offset is defined as (V alueSED −V alueSAM)/(V alueSAM) for
stellar mass, SFR, and age. For redshift, relative offset is defined as (zSED−
zSAM)/(1+zSAM).
bRelative offsets between values derived in SED-fitting with redshifts
fixed and intrinsic values
cRelative offsets between values derived in SED-fitting with redshifts
allowed to vary and intrinsic values
– 84 –
Table 4.
Relative Changesa of the Mean Stellar Masses and Ages with the Limited τ
Range
τs Used for Fitting
Stellar Mass (%)
Mean Age (%)
U-dropouts
τ ≤1.0 Gyr
-4.9
-15
τ = 15.0 Gyr
-8.0
-12
B-dropouts
τ ≤1.0 Gyr
-12
-26
τ = 15.0 Gyr
-3.4
-7.4
V-dropouts
τ ≤1.0 Gyr
-5.4
-16
τ = 15.0 Gyr
-3.2
-6.6
aRelative
change
is
defined
as
(⟨valueτ⟩
−
⟨valueall⟩)/⟨valueall⟩.
⟨valueall⟩is the mean value
of the stellar mass or age derived with the full range of
τ. ⟨valueτ⟩is the mean value of the stellar mass or age
derived with the limited τ range.
– 85 –
Table 5.
Toy Models
Model 1
Model 2
Model 3
input
τ (Gyr)
15.0
0.2
0.2 (young), 0.2 (old)
t (Gyr)
0.01, 0.05, 0.1, 0.5, 1.0
0.1, 0.2, 0.5, 1.0, 1.3
0.1 (young), 1.0 (old)
E(B −V )
0, 0.05, 0.1, 0.2
0, 0.05, 0.1, 0.2
0.0, 0.2
Mass Fractiona
· · ·
· · ·
0.1, 0.26, 0.52, 1.0, 2.6
Number of SEDs
120
120
60
used for fitting
τ (Gyr)
≤1.0
≥8.0
0.2 ≤τ ≤15.0
t (Gyr)
0.01 - 2.3b
0.01 - 2.3b
0.01 - 2.3b
aStellar mass fraction between young component and old component defined as
Myoung/Mold
bt is limited to be smaller than the age of the universe at each galaxy’s redshift. |
vixra | 2603.0037 | Astrophysics | A Cosmological constant problem has been considered in the light of the Standard model ofelementary particles (SM) by account two fundamental fermions in the SM, i.e. the u quark and the electron and their antiparticles. All the movings of these virtual pairs and their orientations in space have been taken into account in the offered model of vacuum. This led to the more precise and accurate estimation of vacuum energy of the Universe on the discovered and examined minimal scale ~1.5×10^-15 m on uncertainty relations, than in the previous work. Comparison of this estimation with the calculation realized by the method and the model used in the previous work was carried out. The current calculation is the full computation, i.e. for the full scale range, which begins from 1.5×10^-15 m and can think terminated for the effect on the maximal linear size2.00008×10^-11 m for the experimental data and the theoretical data on hydrogen and heliumatoms correspondingly. These atoms are considered due to that they are the most common in the entire Universe. Thus, one can say that the vacuum energy near matter in the special effect of the reducing of vacuum by matter has been assessed more precisely as it is in the real world. | Nekrasov Grigory Yu | Astrophysics | https://vixra.org/abs/2603.0037 | 1
A further specification of the Cosmological constant problem by account the
two fermions in the Standard model and an effect of the reducing of vacuum
by matter based on uncertainty relations
Grigory Yu. Nekrasov
Federal State University of Education 141014, Moscow region, Mytishchi, Russia
Abstract
A Cosmological constant problem has been considered in the light of the Standard model of
elementary particles (SM) by account two fundamental fermions in the SM, i.e. the u quark and
the electron and their antiparticles. All the movings of these virtual pairs and their orientations in
space have been taken into account in the offered model of vacuum. This led to the more precise
and accurate estimation of vacuum energy of the Universe on the discovered and examined
minimal scale ~1.5×10-15m on uncertainty relations, than in the previous work. Comparison of
this estimation with the calculation realized by the method and the model used in the previous
work was carried out. The current calculation is the full computation, i.e. for the full scale range,
which begins from 1.5×10-15m and can think terminated for the effect on the maximal linear size
2.00008×10-11m for the experimental data and the theoretical data on hydrogen and helium
atoms correspondingly. These atoms are considered due to that they are the most common in the
entire Universe. Thus, one can say that the vacuum energy near matter in the special effect of the
reducing of vacuum by matter has been assessed more precisely as it is in the real world.
Keywords: Vacuum, Uncertainty relations, Cosmological constant, Virtual fermions, Standard
model of elementary particles, Vacuum energy, Matter, Interaction of vacuum with matter,
Vacuum energy density
Introduction
Like in the previous work which is called “A statement of the Cosmological constant
problem and an effect of the reducing of vacuum by matter based on uncertainty relations” we
can employ Heisenberg uncertainty principle to assess the vacuum energy of all empty space of
the Universe and the vacuum energy near matter of the Universe. Due to the war in Ukraine and
on this cause lack of resources I can compute only the following quantities: all for hydrogen and
helium for u quark and electron, also it is worthy for the all types of neutrino to compute only the
vacuum energy for empty space, i.e. without atoms or the free vacuum energy. Now these
fundamental and, in the time, elementary particles do exist in the Standard model of elementary
particles. All the fermions of the SM, undoubtedly, contribute in the vacuum energy according
the SM itself and the quantum field theories on which the SM is built. Thus, we are going to gain
the part of the full vacuum energy in this case, except the contribution of the boson fields which
is responsible for the interactions, i.e. the matter contribution in vacuum energy. In this paper we
are going to consider the part of only fundamental vacuums, i.e. the vacuums generated by only
the fundamental particles of matter. As well known, matter can be various, namely, low-energy
particles and high-energy particles. We will take into account low-energy electron and positron
and heavier u quark and antiquark in this work. Both present in the ordinary low-energy matter.
This calculation is also going to be comprehensive in the sense that all movings: forward and
2
backward, rotations are going to be taken into account, and all changings of the sizes of wave
packets in location space at the moving of them are going to be considered to make the
calculation as exact as possible. And, of course, this calculation is limited in our work (present
and previous) by the considered effect of the reducing of vacuum near matter. We should note
that this is the one single effect form great number of quantum field effects which govern in
space with matter at subatomic level. Like in the previous paper we are not going to go with the
set course of the quantum field theory (QFT) in the assessment of vacuum energy, but going to
chose simple and evident way, taking from QFT only empirical concept of particle-antiparticle
pairs in the vacuum as a picture of it. Also we say that like in the previous work this will be built
on the theory excluding the interactions of virtual particles. The exact assessment of the full
fermion vacuum energy is going to be done by using uncertainty relations. We delay account of
interactions of virtual particles to next paper, acting gradually.
1. The concept of discrete space, needed for finite vacuum energy
In the previous work we have defined the concept of discrete space, basing on that
vacuum energy cannot be finite but must be infinite in continuous space. This is true due to that
at the accounting all the scales of space, we sum energy being on each space scale. At that if the
number of nonzero elements of the sum is finite, then the sum is going to be finite, and vice
versa. That is why space must be discrete – for the number of energies on all the space scales is
to be finite. In that work we have discussed what is motion of a body in such space, in the
current work we will also be needed in concept of discrete time. Thus, we will construct discrete
space-time of the special theory of relativity. It is needed to say that in the usual theory if space
is discrete, then the special and the general principles of relativity have to be wrong [1]. But, as
author is going to show, this is appearing to be due to not completely right approach to the
problem of discrete space-time.
Let us construct discrete space and time in transformed special theory of relativity for
such purpose. Let velocity v of a body be determined, according to the previous paper [2], as
Pl
v
t
,
(1.1)
where
P
t
t
, here like in [2] we designated: Pl is the Planck length and
Pt is the Planck time,
so that, it is true
P
P
l
c
t
,
(1.2)
where c is the speed of light. The author’s idea can be demonstrated on the simple example of
one-axis motion, i.e. motion along one axis of the Cartesian coordinate system, and it can be
found out that this idea lead to the contrary conclusion of discrete and flat physical manifold than
that is concluded in the article [1]. In such case, according well-known the special theory of
relativity (SR), length contraction is
2
2
1
v
l
l
c
.
(1.3)
3
Now we designate:
P
l
nl
and
P
l
ml
, where n and m are integer numbers. Substituting them
into (1.3), and (1.1) into (1.3), also using (1.2), we got
2
2
1
Pt
m
n
t
.
(1.4)
According (1.4), must be m
n
. On this stage let us to take into account an axiom (1):
Pl
const
for all frames of reference. With this aspect, indeed, there was a problem in [1],
therefore we exclude it at root. Taking into account this axiom, note, the simple logic follows
from (1.4): if
1
n , then
1
m , but must be always
1
m and, as we can see, also, must be
1
n , hence, n cannot be equal to 1. In the all above we already accept that space is discrete,
consisting of the Planck lengths along each dimension or Cartesian axis like in [2], now consider
also discrete time, parting on the Planck times along time axis in the pseudoeuclidean four-
dimensional Minkowski space-time. We quantize time interval as
1
P
t
k
t
,
(1.5)
where
1
k . Then, substituting (1.5) into (1.4), we obtain the formula
2
1
1
1
m
n
k
,
(1.6)
which is valid for one-dimensional length contraction. In (1.6) k is the number of indivisible
time intervals, n is the number of indivisible space segments in a body in its own frame of
reference, m is the number of indivisible space segments in a body in the related frame of
reference. This formula already describes discrete space-time. As one can see, constant
Pt is
invariant at changing frames of reference, that is the containing of an axiom (2):
Pt
const
for
all frames of reference. This is in contrast with [1]: fundamental space-time quanta do not
change in any situation, this thesis is laid in the basis of our theory.
Analogously, for the time dilation in such simple case we have
2
2
1
t
t
v
c
.
(1.7)
We introduce the following formulae for the time intervals in each frame of reference like (1.5)
1
P
t
k
t
,
(1.8)
1
P
t
k
t
(1.9)
and the formula for the velocity like (1.1)
Pl
v
t
,
(1.10)
4
where
1
P
t
f
t
.
(1.11)
Here k , k and f are integer numbers which satisfy the conditions:
1
k ,
1
k,
1
f . So that
the following result is justified for the time dilation in the modified SR
2
1
1
1
1
1
k
k
f
.
(1.12)
In validity of the conditions one can convince, if to take
1
k and
1
f , then one can gain
1.3094
1
k
,
(1.13)
so that the conditions are generally executed. Thus, we can generalize the all, that was performed
above in the words: the fundamental segments of space and time (corresponding quanta) are
always invariant under Lorenz group transformations, i.e. they never change at transition of the
frame of reference, but do change number of such quanta in a body or in a field (as type of
matter), it can increase or decrease. From here it follows at once that if a body has one single
segment of space, then it does not change, i.e. remains constant at any speed of the body.
2. Consideration of exact admissible movings of virtual electron-positron
pairs in the field of the charged center in the vacuum limitation effect
As was noted above, in this work we consider all movings of the particles in virtual pair,
namely, the vacuums will be considered in the dynamics. The movings, apparently, can be in
forward direction and in backward direction, i.e. to annihilation point. We take that these
movings occur at the constant velocity there and back, that is they are uniform. The turn occurs
instantaneously. So, we have enough information to imagine some a vacuum, let it be electron-
positron vacuum for definiteness. Consider enough large limited volume of space. The vacuum is
being represented the virtual electron-positron pairs which at the same time jump out from the
void (empty space) and, as I said, they have the same position on the axis of moving at any
instant of time. In this moment we apply Galileo’s principle of relativity (Classical Relativity,
CR). This means that time is absolute and flows homogeneously in every place in the Universe.
All the pairs create and annihilate simultaneously. However, in reality such situation can never
be, there we have chaos and non-simultaneity. This situation can be reached by the following
approach. It is needed to find an average on the changing time variable on different pairs. By this
approach we will succeed to shift different pairs on different values of the time variable and we
will gain the required characteristic of the vacuum. As we will show below, this approach can be
implemented using the distribution function. And to gain a vacuum energy – a numerical value,
we need to take an average of the determined expression on the all movings of these and each
one pair in space (actually, on each specially orientated axis), happening during time. So,
summing up, we will need two enclosed each other average values to gain the final result, a
vacuum energy.
5
In the calculation, as it was also said in the end of the previous paper [2], we will
consider dispersion of the virtual particles wave packets. Let us at once say about this. According
the carried out investigation which preceded this article, an accounting of the real dispersion of
the real free fermion wave packets, as this is ordered, – from the spinor – the solution of the free
Dirac equation is complicate to do because difficulty of the source of this dispersion – the
integral of the spinor. To extract all possible dispersion functions which, undoubtedly, represent
types of the real dispersion of matter on our level (without interaction of the virtual particles to
each other) to someone would be needed a special separate research which will be very complex.
Also the number of these functions is infinite because the theoretical model limitlessness of the
momentum space. As a result of this circumstance the author is decided in the presented
computation to account dispersion as linear function as an average dispersion on this entire
infinite manifold of the dispersion functions of the wave packets. The tilt of the line
characterizes an average dispersion on all dispersion functions of the real free fermion
dispersion, known from the Dirac spinor.
3. Types of virtual matter that contains the Universe or vacuums
We will consider two types of the particles of the SM in the physical vacuum. Filling
empty space, they appeared to be a media or a virtual matter because it consists of virtual
particles. Wave packets of these particles types are overlapped, that means wave packets of the
virtual particles of one single type are not overlapped but different types of virtual particles wave
packets are in the same place in space between each other. Initially, according the SM, real
physical vacuum consists of zero-point fluctuations of the boson fields and the corresponding
quanta, as it is in the quantum field theory, and interacting with each other fermion pairs by
means of these boson quanta. Fermions are performed the matter of our Universe, and exactly
them we will consider, but this paper is without any consideration of their interactions. It is also
needed to say that vacuum, according QFT, is sufficiently nonlinear structure and has
selfinteraction. However, also we are not going to consider nonlinear component of vacuum in
this paper. We delay its consideration on next paper.
As well known, particles of matter include two general families: quarks and leptons.
Quarks and leptons combine with each other forming families or generations, in the time the 3
families are known. These are: (u, d, e, νe), (c, s, ν𝜇, 𝜇), (t, b, ν𝜏, 𝜏). Here u, d, c, s, t, b are quarks,
first two are stable and building all stable and seeable matter in the Universe, others are unstable
and much more heavy than first two, especially last two: t and b. The rest particles are leptons,
such as: e, 𝜇 and 𝜏 are electron, muon and tau-lepton, correspondingly. Electron e together with
two quarks – up and down (u, d) consist all visible matter in the Universe finally. This matter can
exist at low energy and at high energy. And the other particles except the neutrinos exist only at
high energies in laboratory (accelerator and collider) or in violent processes in stars and black
holes. The neutrinos, such as: νe, ν𝜇, ν𝜏, called electron-neutrino, muon-neutrino and tau-neutrino
correspondingly, together with mentioned above leptons form families by pairs, thus, appear the
3 families. These are: (e, νe), (𝜇, ν𝜇) and (𝜏, ν𝜏). One can unite quarks themselves also in families
in a view: (u, d), (c, s) and (t, b). On the last data in research of the controversial issue on the
time of writing of this article the neutrinos have small nonzero masses; the fact is going to be
important for the current work.
All these fermions enter together with their own antiparticles, so that we have 12 particles
and 12 antiparticles, total: 24 particles and 12 pairs. The last work has the computation that has
6
been carried out for electron-positron pairs on the basis of the consideration of one single
electron-positron pair. The current work, as already was said above, must base on this article,
therefore the computation in that work is applicable for pair and not for isolated particle.
Therefore for us had to be essential number 12 of the types of fermion pairs. For the further
statement we need a list of masses of the particles; the masses in the listing are rounded up to
two meaning digits (for quarks the range is set, in which the corresponding mass hits; via
semicolon the average values for the corresponding ranges are divided, and for u, d, s-quarks the
current masses are given, for c, b-quarks the running masses are given)
Table 1 [3, 4, 5]
Family 1
Family 2
Family 3
Particle
Mass(kg)×10-31
Particle
Mass(kg)×10-31
Particle
Mass(kg)×10-31
Electron (e) 9.11
Muon (𝜇)
1883.56
𝜏-lepton
31675.1
νe
<0.000036
ν𝜇
<3.39
ν𝜏
<324.45
u-quark
26.74 – 58.83;
42.785
s-quark
1247.86
–
2317.46;
1782.66
b-quark
73624 – 95729;
84676.5
d-quark
62.39 – 106.96;
84.675
c-quark
20678.9
–
23887.7;
22283.3
t-quark
3.015×106
–
3.089×106;
3.052×106
4. The theory of virtual particles
Consider the Figure 3 in the paper [2]. For this 2D geometrical scheme the given formula
(2.1.7) is valid, but we need another one formula which follows from this scheme, namely
tan
2
l
s
(4.1)
as it is easy to see. In fact, this is different form of the formula (2.1.6). Following that paper, we
equate these two formulae, in the result we get the formula for the effective scale reducing in the
form
1
l
l
r
s
,
(4.2)
like in the previous paper. Now we need to account the dynamics of the virtual particles in pair.
This can be done by accounting time dependence in the particle’s passed way in both directions,
so let’s take
s
vt
,
(4.3)
so that in the forward direction at
0
t
, the start or creation point of the motion lying on the axis
or even the ray will be
0
s
. To find the velocity v we use (2.1.19), noting that if velocity is
constant at moving in the both directions, then it must depend only on scale l and maybe on
distance from charged center r and nothing else, that is why it is good to use the theory of the
previous paper, i.e. the static, not dynamical theory combining it with this new theory to extract
7
velocity. So, substitute (2.1.19) into (4.3) and the gained expression into (4.2), then having
simplified the result and having brought the gained formula a little, we obtain
2
2
2
2
1
16
1
3
l
l
r
ct
m c l
.
(4.4)
From here we extract the velocity
2
2
2
2
16
1
3
c
v
m c l
.
(4.5)
As we can see, it depends only on scale l . Also, as we can see, with decreasing of l and going
its value to zero, the velocity goes to speed of light c , and with increasing l and unlimited
increasing of it, the velocity of the particle goes to zero value. This result is explainable because
with decreasing of scale, energy of the particle will grow up, that is understood from the simple
uncertainty relation (1.1) in [2], and vice versa.
Now consider formula (2.1.15) in the same work. We have found out that this formula
describes the lifetime of the virtual particle or the pair, therefore to gain a half of the lifetime
which is time of the motion in the one direction or in the other direction (because the speeds are
equal), one needs to substitute in (2.1.15) maximal value of the reducing of scale which is
already known in [2] in (2.1.21) and to divide the gained expression into 1 2. If to do this, one
can gain the following formula
2
2
2
2
2
2
2
2
3 16
3 16
m c l
l
t
m c
c
l
(4.6)
and
2
2
2
2
1
2
3 16
l
T l
c
m c l
.
(4.7)
Combining formulae (4.5) and (4.7) with corresponding coefficients, we can get
2
3
3
vl
t
c
.
(4.8)
Using (2.1.5) and (4.8) together with (4.5), we obtain intermediate result
1 4
6
3
c
s
v
l
.
(4.9)
8
And using here again (4.5) and simplifying, we gain final result on the current stage
2
2
2
2
3
2 3 16
l
s
m c l
.
(4.10)
Let us now analyze this result. If
2
2
2
2
16
1
m c l
,
(4.11)
then
3
3
2 3
6
l
s
l
,
(4.12)
i.e. s
l
. If
2
2
2
2
16
1
m c l
,
(4.13)
then
1
3
2 3
l
s
,
(4.14)
but
1
1
. Let’s perform (4.14) in the view
2
3
2
l
s
,
(4.15)
where
2
1
, hence, s
l
. That means the condition
s
l
(4.16)
is always true. To conclude finally on this stage, we need to do additionally some calculations.
According (2.1.5) and the well-known formula of SR, the momentum of the particle will
have the view
01
2
2
2
2
1
4
1
v
ms
t
s
c
t
p
e ,
(4.17)
where must be s
l
, and
ve is the basis vector of the velocity v, so that
1
v
e
. Now, let us
substitute (4.7) in (4.17), then after simplifying, we gain
9
3
3 2
2
01
2
2
2
2
2
2
2
2
2
2
2
32
6
12
64
3 16
1
v
m c l s
cms
m c
s
m c
l
l
s
l
p
e
.
(4.18)
This expression can be valid only at the condition
2
2
2
2
2
2
12
64
1
0
s
m c s
l
,
(4.19)
so that only positive number would stand under square root and this number must not be zero.
From here we come to the conclusion
2
2
2
2
2
2
3 16
4
m c
l
l
s
,
(4.20)
this is true only if s
l
. So, now we are ready to conclude finally on this stage. According
(4.16), the formula (4.18) is always valid.
We need to make some remarks about the theory. The first, the formula for momentum of
the point particle and the formula for velocity of motion of the wave packet as whole are
justified, that follows from the theory lying in basis of these expressions. The second, the cause
of that the first aspect is true, is that qualitatively new physics of vacuum virtual particles
appears to be in the fact is that the way going by wave packet of the virtual particle as whole is
always less of the linear size of the particle wave packet. Owing to the theory from which this
has been withdrawn these two aspects are connected, the first can be described by the second.
Now we are ready to express this understanding mathematically. Using the well-known formula
of the SR for momentum of a point particle in the view
01
2
2
1
v
mv
v
c
p
e ,
(4.21)
and using (4.5), we can obtain the following formula
01
3
4
v
l
p
e
,
(4.22)
which accords with the basic uncertainty relation for the momentum and the coordinate of the
particle (1.1) in [2] with a condition that the value of the momentum is less than the value of
uncertainty interval of the momentum postponed on the same axis of momentum, i.e. if
2
1
i
i
i
p
p
p
,
(4.23)
then the vector inequality
01
2
1
p
p
p
(4.24)
is justified. According this inequality and the fact is that
01
p must hit the uncertainty interval,
must be
10
01
1
p
p .
(4.25)
Thus, in (4.22) the value of momentum and not the uncertainty, like in (1.1), stands. In the
spherical polar coordinate system the basis vector of the velocity has the well-known expression
via angles
sin
cos
,sin
sin
,cos
v
e
,
(4.26)
where and are polar and azimuthal angles correspondingly.
5. The general mathematical theory needed for computation
As was mentioned afore, we will need in two average values: the first, we need to
average all movings of the virtual particles in the pair on these movings or, that is the same, on
time. It is needed to average them on the all temporal interval of the moving in one and in
another direction. Let us begin from the forward direction or the one direction. The approach will
be correct if the condition is true, namely: the describing effect of the wrapping of vacuum does
not impact the distribution on average values of any quantities of all virtual pairs in all atoms in
the Universe. We have the following average in the first place
1
...
T l
dt
T l
,
(5.1)
where
T l defines by (4.7), and there is the condition for new variable
0
T l
.
(5.2)
The average (5.1) is the average on time value applied for the moving of this virtual particle, it
transforms the segment of the moving trajectory into a dot, i.e. it removes time variable t . But
according what was said in the Section 2, all virtual pairs differ and, correspondingly, all average
values for each pair also differ, therefore we need to distinguish them anyway. The best way to
do this already consists in the offered average value (5.1): this difference in the initial locations
of virtual particles in all pairs in the vacuum in our chosen instant of time makes -parameter.
Actually, this parameter does it perfectly and, thus, -parameter is that parameter which by
unambiguous and by sufficient way characterizes this unique average value of any quantity. An
upper value for one can set by a mode
T l
,
(5.3)
where is a small enough number with dimension of time. Thus, we have one of two average
values.
The second average, as was already said in the Section 2, is required an introducing the
distribution function. So, we need distribution function of averages (or average values) on time
on number of particles in our large volume of space. This function can be determined by the
following equation
11
f
dN
N F
d
,
(5.4)
where
f
N is the full number of all virtual pairs in all atoms in the Universe; N is the number of
pairs which have -parameter in the interval
2
1
,
(5.5)
and, finally,
F is the required distribution function. Of course, number
f
N can relate only to
a separated volume, but globally it must describe the entire Universe. Fortunately, we already
know this number from [2], it must look as
0
f
a
n
n
N
N A
(5.6)
in the terms used in [2]. I remind them;
a
N is the number of all atoms in the Universe, the sum
n
A on n is the sum of all numbers of all wave packets in all ball layers in one single atom (for
each numerator n this is the number of all wave packets in the given ball layer in one single
atom). Thus, according the concepts of average value and distribution function, the probability of
the event that this chosen pair will have -parameter in the interval (5.5) is going to have the
view
2
1
2
1
,
f
dN
P
F
d
N
.
(5.7)
At that the distribution function is normalized as
0
0
lim
1
T l
F
d
.
(5.8)
If virtual pairs distribute on average values identically and uniformly, then
F does not
depend on and equals
1
F
T l
.
(5.9)
This one can easily check by substitution (5.9) in the normalization (5.8), so that one reads that
(5.8) is executed, but this can be done with one single distribution function which is (5.9).
Thus, the full average value consisted of two enclosed each other averages at the
condition, if we decided to use (5.9) in quality of the distribution function on the basis of the
corresponding fact above, now has the view
N1
0
0
0
0
0
1
1
1
...
lim
...
lim
...
T l
T l
T l
T l
a
n
n
dtdN
N A
dtd
T l
T l
T l
. (5.10)
12
Here we have neglected all of scales that is larger than any atom, that is why in (5.10) in the sum
the finite number of summands stands in contrast (5.6). The upper limit of the sum N1 just
determines this finite number.
Here we also give the first average value for another direction or the backward direction.
It has the view
2
1
...
...
...
T l
n
T l
l
t
l t
dt
T l
,
(5.11)
where we have given at once the dependence of the integrand on the effective reducing of scale
(4.4) which must have reversal dependence on time t because it thinks moving in backward
direction. Here only the time dependence of the effective scale reducing is shown. We also have
included in the integrand the dependence on dispersion that is expressed by the function
l t
which must have direct dependence on time because with flowing of time dispersion only
increases. Going back to the beginning of this Section analogously the dependence in (5.11), the
dependence of (5.1) must look so
...
...
...
nl
t
l t
.
(5.12)
Here only the time dependence of the effective scale reducing is shown. That is, we must allow
time to flow in direct, normal direction for the forward direction of motion of the particle and we
must turn time flow direction for the backward direction of motion of the particle, that describes
well and enough the backward motion in the offered model. Thus, the full average value for the
backward direction at analogous the forward motion conditions will have the view
2
1
0
0
0
1
1
...
lim
...
T l
T l
N
a
n
n
T l
N A
dtd
T l
T l
.
(5.13)
In any case we must additionally integrate the first average on dN , using this expression from
(5.4) with the upper and the lower limits:
0,T l
, thinking that is enough small constant.
Now let us consider the full integrand in the both cases of the motion. It will have
changed in comparison with the integrand in [2], as we already said above, on the two causes:
the dynamics of the task and dispersion. The main multiplier will have the view
2
2
2
4
2
3
16
,
cell
v m
k
c
E
m c
l
l
l t
l t
.
(5.14)
The number multiplier
fk
N
also will have changed due to consideration of the dynamics and
dispersion:
2
,
,
,
,
2
2
k
fk
a
k
N l r t
N
N
N l r t
,
(5.15)
13
where the number of the wave packets going in the one single arbitrary ball layer
kA in the
second multiplier of (5.15) depends on the number of balls or wave packets placed only on the
meridian slash of the one layer of the concentric ball layer packaging which has now the view
2
2
,
,
,
,
arctan
2
0.25
,
,
k
k
k
k
N l r t
l
l l r t
l t
r
l
l l r t
l t
.
(5.16)
Therefore the full integrand now takes the form
2
2
max
2
4
2
0
3
,
16
,
n
fk
k
k
c
N
l t
m c
l
l
l t
l t
.
(5.17)
For the backward direction we also need the modified effective scale reducing in comparison
with (4.4); it has the form
1
1
2
k
l
lm
rm
v l
t l
t
,
(5.18)
which is given taking into account the discretization of the radii of spheres layers (denoted by the
k numerator) and the reversal run of time (it already has minus sign and it is thought positive).
And, therefore, at accounting of the backward motion in the all formulae must be done
substitutions
l
lm
,
(5.19)
k
k
r
rm
.
(5.20)
(The letter m entering in these designations means ‘minus sign’ which appears at the description
of the backward motion.) Of course, we should not forget to integrate the gained expression on
all the scales l to gain an estimation of the vacuum energy of this given vacuum. The dispersion
function, as was already said in the Section 2, will look so
l t
qt
,
(5.21)
the tilt of the line must correspond an average function of the all real dispersion functions. In
accordance with this imaginable existing average function we have chosen the value of the
constant
2
3
q
.
(5.22)
This choice is caused by an intuitive view that an average value of the angle
2
, in the range of
which the curves of the dispersion functions must vary, is the angle
4
. Therefore we have
14
chosen the number is close to 1 but it is smaller than unity because it is reasonable to think that
the curves of the dispersion functions are nestled to the axis of the trajectory. Note that at
0
k
min
kr
r
, this will account in the computation given in the computer program below. Like in the
article [2] the computer program is written in the “Wolfram language®”.
Like that is in [2] we need the system of the equations for the radii
kr and
k
rm for the
forward direction and the backward direction correspondingly. In the computational view they
are
1
1
1
1
2
2
1
1
k
k
k
k
l
l
r
l
l t
r
l
l t
r
r
v l t
v l t
,
(5.23)
1
1
1
1
2
2
1
1
2
k
k
k
l
rm
l
l
t l
t
rm
rm
v l
t l
t
(5.24)
1
1
2
2
1
1
2
k
l
l
l
t l
t
rm
v l
t l
t
.
The system of the equations appears to be when the values, beginning from zero to some
maximal value, appropriate to index k and the gained different equations are written down.
Actually, these equations are the quadratic equations, and we should to resolve them relatively
1
kr and
1
k
rm to write fastest algorithm. The maximal value for radius, actually, is initially set
(it is going in the external data of the task). And the computer program must solve as much of
the equations as it is needed for the radius, depending on the minimal scale and at maximal
reducing of the scale (in the maximal time, i.e. in the most distant point from the creation point
of the pair), would be smaller or equal to the maximal its value, set as the external parameter.
This condition is right because if it is executed, then the most number of wave packets (balls) of
the minimal volume at given dispersion (the coefficient of the linear dispersion equals 2 3 ) can
go in the all ball layers in whole atom. I.e. in this case we must judge maximally. However, the
maximal scale must not excess the maximal value of the radius of the atom, to stay in the frames
of the task. Thus, the radii are the functions and they have the following dependences
,
kr l t ,
(5.25)
,
k
rm
l t .
(5.26)
15
The maximal value of scale l can be found by solving the inequalities for l
min
1
2
r
l
q t l
,
(5.27)
min
min
1
1
2
2
1
1
2
l
r
l
q t l
r
v l
t l
.
(5.28)
The solutions of these inequalities are intervals of l , therefore we need to choose maximal
boundaries of the intervals and then to compare them to choose the maximal value of scale. All
the computations applied below are realized by the lattice method (numerically) to add the
computation in [2] and to fill its disadvantages. The lattice means lt flat grid, or scale-time plane
consisting rectangular grid. The inequalities (5.27) and (5.28) consist of a step for time, of the
lattice; it determines frequency of the grid and simultaneously maximal scale of the task, as it
goes in these inequalities. We also must remember that integration on scale l and division into
minimal length or scale – the Planck length of the all expression are needed. I.e.
max
min
1
...
l
P l
dl
l
,
(5.29)
where the full expression described above must stay in the brackets.
Summing up, the estimation of the vacuum energy near matter without any interactions
now takes the form
max
min
2
max
0
0
1
0
1
1
1
lim
2
2
T l
T l
l
n
cell
v m
a
v m
k
P l
N
E
N
N
E
dtd dl
l
T l T l
(5.30)
max
min
2
2
max
0
1
0
1
1
1
lim
2
2
T l
T l
l
n
cell
a
v m
k
P l
T l
Nm
N
Nm
Em
dtd dl
l
T l T l
.
Here we omitted the dependence of
,
,
k
N l r t and we designated by letter m all the functions
that relate to the backward motion having their corresponding modifications. The different from
(5.19) and (5.20) modifications are that we must replace everywhere in the dispersion function
1
2
t
t l
t
,
(5.31)
at that
0
t
, as it is always. Let us compare the real lower boundary of the vacuum energy with
the free vacuum energy of the same volume occupied by empty space in relation of real particles,
but it contains virtual particles. To do this, we need to put
,
,
0
k
l l r t
,
(5.32)
16
,
,
0
k
lm l rm t
(5.33)
in all the formulae. I.e. there is no the vacuum limitation effect anymore. Therefore the all above
formulae are going to change. We begin to perform them:
2
2
2
4
2
3
16
cell
v
c
E
m c
l
l t
,
(5.34)
2
2
,
,
arctan
2
0.25
k
k
N l r t
l
l t
r
l
l t
,
(5.35)
2
2
max
2
4
2
0
3
,
16
n
fk
k
c
N
l t
m c
l
l t
,
(5.36)
1
1
1
2
2
k
k
r
l
l t
r
l
l t
,
(5.37)
1
1
1
1
1
2
2
2
2
k
k
rm
l
l
t l
t
rm
l
l
t l
t
.
(5.38)
Here in (5.36) the mark means that the corresponding number of balls (wave packets) accounts
absence of the effect, i.e. it contains the number of balls on the meridian slash in the form (5.35),
and the mark that has the upper limit of the sum means difference of the number of ball layers
(of the whole ball of the atom) without the reducing from the theory of the effect. In this case the
inequality (5.27) remains, and the inequality (5.28) transforms in the following one
min
1
1
2
2
r
l
q t l
.
(5.39)
Thus, the estimation of the vacuum energy of the free space in atoms without nuclei
corresponding to atoms which have their nuclei takes the form
max
min
2
max
0
1
0
1
1
1
lim
2
2
T l
T l
l
n
a
cell
v
a
v
k
P l
N
E
N
N
E
dtd dl
l
T l T l
(5.40)
max
min
2
2
max
0
1
0
1
1
1
lim
2
2
T l
T l
l
n
cell
a
v
k
P l
T l
Nm
N
Nm
Em
dtd dl
l
T l T l
,
where for N we have expression (5.35), the index a in brackets means ‘atoms’ and
2
2
2
4
2
3
16
1
2
cell
v
c
Em
m c
l
l
t l
t
.
(5.41)
17
Also it is needed to assess the free vacuum energy of all empty space of the Universe.
This must be done because there have been introduced the dynamics and the dispersion which
are absented in [2]. Formulae (2.2.12) and (2.2.13) are no longer valid in this approach. Let us
imagine all empty space of the Universe as a cube, at that the volume of the space and the
volume of this cube are equal. Now, we divide this enormous cube into a large number of small
cubes – cells, in each of which the wave packet is. Such approach will allow estimating the free
vacuum energy by our method. Thus, the full number of the wave packets in all empty space in
the Universe will be
3
,
s
s
V
N
l t
l
l t
(5.42)
at account of dispersion. Knowing this number, we can use our new method with average values;
thus, the free vacuum energy takes the form
min
0
0
0
1
1
1
lim
,
T l
T l
cell
v
s
v
P l
E
N
l t E
dtd dl
l
T l T l
(5.43)
min
2
0
0
1
1
1
lim
,
T l
T l
cell
s
v
P l
T l
Nm
l t Em
dtd dl
l
T l T l
,
where
min
l
is the ultraviolet cutoff; the full number of the wave packets at the backward motion
in empty space is
3
,
1
2
s
s
V
Nm
l t
l
l
t l
t
.
(5.44)
The required infinite upper limit of the integrals on l in (5.43) can be realized in practical
calculation only for analytical integration, and for the numerical integration we will use it must
be replaced by finite limit, actually, the same as in (5.30) – max
l
.
At the next stage the wrapping vacuum coefficients have to be calculated to show that the
vacuum really reduces by matter and how much that happens. We already have everything to do
this. Let it be designated
max
min
2
max
0
1
0
1
1
1
lim
2
2
T l
T l
l
n
cell
a
v m
k
P l
N
BVM
N
N
E
dtd dl
l
T l T l
(5.45)
max
min
2
2
max
0
1
0
1
1
1
lim
2
2
T l
T l
l
n
cell
a
v m
k
P l
T l
Nm
N
Nm
Em
dtd dl
l
T l T l
,
max
min
2
max
0
1
0
1
1
1
lim
2
2
T l
T l
l
n
cell
a
v
k
P l
N
BV
N
N
E
dtd dl
l
T l T l
(5.46)
18
max
min
2
2
max
0
1
0
1
1
1
lim
2
2
T l
T l
l
n
cell
a
v
k
P l
T l
Nm
N
Nm
Em
dtd dl
l
T l T l
,
where ‘BVM’ means ‘boundary vacuum plus matter’ and it is the right hand side of (5.30) and
‘BV’ means ‘boundary vacuum’ and it is the right hand side of (5.39). Then the general wrapping
vacuum coefficient in this model takes the form
BVM
BV
.
(5.47)
Actually, we divide the inequalities (5.30) and (5.40) into each other and equate the fraction of
the energies in the left hand side to .
6. The final results of the computation
This is final stage of the work and here we provide all numerical values of every
parameter of the computation and its results before presenting of the computer program itself.
All the data will be resulted in the following Table 3, Table 4, Table 5 and Table 6; for the
neutrinos results are given in the opposite form in comparison with the masses are, i.e. the lower
limit, besides our usual estimation scheme (where all estimations are lower limits or boundaries),
this is done, proceeding from the complex dependence found in the computation: the free
vacuum energy of the specific matter (particle) is not always directly proportional or changes
like the mass of the corresponding real particle, it can sometimes have an opposite dependence;
(pm means picometer,
12
1pm
10
m
). The Table 2 contains the information needed for the
calculation of the volume of empty space of the Universe. The computation has been carried out
at the following values of the fundamental constants (1) in SI units and the values of the spatial
and the temporal grids (2), needed for the numerical computation: (1)
34
6.6260755 10
h
is the
Planck constant;
299792458
c
is the speed of light in vacuum;
35
1.616255 10
Pl
is the
Planck length;
80
3.6 10
U
V
is the volume of the observable Universe;
79
7.39 10
a
N
is the
number of atoms in the Universe, also called the Eddington number;
10
1
s
U
Ai
i
a
i
V
V
V
N
is
the volume of the empty Universe, i.e. the volume of the empty space of the Universe, which
defines via the volumes of each type of atoms
Ai
V , the fractions of presence in number relation
of each atom in the full number of atoms
i
and the full number of atoms in the Universe; here
we considered 10 most common atoms in the Milky Way Galaxy or in the Universe in order of
decreasing of their abundance,
1,2,...,10
i
;
Table 2 (Ten most common elements in the Universe) [6, 7, 8]
Symbol
3
32
m
10
Ai
V
i
1
1H
6.12611
0.739
4
2He
12.0599
0.24
15
8O
90.059
0.0104
12
6C
143.257
0.0046
19
20
10Ne
22.5659
0.0013
55
26Fe
1148.99
0.0011
14
7 N
114.616
0.00096
28
14Si
557.109
0.00065
24
12Mg
1413.3
0.00058
32
16S
418.46
0.00044
Total:
0.99906
(2)
50
NMAX
,
104
MMAX
are the number of segments on the scale and the time
axes correspondingly;
24
max
1.54803 10
s
e l
,
25
max
3.29625 10
s
u l
are the duration of
one single time interval for the electron and the u quark correspondingly;
11
max
2.00008 10
m
l
,
11
max
2 10
m
l
are the maximal values of the scale variable for the electron and the u quark
correspondingly, and all remains the same whatever the effect is or it is no;
20
min
10
P
l
l
is the
general minimal scale;
11
min
10
m
r
is the minimal radius of the sphere, which is the minimal
boundary of the ball layer, where the limitation vacuum effect takes place in atom. The volumes
of the atoms, where the effect takes place can be computed by the formula
3
3
max
min
4
3
A
V
r
r
,
and they have the following values:
1
32
3
1H
6.12611 10
m
A
V
,
4
31
3
2He
1.20599 10
m
A
V
.
The program is separated on the six subprograms, therefore in each subprogram its own values
of the spatial and temporal grids are used. In the computer program the following values for the
grids have been used:
2
3
4
l
d
d
d
d
NMAX
, and
2
3
4
1
2
X
X
X
X
t
MMAX
, where
max
min
l
l
l
and
,
X
e u
. The numbers in the indices and the absence of them mean
belonging to one of these subprograms. More accurately one can see that in the program itself.
For we have two values for the maximal scale: for the electron and for the u quark, there are the
two considering scale range for each particle:
max
min
l
l
l
and
2
max 2
min
l
l
l
. Also for the
other components of the program:
max3
max4
l
l
and it can be equal
max
l
or
max 2
l
. Therefore there
are the few values of the small numerical constant , which has dimensionality of time on each
stage of the computation in the computer program:
min
1
T l
NMAXd
MMAX
,
2
min
2
2
1
X
T l
NMAXd
MMAX
,
3
min
3
3
1
X
T l
NMAXd
MMAX
,
4
min
4
4
1
X
T l
NMAXd
MMAX
. All the values are rounded up to five meaning digits:
Table 3 (Two most common elements in the Universe) [6, 7, 8]
Symbol
Exp
max
r
(pm)
error
±5pm, *
Theor
max
r
(pm)
Abundance
on number
in %
Abundance
on number
in quantity
×
75
10
Particles
pairs ( p
means
p p )
0
133
J
10
v
E
0
103
J
10
v m
E
103
J
10
BV
1
1H
25
73.9
73900
u
1004.48
40.6758
40.6434
20
e
7.07064
7.91452
7.90539
4
2He
*31
24
24000
u
-
32.6181
32.5837
e
-
5.18525
5.18142
Total:
97.9
97900
1011.55
86.3937
86.3139
These two elements were estimated spectroscopically in the Milky Way Galaxy, but if we take a
condition that our Galaxy is common, typical and usual in the Universe, and according the theory
of the development of the Universe [9 – 14], this is plausible, we can generalize them and the all
data on the entire Universe.
Table 4 (The estimation for the neutrinos)
e
Total:
0
136
J
10
v
E
150.26
2.12968
1.30695
153.697
0
133
2
J
10
v
E
2.40735
2.4074
2.74955
7.5643
0
0
136
J
2
J
10
v
v
E
E
150.258
2.12727
1.3042
153.689
Table 5 (The previous computation for the elements and comparison of the two calculations)
Symbol
Particles
pairs ( p
means
p p )
[2]
0
133
J
10
v
E
[2]
0
103
J
10
v m
E
[2]
101
J
10
BV
0
0
133
[2]
J
10
v
v
E
E
0
0
103
[2]
J
10
v m
v m
E
E
103
2
J
10
BV
BV
1
1H
u
2.4145
7
13.8119
5.3125
7
1002.07
26.8639
40.5903
e
2.4076
9
3.08206
5.2689
5
4.66295
4.83246
7.8527
4
2He
u
-
12.288
3.3964
-
20.3301
32.5497
e
-
2.72712
3.3685
-
2.45813
5.14774
Total:
4.8222
6
31.9091
17.346
4
1006.73
54.4846
86.1404
Table 6 (The wrapping vacuum coefficients)
Symbol
Particles pairs ( p
means p p )
2
1
1H
u
1.0008
259.985
e
1.00115
58.4948
4
2He
u
1.00106
361.795
e
1.00074
80.9595
Total:
1.00092
183.952
I n [] : = Tm1 =
абсолютное значение
AbsoluteTime[];
I n [] : = me = 9.11 × 10-31;
I n [] : = mμ = 1.88356 × 10-28;
mτ = 3.16751 × 10-27;
uplmνe = 3.6 × 10-36;
I n [] : = uplmνμ = 3.39 × 10-31;
I n [] : = uplmντ = 3.2445 × 10-29;
I n [] : = mu = 4.2785 × 10-30;
I n [] : = md = 8.4675 × 10-30;
I n [] : = ms = 1.78266 × 10-28;
I n [] : = mc = 2.22833 × 10-27;
I n [] : = mb = 8.46765 × 10-27;
I n [] : = mt = 3.052 × 10-25;
I n [] : = VU = 3.6 × 1080;
Na = 7.39 × 1079;
VA =
4
3
π rmax
3
- rmin
3
;
m0 = me;
c = 299 792 458;
h = 6.6260755 × 10-34;
ℏ=
h
2 π
;
I n [] : = VAp1 = 6.12611 × 10-32;
VAp2 = 1.20599 × 10-31;
VAp3 = 9.0059 × 10-31;
VAp4 = 1.43257 × 10-30;
VAp5 = 2.25659 × 10-31;
VAp6 = 1.14899 × 10-29;
VAp7 = 1.14616 × 10-30;
VAp8 = 5.57109 × 10-30;
VAp9 = 1.4133 × 10-29;
VAp10 = 4.1846 × 10-30;
I n [] : = Δ1 = 0.739;
Δ2 = 0.24;
Δ3 = 0.0104;
Δ4 = 0.0046;
Δ5 = 0.0013;
Δ6 = 0.0011;
Δ7 = 0.00096;
Δ8 = 0.00065;
Δ9 = 0.00058;
Δ10 = 0.00044;
I n [] : = rmin := 10-11
I n [] : = lmin := 1020 lP
I n [] : = lP := 1.616255 × 10-35
Note: lmax depends on δ and vice versa (including dependence on MMAX), see the inequalities
below.
I n [] : = lmax := 2.00008 × 10-11
I n [] : = lmax2 := 2 × 10-11
I n [] : = ϵ = T[lmin + NMAX d] - (MMAX - 1) δ;
I n [] : = ϵ2 = T[lmin + NMAX d2] - (MMAX - 1) δ2;
I n [] : = ϵ3 = T[lmin + NMAX d3] - (MMAX - 1) δ3;
I n [] : = ϵ4 = T[lmin + NMAX d4] - (MMAX - 1) δ4;
I n [] : = rHmax = 2.5 × 10-11;
I n [] : = rHemax = 3.1 × 10-11;
I n [] : = rOmax = 6 × 10-11;
I n [] : = rCmax = 7 × 10-11;
I n [] : = rNemax = 3.8 × 10-11;
I n [] : = rFemax = 1.4 × 10-10;
I n [] : = rNmax = 6.5 × 10-11;
I n [] : = rSimax = 1.1 × 10-10;
I n [] : = rMgmax = 1.5 × 10-10;
I n [] : = rSmax = 10-10;
I n [] : = rmax = rHmax;
I n [] : = q :=
2
3
I n [] : = Δt[l_] :=
l
c
×
3 + 16 × m02 c2
ℏ2
l2
3 + 16 × m02 c2
ℏ2
l2
2 The First Part of the Program New SCCPERVMUR.nb
I n [] : = v[l_] :=
c
1 + 16
3 × m02 c2
ℏ2
l2
I n [] : = δl[t_] := q t
I n [] : = ΔL := lmax - lmin
I n [] : = ΔL2 := lmax2 - lmin
I n [] : = d :=
ΔL
50
I n [] : = d2 :=
ΔL2
50
MMAX:=104
I n [] : = δ := 1.54803 × 10-24
I n [] : = δ2 := 1.54803 × 10-24
I n [] : = δ =
1
2
×
1
104
×
lmin + ΔL
c
×
3 + 16 × m02 c2
ℏ2
(lmin + ΔL)2
3 + 16 × m02 c2
ℏ2
(lmin + ΔL)2
I n [] : = δ2 =
1
2
×
1
104
×
lmin + ΔL2
c
×
3 + 16 × m02 c2
ℏ2
(lmin + ΔL2)2
3 + 16 × m02 c2
ℏ2
(lmin + ΔL2)2
I n [] : = NMAX =
округление вн
Floor
ΔL
d
O u t [] =
50
I n [] : = "NMAX2==NMAX"
I n [] : = MMAX =
округление вверх
Ceiling
1
2
1
δ
×
lmin + ΔL
c
×
3 + 16 × m02 c2
ℏ2
(lmin + ΔL)2
3 + 16 × m02 c2
ℏ2
(lmin + ΔL)2
O u t [] =
104
"MMAX2==MMAX"
I n [] : = MMAX δ;
I n [] : = n0 := 0
I n [] : = MMAXN0 =
1
2
1
δ
×
lmin
c
×
3 + 16 × m02 c2
ℏ2
lmin2
3 + 16 × m02 c2
ℏ2
lmin2
O u t [] =
1.00531
The First Part of the Program New SCCPERVMUR.nb 3
I n [] : = m00 =
округление вниз
Floor[MMAXN0]
O u t [] =
1
I n [] : =
привести
Reducermin ≥
1
2 абсолютное значение
Abs[l + q Δt[l]], l,
множеств
Reals
Reduce: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
O u t [] =
-2. × 10-11 ≤l ≤2. × 10-11
привести
Reducermin ≥
1
2 абсолютное значение
Absl -
l
1 +
rmin
v[l]
1
2 Δt[l]
+
1
2
q Δt[l], l,
множеств
Reals
Reduce: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
O u t [] =
-1.99992 × 10-11 ≤l ≤2.00008 × 10-11
I n [] : =
привести
Reducermax ≥
1
2 абсолютное значение
Abs[l + q Δt[l]], l,
множеств
Reals
Reduce: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
O u t [] =
-5. × 10-11 ≤l ≤5. × 10-11
привести
Reducermax ≥
1
2 абсолютное значение
Absl -
l
1 +
rmax
v[l]
1
2 Δt[l]
+
1
2
q Δt[l], l,
множеств
Reals
Reduce: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
O u t [] =
-4.99997 × 10-11 ≤l ≤5.00003 × 10-11
I n [] : =
⋯
Do[
условный⋯
If[i 0,
печатать
Print[1];
j0 = 1;,
пустой
Null];
условный оператор
If[i ≥1 && i ≤10,
печатать
Print[928 i];
ji = 928 i;,
пустой
Null], {i, 0, 10}]
4 The First Part of the Program New SCCPERVMUR.nb
1
928
1856
2784
3712
4640
5568
6496
7424
8352
9280
I n [] : =
таблица значений
Table[r10,n,m = rmin;, {n, 0, NMAX}, {m, 0, MMAX}];
I n [] : =
таблица значений
Table[r0,n,m = rmin;, {n, 0, NMAX}, {m, 0, MMAX}];
I n [] : =
цикл ДЛЯ
Fora = 0, ra,n0,m00 ≤rmax, a++,
условный⋯
If[a 0,
печатать
Print["a=", 0, " ", "ra=",
численно⋯
N[rmin]],
пустой
Null];
оператор цикла
Doln = lmin + n d;
оператор цикла
Dotm = m δ;
r1a+1,n,m =
1
4 (r1a,n,m + tm v[ln])
ln (2 r1a,n,m + tm v[ln]) + 2 (r1a,n,m + tm v[ln]) (r1a,n,m - tm v[ln] + δl[tm]) + √8 tm
v[ln] (r1a,n,m + tm v[ln]) (ln r1a,n,m + 2 (r1a,n,m + tm v[ln]) (r1a,n,m + δl[tm])) +
(ln (2 r1a,n,m + tm v[ln]) + 2 (r1a,n,m + tm v[ln]) (r1a,n,m - tm v[ln] + δl[tm]))2;
ra+1,n,m = r1a+1,n,m;
r1a,n,m =.;
tm =., {m, 0, MMAX};
ln =., {n, 0, NMAX};
условный оператор
If[a > 0 && (a j0 || a j1 || a j2 || a j3 || a j4 || a j5 || a j6 || a j7 || a
j8 || a j9 || a j10),
печатать
Print["a=", a, " ", "ra,n0,m00=", ra,n0,m00],
пустой
Null]//
затраченно
Timing
The First Part of the Program New SCCPERVMUR.nb 5
a=0 ra=1. × 10-11
a=1 ra,n0,m00=1.00016 × 10-11
a=928 ra,n0,m00=1.14998 × 10-11
a=1856 ra,n0,m00=1.29996 × 10-11
a=2784 ra,n0,m00=1.44995 × 10-11
a=3712 ra,n0,m00=1.59993 × 10-11
a=4640 ra,n0,m00=1.74992 × 10-11
a=5568 ra,n0,m00=1.8999 × 10-11
a=6496 ra,n0,m00=2.04989 × 10-11
a=7424 ra,n0,m00=2.19987 × 10-11
a=8352 ra,n0,m00=2.34986 × 10-11
a=9280 ra,n0,m00=2.49984 × 10-11
O u t [] =
{5042.39, Null}
I n [] : = s = a - 1
O u t [] =
9280
I n [] : = r9280,n0,m00
O u t [] =
2.49984 × 10-11
I n [] : =
таблица зн⋯
Table[Rk1 =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + n1 d, m1 δ}, rk1,n1,m1}, {n1, 0, NMAX}, {m1, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];, {k1, 1, s}];
Rk1[l, t]
I n [] : =
⋯
Do[
печатать
Print[10 i]; fi = 10 i;, {i, 1, 5}]
10
20
30
40
50
I n [] : =
оператор цикла
Dola1 = lmin + a1 d;
условный оператор
If[a1 0 || a1 1,
печатать
Print["a1=", a1, " ", "la1=", la1],
условный оператор
If[a1 f1 || a1 f2 ||
a1 f3 || a1 f4 || a1 f5,
печатать
Print["a1=", a1, " ", "la1=", la1],
пустой
Null],
пустой
Null];
оператор цикла
Dotb1 = b1 δ;
6 The First Part of the Program New SCCPERVMUR.nb
условный оператор
If[(b1 0 || b1 1 || b1 MMAX) && (a1 0 || a1 1 || a1 f1 || a1 f2 ||
a1 f3 || a1 f4 || a1 f5),
печатать
Print["b1=", b1, " ", "tb1=", tb1],
пустой
Null];
IdJ1a1,b1 =
1
2
π
ArcTan
la1-
la1 tb1
tb1+
rmin
v[la1]
+δl[tb1]
2
rmin
2
-
1
4
la1-
la1 tb1
tb1+
rmin
v[la1]
+δl[tb1]
2
2
-
π
ArcTan
la1-
la1 tb1
tb1+
rmin
v[la1]
+δl[tb1]
2
rmin
2
-
1
4
la1-
la1 tb1
tb1+
rmin
v[la1]
+δl[tb1]
2
+
2
ℏ2 c2
16
×
3
la1 -
la1 tb1
tb1+
rmin
v[la1]
+ δl[tb1]
2 + m02 c4 +
k2=1
s
1
2
π
ArcTan
la1-
la1 tb1
tb1+
Rk2[la1,tb1]
v[la1]
+δl[tb1]
2
(Rk2[la1,tb1])2-
1
4
la1-
la1 tb1
tb1+
Rk2[la1,tb1]
v[la1]
+δl[tb1]
2
2
-
π
ArcTan
la1-
la1 tb1
tb1+
Rk2[la1,tb1]
v[la1]
+δl[tb1]
2
(Rk2[la1,tb1])2-
1
4
la1-
la1 tb1
tb1+
Rk2[la1,tb1]
v[la1]
+δl[tb1]
2
+ 2
ℏ2 c2
16
×
3
la1 -
la1 tb1
tb1+
Rk2[la1,tb1]
v[la1]
+ δl[tb1]
2 + m02 c4
;
tb1 =., {b1, 0, MMAX};
l
{a1
0
NMAX}
The First Part of the Program New SCCPERVMUR.nb 7
a1=0 la1=1.61626 × 10-15
b1=0 tb1=0.
b1=1 tb1=1.54803 × 10-24
b1=104 tb1=1.60995 × 10-22
a1=1 la1=4.016 × 10-13
b1=0 tb1=0.
b1=1 tb1=1.54803 × 10-24
b1=104 tb1=1.60995 × 10-22
a1=10 la1=4.00145 × 10-12
b1=0 tb1=0.
b1=1 tb1=1.54803 × 10-24
b1=104 tb1=1.60995 × 10-22
a1=20 la1=8.00129 × 10-12
b1=0 tb1=0.
b1=1 tb1=1.54803 × 10-24
b1=104 tb1=1.60995 × 10-22
a1=30 la1=1.20011 × 10-11
b1=0 tb1=0.
b1=1 tb1=1.54803 × 10-24
b1=104 tb1=1.60995 × 10-22
a1=40 la1=1.6001 × 10-11
b1=0 tb1=0.
b1=1 tb1=1.54803 × 10-24
b1=104 tb1=1.60995 × 10-22
a1=50 la1=2.00008 × 10-11
b1=0 tb1=0.
b1=1 tb1=1.54803 × 10-24
b1=104 tb1=1.60995 × 10-22
I n [] : = IdJ11 =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + a11 d, b11 δ}, IdJ1a11,b11}, {a11, 0, NMAX}, {b11, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];
IdJ11[l, t]
I n [] : = t := 4 δ
8 The First Part of the Program New SCCPERVMUR.nb
I n [] : =
график функции
Plot[IdJ11[l, t], {l, lmin, lmax},
отображаемый диапазон граф
PlotRange {-10, 10}]
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-10
-5
5
10
From here we can see that this computation is correct.
I n [] : = t =.
I n [] : = T[l_] :=
1
2
Δt[l]
"t>0"
I n [] : = Δlmin :=
l t
t +
rmin
v[l]
I n [] : = Δlmmin :=
l 1
2 Δt[l] - t
1
2 Δt[l] - t +
rmin
v[l]
I n [] : = N1min :=
π
ArcTan
l-Δlmin+δl[t]
2
rmin
2
-
1
4 (l-Δlmin+δl[t])2
I n [] : = N1mmin :=
π
ArcTan
l-Δlmmin+δl
1
2 Δt[l]+t
2
rmin
2
-
1
4 l-Δlmmin+δl
1
2 Δt[l]+t
2
The First Part of the Program New SCCPERVMUR.nb 9
I n [] : = l = lmax;
I n [] : = t =
1
2
Δt[l];
I n [] : = N1min
O u t [] =
2.00106
I n [] : = l =.
I n [] : = l = lmin;
I n [] : = N1min
O u t [] =
39 062.6
I n [] : = l =.
I n [] : = t =.
I n [] : = l = 1.99999 × 10-11;
I n [] : = t =
1
2
Δt[l];
I n [] : = N1mmin
O u t [] =
2.00403
It must be equal to 2.
I n [] : = l =.
I n [] : = l = lmin;
I n [] : = N1mmin
O u t [] =
38 689.2
I n [] : = l =.
I n [] : = t =.
The results are different due to the dispersion.
I n [] : =
привести
Reducermin
2
-
1
4
l - Δlmmin + δl
1
2
Δt[l] + t
2
≥0, l
Reduce: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
O u t [] =
-2. × 10-11 ≤l ≤2. × 10-11
The extremal upper value is to be when the equality to zero is executed.
10 The First Part of the Program New SCCPERVMUR.nb
I n [] : =
численное решение уравнений
NSolvermin
2
-
1
4
l - Δlmmin + δl
1
2
Δt[l] + t
2
0, l,
множеств
Reals
O u t [] =
l -2.10199 × 10-11, l 2.10199 × 10-11
This result is abnormal.
I n [] : = l = 1.99999 × 10-11;
I n [] : =
упростить
Simplifyrmin
2
-
1
4
l - Δlmmin + δl
1
2
Δt[l] + t
2
≥0
O u t [] =
True
I n [] : =
оператор цикла
Dola2 = lmin + a2 d;
условный оператор
If[a2 0 || a2 1,
печатать
Print["a2=", a2, " ", "la2=", la2],
условный оператор
If[a2 f1 || a2 f2 ||
a2 f3 || a2 f4 || a2 f5,
печатать
Print["a2=", a2, " ", "la2=", la2],
пустой
Null]];
оператор цикла
Doτb2 = b2 δ;
условный оператор
If[(a2 0 || a2 1) && (b2 0 || b2 1 || b2 MMAX),
печатать
Print["b2=", b2,
" ", "τb2=", τb2],
условный оператор
If[(a2 f1 || a2 f2 || a2 f3 || a2 f4 || a2 f5) &&
(b2 0 || b2 1 || b2 MMAX),
печатать
Print["b2=", b2, " ", "τb2=", τb2],
пустой
Null]];
J1a2,b2 =
1
T[la2] - τb2
1
T[la2] квадратурное интегрирование
NIntegrate[IdJ11[la2, t], {t, τb2, T[la2]}];
τb2 =., {b2, 0, MMAX};
la2 =., {a2, 0, NMAX}
The First Part of the Program New SCCPERVMUR.nb 11
a2=0 la2=1.61626 × 10-15
b2=0 τb2=0.
b2=1 τb2=1.54803 × 10-24
b2=104 τb2=1.60995 × 10-22
a2=1 la2=4.016 × 10-13
b2=0 τb2=0.
b2=1 τb2=1.54803 × 10-24
b2=104 τb2=1.60995 × 10-22
a2=10 la2=4.00145 × 10-12
b2=0 τb2=0.
b2=1 τb2=1.54803 × 10-24
b2=104 τb2=1.60995 × 10-22
a2=20 la2=8.00129 × 10-12
b2=0 τb2=0.
b2=1 τb2=1.54803 × 10-24
b2=104 τb2=1.60995 × 10-22
a2=30 la2=1.20011 × 10-11
b2=0 τb2=0.
b2=1 τb2=1.54803 × 10-24
b2=104 τb2=1.60995 × 10-22
a2=40 la2=1.6001 × 10-11
b2=0 τb2=0.
b2=1 τb2=1.54803 × 10-24
b2=104 τb2=1.60995 × 10-22
a2=50 la2=2.00008 × 10-11
b2=0 τb2=0.
b2=1 τb2=1.54803 × 10-24
b2=104 τb2=1.60995 × 10-22
I n [] : = δ
O u t [] =
1.54803 × 10-24
I n [] : = T[lmin]
O u t [] =
1.55625 × 10-24
I n [] : = J11 =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + a22 d, b22 δ}, J1a22,b22}, {a22, 0, NMAX}, {b22, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];
12 The First Part of the Program New SCCPERVMUR.nb
J11[l, τ]
I n [] : =
⋯
Do[
условный оператор
If[(a f1 || a f2 || a f3 || a f4 || a f5),
печатать
Print["a=", a],
пустой
Null];
la = lmin + a d;
Int1a =
квадратурное интегрирование
NIntegrate[J11[la, τ], {τ, 0, T[la] - ϵ}];
la =., {a, 0, NMAX}]
a=10
a=20
a=30
a=40
a=50
I n [] : = F1 =
интерполировать
Interpolation[
таблица значений
Table[{lmin + a3 d, Int1a3}, {a3, 0, NMAX}],
порядок интерполяции
InterpolationOrder 5];
I n [] : = I1 =
квадратурное интегрирование
NIntegrate[F1[l], {l, lmin, lmax}]
O u t [] =
8.64509 × 10-12
I n [] : =
численное приближение
N
1
lP
Na I1
O u t [] =
3.95279 × 10103
A new interesting structure is found, it must be investigated.
I n [] : = t := 4 δ
I n [] : =
график функции
PlotIdJ11[l, t], l, lmin, 10-12,
отображаемый диапазон граф
PlotRange {-10, 10}
O u t [] =
2×10-13
4×10-13
6×10-13
8×10-13
1×10-12
-10
-5
5
10
Following from the plot, we must write
I n [] : =
найти корень
FindRootIdJ11[l, t] 0, l, 4 × 10-13
O u t [] =
l 4.06607 × 10-13 + 0.
The First Part of the Program New SCCPERVMUR.nb 13
I n [] : =
найти корень
FindRootIdJ11[l, t] 0, l, 8 × 10-13
O u t [] =
l 7.83835 × 10-13 + 0.
I n [] : = t =.
I n [] : =
график функции
Plot[F1[l], {l, lmin, lmax},
отображаемый диапазон графика
PlotRange {-1.5, 1.5}]
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-1.5
-1.0
-0.5
0.5
1.0
1.5
I n [] : =
график функции
PlotF1[l], l, lmin, 2 × 10-12,
отображаемый диапазон графика
PlotRange {-1.5, 1.5}
O u t [] =
5.0×10-13
1.0×10-12
1.5×10-12
2.0×10-12
-1.5
-1.0
-0.5
0.5
1.0
1.5
I n [] : = u0 := n0
I n [] : = w00 := m00
I n [] : =
таблица значений
Table[r1m0,u,w = rmin;, {u, 0, NMAX}, {w, 0, MMAX}];
I n [] : =
таблица значений
Table[rm0,u,w = rmin;, {u, 0, NMAX}, {w, 0, MMAX}];
14 The First Part of the Program New SCCPERVMUR.nb
I n [] : =
цикл ДЛЯ
Fora4 = 0, rma4,u0,w00 ≤rmax, a4++,
условный ⋯
If[a4 0,
печатать
Print["a4=", 0, " ", "rma4=",
численно⋯
N[rmin]],
пустой
Null];
оператор цикла
Dolmu = lmin + u d2;
оператор цикла
Dotmw = -w δ2;
r1m1+a4,u,w =
1
8 r1ma4,u,w + 4 v[lmu] (2 tmw + Δt[lmu])
4 r1ma4,u,w
2
+ 4 r1ma4,u,w
lmu + δl-tmw +
Δt[lmu]
2
+ v[lmu] (2 tmw + Δt[lmu])
lmu +
2 δl-tmw +
Δt[lmu]
2
- v[lmu] (2 tmw + Δt[lmu])
+ 8 v[lmu] (2 tmw + Δt[lmu])
(2 r1ma4,u,w + v[lmu] (2 tmw + Δt[lmu]))
lmu r1ma4,u,w +
r1ma4,u,w + δl-tmw +
Δt[lmu]
2
(2 r1ma4,u,w + v[lmu] (2 tmw + Δt[lmu]))
+
2 r1ma4,u,w +
2 δl-tmw +
Δt[lmu]
2
- v[lmu] (2 tmw + Δt[lmu])
(2 r1ma4,u,w + v[lmu]
(2 tmw + Δt[lmu])) + lmu (4 r1ma4,u,w + v[lmu] (2 tmw + Δt[lmu]))
2
;
rma4+1,u,w = r1ma4+1,u,w;
r1ma4,u,w =.;
tmw =., {w, 0, MMAX};
lmu =., {u, 0, NMAX};
условный оператор
If[a4 > 0 && (a4 j0 || a4 j1 || a4 j2 || a4 j3 ||
a4 j4 || a4 j5 || a4 j6 || a4 j7 || a4 j8 || a4 j9 || a4 j10),
печатать
Print["a4=", a4, " ", "rma4,u0,w00=", rma4,u0,w00],
пустой
Null]//
затраченное время
Timing
The First Part of the Program New SCCPERVMUR.nb 15
a4=0 rma4=1. × 10-11
a4=1 rma4,u0,w00=1.00016 × 10-11
a4=928 rma4,u0,w00=1.14999 × 10-11
a4=1856 rma4,u0,w00=1.29998 × 10-11
a4=2784 rma4,u0,w00=1.44997 × 10-11
a4=3712 rma4,u0,w00=1.59995 × 10-11
a4=4640 rma4,u0,w00=1.74994 × 10-11
a4=5568 rma4,u0,w00=1.89993 × 10-11
a4=6496 rma4,u0,w00=2.04992 × 10-11
a4=7424 rma4,u0,w00=2.19991 × 10-11
a4=8352 rma4,u0,w00=2.3499 × 10-11
a4=9280 rma4,u0,w00=2.49988 × 10-11
O u t [] =
{9193.53, Null}
I n [] : = sm = a4 - 1
O u t [] =
9280
I n [] : = rm9280,u0,w00
O u t [] =
2.49988 × 10-11
I n [] : =
таблица значений
Table[
Rmk3 =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + u1 d2, -w1 δ2}, rmk3,u1,w1}, {u1, 0, NMAX},
{w1, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];, {k3, 1, sm}];
Rmk3[l, t]
I n [] : =
оператор цикла
Dola5 = lmin + a5 d2;
условный оператор
If[a5 0 || a5 1,
печатать
Print["a5=", a5, " ", "la5=", la5],
условный оператор
If[a5 f1 || a5 f2 ||
a5 f3 || a5 f4 || a5 f5,
печатать
Print["a5=", a5, " ", "la5=", la5],
пустой
Null],
пустой
Null];
оператор цикла
Dotb5 = -b5 δ2;
условный оператор
If[(b5 0 || b5 1 || b5 MMAX) && (a5 0 || a5 1 || a5 f1 || a5 f2 ||
a5 f3 || a5 f4 || a5 f5),
печатать
Print["b5=", b5, " ", "tb5=", tb5],
пустой
Null];
16 The First Part of the Program New SCCPERVMUR.nb
IdJ2a5,b5 =
1
2
π
ArcTan
la5-
la5
1
2 Δt[la5]+tb5
1
2 Δt[la5]+tb5+
rmin
v[la5]
+δl
1
2 Δt[la5]-tb5
2
rmin
2
-
1
4
la5-
la5
1
2 Δt[la5]+tb5
1
2 Δt[la5]+tb5+
rmin
v[la5]
+δl
1
2 Δt[la5]-tb5
2
2
-
π
ArcTan
la5-
la5
1
2 Δt[la5]+tb5
1
2 Δt[la5]+tb5+
rmin
v[la5]
+δl
1
2 Δt[la5]-tb5
2
rmin
2
-
1
4
la5-
la5
1
2 Δt[la5]+tb5
1
2 Δt[la5]+tb5+
rmin
v[la5]
+δl
1
2 Δt[la5]-tb5
2
+ 2
ℏ2 c2
16
×
3
la5 -
la5
1
2 Δt[la5]+tb5
1
2 Δt[la5]+tb5+
rmin
v[la5]
+ δl1
2 Δt[la5] - tb5
2 + m02 c4 +
k4=1
sm
1
2
π
ArcTan
la5-
la5
1
2 Δt[la5]+tb5
1
2 Δt[la5]+tb5+
Rmk4[la5,tb5]
v[la5]
+δl
1
2 Δt[la5]-tb5
2
(Rmk4[la5,tb5])2-
1
4
la5-
la5
1
2 Δt[la5]+tb5
1
2 Δt[la5]+tb5+
Rmk4[la5,tb5]
v[la5]
+δl
1
2 Δt[la5]-tb5
2
2
-
π
ArcTan
la5-
la5
1
2 Δt[la5]+tb5
1
2 Δt[la5]+tb5+
Rmk4[la5,tb5]
v[la5]
+δl
1
2 Δt[la5]-tb5
2
(Rmk4[la5,tb5])2-
1
4
la5-
la5
1
2 Δt[la5]+tb5
1
2 Δt[la5]+tb5+
Rmk4[la5,tb5]
v[la5]
+δl
1
2 Δt[la5]-tb5
2
+ 2
The First Part of the Program New SCCPERVMUR.nb 17
ℏ2 c2
16
×
3
la5 -
la5
1
2 Δt[la5]+tb5
1
2 Δt[la5]+tb5+
Rmk4[la5,tb5]
v[la5]
+ δl1
2 Δt[la5] - tb5
2 + m02 c4
;
tb5 =., {b5, 0, MMAX};
la5 =., {a5, 0, NMAX}
a5=0 la5=1.61626 × 10-15
b5=0 tb5=0.
b5=1 tb5=-1.54803 × 10-24
b5=104 tb5=-1.60995 × 10-22
a5=1 la5=4.01584 × 10-13
b5=0 tb5=0.
b5=1 tb5=-1.54803 × 10-24
b5=104 tb5=-1.60995 × 10-22
a5=10 la5=4.00129 × 10-12
b5=0 tb5=0.
b5=1 tb5=-1.54803 × 10-24
b5=104 tb5=-1.60995 × 10-22
a5=20 la5=8.00097 × 10-12
b5=0 tb5=0.
b5=1 tb5=-1.54803 × 10-24
b5=104 tb5=-1.60995 × 10-22
a5=30 la5=1.20006 × 10-11
b5=0 tb5=0.
b5=1 tb5=-1.54803 × 10-24
b5=104 tb5=-1.60995 × 10-22
a5=40 la5=1.60003 × 10-11
b5=0 tb5=0.
b5=1 tb5=-1.54803 × 10-24
b5=104 tb5=-1.60995 × 10-22
a5=50 la5=2. × 10-11
b5=0 tb5=0.
b5=1 tb5=-1.54803 × 10-24
b5=104 tb5=-1.60995 × 10-22
18 The First Part of the Program New SCCPERVMUR.nb
I n [] : = IdJ22 =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + a55 d2, -b55 δ2}, IdJ2a55,b55},
{a55, 0, NMAX}, {b55, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];
IdJ22[l, t]
I n [] : = t := -4 δ2
I n [] : =
график функции
Plot[IdJ22[l, t], {l, lmin, lmax2},
отображаемый диапазон граф
PlotRange {-10, 10}]
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-10
-5
5
10
From here we can see that this computation is correct.
I n [] : = t =.
I n [] : =
оператор цикла
Dola6 = lmin + a6 d2;
условный оператор
If[a6 0 || a6 1,
печатать
Print["a6=", a6, " ", "la6=", la6],
условный оператор
If[a6 f1 || a6 f2 ||
a6 f3 || a6 f4 || a6 f5,
печатать
Print["a6=", a6, " ", "la6=", la6],
пустой
Null]];
оператор цикла
Doτb6 = b6 δ2;
условный оператор
If[(a6 0 || a6 1) && (b6 0 || b6 1 || b6 MMAX),
печатать
Print["b6=", b6,
" ", "τb6=", τb6],
условный оператор
If[(a6 f1 || a6 f2 || a6 f3 || a6 f4 || a6 f5) &&
(b6 0 || b6 1 || b6 MMAX),
печатать
Print["b6=", b6, " ", "τb6=", τb6],
пустой
Null]];
J2a6,b6 =
1
T[la6] - τb6
1
T[la6] квадратурное интегрирование
NIntegrate[IdJ22[la6, t], {t, T[la6] + τb6, 2 T[la6]}];
τb6 =., {b6, 0, MMAX};
la6 =., {a6, 0, NMAX}
The First Part of the Program New SCCPERVMUR.nb 19
a6=0 la6=1.61626 × 10-15
b6=0 τb6=0.
b6=1 τb6=1.54803 × 10-24
b6=104 τb6=1.60995 × 10-22
a6=1 la6=4.01584 × 10-13
b6=0 τb6=0.
b6=1 τb6=1.54803 × 10-24
b6=104 τb6=1.60995 × 10-22
a6=10 la6=4.00129 × 10-12
b6=0 τb6=0.
b6=1 τb6=1.54803 × 10-24
b6=104 τb6=1.60995 × 10-22
a6=20 la6=8.00097 × 10-12
b6=0 τb6=0.
b6=1 τb6=1.54803 × 10-24
b6=104 τb6=1.60995 × 10-22
a6=30 la6=1.20006 × 10-11
b6=0 τb6=0.
b6=1 τb6=1.54803 × 10-24
b6=104 τb6=1.60995 × 10-22
a6=40 la6=1.60003 × 10-11
b6=0 τb6=0.
b6=1 τb6=1.54803 × 10-24
b6=104 τb6=1.60995 × 10-22
a6=50 la6=2. × 10-11
b6=0 τb6=0.
b6=1 τb6=1.54803 × 10-24
b6=104 τb6=1.60995 × 10-22
I n [] : = J22 =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + a66 d2, b66 δ2}, J2a66,b66}, {a66, 0, NMAX}, {b66, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];
J22[l, τ]
I n [] : =
⋯
Do[
условный оператор
If[(am f1 || am f2 || am f3 || am f4 || am f5),
печатать
Print["am=", am],
пустой
Null];
lam = lmin + am d2;
Int2am =
квадратурное интегрирование
NIntegrate[J22[lam, τ], {τ, 0, T[lam] - ϵ2}];
lam =., {am, 0, NMAX}]
20 The First Part of the Program New SCCPERVMUR.nb
am=10
am=20
am=30
am=40
am=50
I n [] : = F2 =
интерполировать
Interpolation[
таблица значений
Table[{lmin + a7 d2, Int2a7}, {a7, 0, NMAX}],
порядок интерполяции
InterpolationOrder 5];
I n [] : = I2 =
квадратурное интегрирование
NIntegrate[F2[l], {l, lmin, lmax2}]
O u t [] =
8.65711 × 10-12
I n [] : =
численное приближение
N
1
lP
Na I2
O u t [] =
3.95829 × 10103
A new interesting structure is found, it must be investigated.
I n [] : = t := -4 δ2
I n [] : =
график функции
PlotIdJ22[l, t], l, lmin, 10-12,
отображаемый диапазон граф
PlotRange {-10, 10}
O u t [] =
2×10-13
4×10-13
6×10-13
8×10-13
1×10-12
-10
-5
5
10
Following from the plot, we must write
I n [] : =
найти корень
FindRootIdJ22[l, t] 0, l, 4 × 10-13
O u t [] =
l 4.06592 × 10-13
I n [] : =
найти корень
FindRootIdJ22[l, t] 0, l, 8 × 10-13
O u t [] =
l 7.83798 × 10-13
I n [] : = t =.
The First Part of the Program New SCCPERVMUR.nb 21
I n [] : =
график функции
Plot[F2[l], {l, lmin, lmax2},
отображаемый диапазон графика
PlotRange {-1.5, 1.5}]
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-1.5
-1.0
-0.5
0.5
1.0
1.5
I n [] : =
график функции
PlotF2[l], l, lmin, 2 × 10-12,
отображаемый диапазон графика
PlotRange {-1.5, 1.5}
O u t [] =
5.0×10-13
1.0×10-12
1.5×10-12
2.0×10-12
-1.5
-1.0
-0.5
0.5
1.0
1.5
I n [] : = I0 = I1 + I2
O u t [] =
1.73022 × 10-11
I n [] : =
численное приближение
N
1
lP
Δ1 Na I0
O u t [] =
5.84629 × 10103
численное приближение
N
1
lP
Δ2 Na I0
I n [] : = Tm2 =
абсолютное значение
AbsoluteTime[];
I n [] : = ComputationTime = Tm2 - Tm1
O u t [] =
40 279.5239986
22 The First Part of the Program New SCCPERVMUR.nb
The free vacuum energy
I n [] : = Vs := VU -
i=1
10
(VApi Δi Na)
The forward motion
I n [] : =
оператор цикла
Dolα1 = lmin + α1 d;
условный оператор
If[α1 0 || α1 1,
печатать
Print["α1=", α1, " ", "lα1=", lα1],
условный оператор
If[α1 f1 || α1 f2 ||
α1 f3 || α1 f4 || α1 f5,
печатать
Print["α1=", α1, " ", "lα1=", lα1],
пустой
Null]];
оператор цикла
Doτβ1 = β1 δ;
условный оператор
If[(α1 0 || α1 1) && (β1 0 || β1 1 || β1 MMAX),
печатать
Print["β1=", β1,
" ", "τβ1=", τβ1],
условный оператор
If[(α1 f1 || α1 f2 || α1 f3 || α1 f4 || α1 f5) &&
(β1 0 || β1 1 || β1 MMAX),
печатать
Print["β1=", β1, " ", "τβ1=", τβ1],
пустой
Null]];
J3α1,β1 =
1
T[lα1] - τβ1 квадратурное интегрирование
NIntegrate
Vs
(lα1 + δl[t])3
ℏ2 c2
16
×
3
(lα1 + δl[t])2 + m02 c4 ,
{t, τβ1, T[lα1]}×
1
T[lα1]
;
τβ1 =., {β1, 0, MMAX};
lα1 =., {α1, 0, NMAX}
α1=0 lα1=1.61626 × 10-15
β1=0 τβ1=0.
β1=1 τβ1=1.54803 × 10-24
β1=104 τβ1=1.60995 × 10-22
α1=1 lα1=4.016 × 10-13
β1=0 τβ1=0.
β1=1 τβ1=1.54803 × 10-24
β1=104 τβ1=1.60995 × 10-22
α1=10 lα1=4.00145 × 10-12
β1=0 τβ1=0.
β1=1 τβ1=1.54803 × 10-24
β1=104 τβ1=1.60995 × 10-22
α1=20 lα1=8.00129 × 10-12
β1=0 τβ1=0.
β1=1 τβ1=1.54803 × 10-24
β1=104 τβ1=1.60995 × 10-22
α1=30 lα1=1.20011 × 10-11
β1=0 τβ1=0.
β1=1 τβ1=1.54803 × 10-24
β1=104 τβ1=1.60995 × 10-22
α1=40 lα1=1.6001 × 10-11
β1=0 τβ1=0.
β1=1 τβ1=1.54803 × 10-24
β1=104 τβ1=1.60995 × 10-22
α1=50 lα1=2.00008 × 10-11
β1=0 τβ1=0.
β1=1 τβ1=1.54803 × 10-24
β1=104 τβ1=1.60995 × 10-22
I n [] : = J33 =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + α11 d, β11 δ}, J3α11,β11}, {α11, 0, NMAX}, {β11, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];
J33[l, τ]
I n [] : =
⋯
Do[
условный оператор
If[(α f1 || α f2 || α f3 || α f4 || α f5),
печатать
Print["α=", α],
пустой
Null];
lα = lmin + α d;
Int3α =
квадратурное интегрирование
NIntegrate[J33[lα, τ], {τ, 0, T[lα] - ϵ}];
lα =., {α, 0, NMAX}]
2 The Second Part of the Program New SCCPERVMUR.nb
α=10
α=20
α=30
α=40
α=50
I n [] : = F3 =
интерполировать
Interpolation[
таблица значений
Table[{lmin + α3 d, Int3α3}, {α3, 0, NMAX}],
порядок интерполяции
InterpolationOrder 5];
I n [] : = I3 =
квадратурное интегрирование
NIntegrate[F3[l], {l, lmin, lmax}]
O u t [] =
5.11251 × 1098
I n [] : =
численное приближение
N
1
lP
I3
O u t [] =
3.16318 × 10133
It is needed to check for a new interesting structure may be found, it must be investigated.
I n [] : =
график функции
PlotF3[l], {l, lmin, lmax},
отображаемый диапазон графика
PlotRange -5 × 10111, 5 × 10111
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-4×10111
-2×10111
2×10111
4×10111
I n [] : =
график функции
PlotF3[l], {l, lmin, lmax},
отображаемый диапазон графика
PlotRange -2 × 10111, 2 × 10111
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-1×10111
1×10111
The Second Part of the Program New SCCPERVMUR.nb 3
I n [] : =
график функции
PlotF3[l], {l, lmin, lmax},
отображаемый диапазон графика
PlotRange -2 × 10110, 2 × 10110
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-2×10110
-1×10110
1×10110
2×10110
I n [] : =
график функции
PlotF3[l], l, lmin, 10-12,
отображаемый диапазон графика
PlotRange -2 × 10110, 2 × 10110
O u t [] =
2×10-13
4×10-13
6×10-13
8×10-13
1×10-12
-2×10110
-1×10110
1×10110
2×10110
I n [] : =
график функции
PlotF3[l], l, 8 × 10-13, 2 × 10-12,
отображаемый диапазон графика
PlotRange -10110, 10110
O u t [] =
1.0×10-12
1.2×10-12
1.4×10-12
1.6×10-12
1.8×10-12
2.0×10-12
-1×10110
-5×10109
5×10109
1×10110
4 The Second Part of the Program New SCCPERVMUR.nb
I n [] : =
график функции
PlotF3[l], l, 10-12, lmax,
отображаемый диапазон графика
PlotRange -5 × 10101, 5 × 10101
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-4×10101
-2×10101
2×10101
4×10101
Following from the plots, we must write
I n [] : =
найти корень
FindRootF3[l] 0, l, 4 × 10-13
FindRoot: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was
unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working
precision to meet these tolerances.
O u t [] =
l 4.016 × 10-13
I n [] : =
найти корень
FindRootF3[l] 0, l, 8 × 10-13
FindRoot: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was
unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working
precision to meet these tolerances.
O u t [] =
l 8.01583 × 10-13
I n [] : = Ir3 =
квадратурное интегрирование
NIntegrateF3[l], l, lmin, 4.016 × 10-13-
квадратурное интегрирование
NIntegrateF3[l], l, 4.016 × 10-13, 8.01583 × 10-13+
квадратурное интегрирование
NIntegrateF3[l], l, 8.01583 × 10-13, lmax
O u t [] =
5.71398 × 1098
I n [] : =
численное приближение
N
1
lP
Ir3
O u t [] =
3.53532 × 10133
The backward motion
The Second Part of the Program New SCCPERVMUR.nb 5
I n [] : =
оператор цикла
Dolα2 = lmin + α2 d;
условный оператор
If[α2 0 || α2 1,
печатать
Print["α2=", α2, " ", "lα2=", lα2],
условный оператор
If[α2 f1 || α2 f2 ||
α2 f3 || α2 f4 || α2 f5,
печатать
Print["α2=", α2, " ", "lα2=", lα2],
пустой
Null]];
оператор цикла
Doτβ2 = -β2 δ;
условный оператор
If[(α2 0 || α2 1) && (β2 0 || β2 1 || β2 MMAX),
печатать
Print["β2=", β2,
" ", "τβ2=", τβ2],
условный оператор
If[(α2 f1 || α2 f2 || α2 f3 || α2 f4 || α2 f5) &&
(β2 0 || β2 1 || β2 MMAX),
печатать
Print["β2=", β2, " ", "τβ2=", τβ2],
пустой
Null]];
J4α2,β2 =
1
T[lα2] + τβ2 квадратурное интегрирование
NIntegrate
Vs
lα2 + δl1
2 Δt[lα2] - t
3
ℏ2 c2
16
×
3
lα2 + δl1
2 Δt[lα2] - t
2 + m02 c4 , {t, T[lα2] - τβ2, 2 T[lα2]}×
1
T[lα2]
;
τβ2 =., {β2, 0, MMAX};
lα2 =., {α2, 0, NMAX}
6 The Second Part of the Program New SCCPERVMUR.nb
α2=0 lα2=1.61626 × 10-15
β2=0 τβ2=0.
β2=1 τβ2=-1.54803 × 10-24
β2=104 τβ2=-1.60995 × 10-22
α2=1 lα2=4.016 × 10-13
β2=0 τβ2=0.
β2=1 τβ2=-1.54803 × 10-24
β2=104 τβ2=-1.60995 × 10-22
α2=10 lα2=4.00145 × 10-12
β2=0 τβ2=0.
β2=1 τβ2=-1.54803 × 10-24
β2=104 τβ2=-1.60995 × 10-22
α2=20 lα2=8.00129 × 10-12
β2=0 τβ2=0.
β2=1 τβ2=-1.54803 × 10-24
β2=104 τβ2=-1.60995 × 10-22
α2=30 lα2=1.20011 × 10-11
β2=0 τβ2=0.
β2=1 τβ2=-1.54803 × 10-24
β2=104 τβ2=-1.60995 × 10-22
α2=40 lα2=1.6001 × 10-11
β2=0 τβ2=0.
β2=1 τβ2=-1.54803 × 10-24
β2=104 τβ2=-1.60995 × 10-22
α2=50 lα2=2.00008 × 10-11
β2=0 τβ2=0.
β2=1 τβ2=-1.54803 × 10-24
β2=104 τβ2=-1.60995 × 10-22
I n [] : = J44 =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + α22 d, -β22 δ}, J4α22,β22}, {α22, 0, NMAX}, {β22, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];
J44[l, τ]
I n [] : =
⋯
Do[
условный оператор
If[(αm f1 || αm f2 || αm f3 || αm f4 || αm f5),
печатать
Print["αm=", αm],
пустой
Null];
lαm = lmin + αm d;
Int4αm =
квадратурное интегрирование
NIntegrate[J44[lαm, τ], {τ, -(T[lαm] - ϵ), 0}];
lαm =., {αm, 0, NMAX}]
The Second Part of the Program New SCCPERVMUR.nb 7
αm=10
αm=20
αm=30
αm=40
αm=50
I n [] : = F4 =
интерполировать
Interpolation[
таблица значений
Table[{lmin + α4 d, Int4α4}, {α4, 0, NMAX}],
порядок интерполяции
InterpolationOrder 5];
I n [] : = I4 =
квадратурное интегрирование
NIntegrate[F4[l], {l, lmin, lmax}]
O u t [] =
5.11251 × 1098
I n [] : =
численное приближение
N
1
lP
I4
O u t [] =
3.16318 × 10133
I n [] : = I00 = I3 + I4
O u t [] =
1.0225 × 1099
I n [] : =
численное приближение
N
1
lP
I00
O u t [] =
6.32636 × 10133
It is needed to check for a new interesting structure may be found, it must be investigated.
I n [] : =
график функции
PlotF4[l], {l, lmin, lmax},
отображаемый диапазон графика
PlotRange -4 × 10110, 4 × 10110
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-4×10110
-2×10110
2×10110
4×10110
8 The Second Part of the Program New SCCPERVMUR.nb
I n [] : =
график функции
PlotF4[l], {l, lmin, lmax},
отображаемый диапазон графика
PlotRange -2 × 10110, 2 × 10110
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-2×10110
-1×10110
1×10110
2×10110
I n [] : =
график функции
PlotF4[l], {l, lmin, lmax},
отображаемый диапазон графика
PlotRange -1.5 × 10110, 1.5 × 10110
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-1.5×10110
-1.0×10110
-5.0×10109
5.0×10109
1.0×10110
1.5×10110
I n [] : =
график функции
PlotF4[l], l, lmin, 5 × 10-12,
отображаемый диапазон графика
PlotRange -1.5 × 10110, 1.5 × 10110
O u t [] =
1×10-12
2×10-12
3×10-12
4×10-12
5×10-12
-1.5×10110
-1.0×10110
-5.0×10109
5.0×10109
1.0×10110
1.5×10110
The Second Part of the Program New SCCPERVMUR.nb 9
I n [] : =
график функции
PlotF4[l], l, lmin, 2 × 10-12,
отображаемый диапазон графика
PlotRange -1.5 × 10110, 1.5 × 10110
O u t [] =
5.0×10-13
1.0×10-12
1.5×10-12
2.0×10-12
-1.5×10110
-1.0×10110
-5.0×10109
5.0×10109
1.0×10110
1.5×10110
I n [] : =
график функции
PlotF4[l], l, 10-12, lmax,
отображаемый диапазон графика
PlotRange -5 × 10101, 5 × 10101
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-4×10101
-2×10101
2×10101
4×10101
Following from the plots, we must write
I n [] : =
найти корень
FindRootF4[l] 0, l, 4 × 10-13
FindRoot: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was
unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working
precision to meet these tolerances.
O u t [] =
l 4.016 × 10-13
I n [] : =
найти корень
FindRootF4[l] 0, l, 8 × 10-13
FindRoot: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was
unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working
precision to meet these tolerances.
O u t [] =
l 8.01583 × 10-13
10 The Second Part of the Program New SCCPERVMUR.nb
I n [] : = Ir4 =
квадратурное интегрирование
NIntegrateF4[l], l, lmin, 4.016 × 10-13-
квадратурное интегрирование
NIntegrateF4[l], l, 4.016 × 10-13, 8.01583 × 10-13+
квадратурное интегрирование
NIntegrateF4[l], l, 8.01583 × 10-13, lmax
O u t [] =
5.71398 × 1098
I n [] : =
численное приближение
N
1
lP
Ir4
O u t [] =
3.53532 × 10133
I n [] : = Ir00 = Ir3 + Ir4
O u t [] =
1.1428 × 1099
I n [] : =
численное приближение
N
1
lP
Ir00
O u t [] =
7.07064 × 10133
I n [] : = Tm2 =
абсолютное значение
AbsoluteTime[];
I n [] : = ComputationTime = Tm2 - Tm1
O u t [] =
41.6397715
The Second Part of the Program New SCCPERVMUR.nb 11
Below is the begining of the computation of the free vacuum energy in atoms without the limita-
tion vacuum effect.
These are linear equations.
This is fastest algorithm, as the examination has shown.
I n [] : = lmax3 := 2 × 10-11
I n [] : = ΔL3 := lmax3 - lmin
d3 :=
ΔL
50
I n [] : = δ3 := 1.54803 × 10-24
δ3 =
1
2
×
1
104
×
lmin + ΔL3
c
×
1
3 + 16 × m02 c2
ℏ2
(lmin + ΔL3)2
"NMAX3==NMAX"
"MMAX3==MMAX"
I n [] : = lmax4 := 2 × 10-11
I n [] : = ΔL4 := lmax4 - lmin
I n [] : = d4 :=
ΔL4
50
I n [] : = δ4 := 1.54803 × 10-24
δ4 =
1
2
×
1
104
×
lmin + ΔL4
c
×
1
3 + 16 × m02 c2
ℏ2
(lmin + ΔL4)2
"NMAX4==NMAX"
"MMAX4==MMAX"
I n [] : =
таблица значений
Table[r20,x1,y1 = rmin;, {y1, 0, MMAX}, {x1, 0, NMAX}];
I n [] : =
таблица значений
Table[ra0,x1,y1 = rmin;, {y1, 0, MMAX}, {x1, 0, NMAX}];
цикл ДЛЯ
For[s1 = 0, ras1,n0,m00 ≤rmax, s1++,
условный ⋯
If[s1 0,
печатать
Print["s1=", 0, " ", "ras1=",
численно⋯
N[rmin]],
пустой
Null];
оператор цикла
Do[lx1 = lmin + x1 d3;
оператор цикла
Do[ty1 = y1 δ3;
r2s1+1,x1,y1 = r2s1,x1,y1 + lx1 + δl[ty1];
ras1+1,x1,y1 = r2s1+1,x1,y1;
r2s1,x1,y1 =.;
ty1 =., {y1, 0, MMAX}];
lx1 =., {x1, 0, NMAX}];
условный оператор
If[s1 > 0 && (s1 j0 || s1 j1 || s1 j2 || s1 j3 ||
s1 j4 || s1 j5 || s1 j6 || s1 j7 || s1 j8 || s1 j9 || s1 j10),
печатать
Print["s1=", s1, " ", "ras1,n0,m00=", ras1,n0,m00],
пустой
Null]] //
затраченное время
Timing
s1=0 ras1=1. × 10-11
s1=1 ras1,n0,m00=1.00016 × 10-11
s1=928 ras1,n0,m00=1.14999 × 10-11
s1=1856 ras1,n0,m00=1.29998 × 10-11
s1=2784 ras1,n0,m00=1.44997 × 10-11
s1=3712 ras1,n0,m00=1.59995 × 10-11
s1=4640 ras1,n0,m00=1.74994 × 10-11
s1=5568 ras1,n0,m00=1.89993 × 10-11
s1=6496 ras1,n0,m00=2.04992 × 10-11
s1=7424 ras1,n0,m00=2.19991 × 10-11
s1=8352 ras1,n0,m00=2.3499 × 10-11
s1=9280 ras1,n0,m00=2.49988 × 10-11
O u t [] =
{1280.97, Null}
I n [] : = p1 = s1 - 1;
I n [] : =
таблица значений
Table[
rak5 =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + x11 d3, y11 δ3}, rak5,x11,y11}, {x11, 0, NMAX},
{y11, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];, {k5, 1, p1}];
rak5[l, t]
I n [] : =
таблица значений
Table[r2m0,x2,y2 = rmin;, {y2, 0, MMAX}, {x2, 0, NMAX}];
I n [] : =
таблица значений
Table[ram0,x2,y2 = rmin;, {y2, 0, MMAX}, {x2, 0, NMAX}];
2 The Third Part of the Program New SCCPERVMUR.nb
I n [] : =
цикл ДЛЯ
Fors2 = 0, rams2,n0,m00 ≤rmax, s2++,
условный ⋯
If[s2 0,
печатать
Print["s2=", 0, " ", "rams2=",
численно⋯
N[rmin]],
пустой
Null];
оператор цикла
Dolx2 = lmin + x2 d4;
оператор цикла
Doty2 = y2 δ4;
r2ms2+1,x2,y2 = r2ms2,x2,y2 + lx2 + δl
1
2
Δt[lx2] + ty2;
rams2+1,x2,y2 = r2ms2+1,x2,y2;
r2ms2,x2,y2 =.;
ty2 =., {y2, 0, MMAX};
lx2 =., {x2, 0, NMAX};
условный оператор
If[s2 > 0 && (s2 j0 || s2 j1 || s2 j2 || s2 j3 ||
s2 j4 || s2 j5 || s2 j6 || s2 j7 || s2 j8 || s2 j9 || s2 j10),
печатать
Print["s2=", s2, " ", "rams2,n0,m00=", rams2,n0,m00],
пустой
Null]//
затраченное время
Timing
s2=0 rams2=1. × 10-11
s2=1 rams2,n0,m00=1.00016 × 10-11
s2=928 rams2,n0,m00=1.14999 × 10-11
s2=1856 rams2,n0,m00=1.29998 × 10-11
s2=2784 rams2,n0,m00=1.44997 × 10-11
s2=3712 rams2,n0,m00=1.59995 × 10-11
s2=4640 rams2,n0,m00=1.74994 × 10-11
s2=5568 rams2,n0,m00=1.89993 × 10-11
s2=6496 rams2,n0,m00=2.04992 × 10-11
s2=7424 rams2,n0,m00=2.19991 × 10-11
s2=8352 rams2,n0,m00=2.3499 × 10-11
s2=9280 rams2,n0,m00=2.49988 × 10-11
O u t [] =
{1798.58, Null}
I n [] : = p2 = s2 - 1;
I n [] : =
таблица знач⋯
Table[ramk6 =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + x22 d4, y22 δ4}, ramk6,x22,y22},
{x22, 0, NMAX}, {y22, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];, {k6, 1, p2}];
ramk6[l, t]
The Third Part of the Program New SCCPERVMUR.nb 3
I n [] : =
привести
Reducermin ≥
1
2 абсолютное значение
Absl +
1
2
q Δt[l], l,
множеств
Reals
Reduce: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
O u t [] =
-2. × 10-11 ≤l ≤2. × 10-11
I n [] : =
привести
Reducermin ≥
1
2 абсолютное значение
Abs[l + q Δt[l]], l,
множеств
Reals
Reduce: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
O u t [] =
-2. × 10-11 ≤l ≤2. × 10-11
I n [] : =
привести
Reducermax ≥
1
2 абсолютное значение
Absl +
1
2
q Δt[l], l,
множеств
Reals
Reduce: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
O u t [] =
-5. × 10-11 ≤l ≤5. × 10-11
I n [] : =
привести
Reducermax ≥
1
2 абсолютное значение
Abs[l + q Δt[l]], l,
множеств
Reals
Reduce: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
O u t [] =
-5. × 10-11 ≤l ≤5. × 10-11
From here the maximal scale for the calculations without the limitation vacuum effect can be
found.
The forward direction
I n [] : = sa = s1 - 1;
4 The Third Part of the Program New SCCPERVMUR.nb
I n [] : =
оператор цикла
Dolsa1 = lmin + sa1 d3;
условный оператор
If[sa1 0 || sa1 1,
печатать
Print["sa1=", sa1, " ", "lsa1=", lsa1],
условный оператор
If[sa1 f1 || sa1 f2 ||
sa1 f3 || sa1 f4 || sa1 f5,
печатать
Print["sa1=", sa1, " ", "lsa1=", lsa1],
пустой
Null],
пустой
Null];
оператор цикла
Dotsb1 = sb1 δ3;
условный оператор
If[(sb1 0 || sb1 1 || sb1 MMAX) && (sa1 0 || sa1 1 || sa1 f1 || sa1 f2 ||
sa1 f3 || sa1 f4 || sa1 f5),
печатать
Print["sb1=", sb1, " ", "tsb1=", tsb1],
пустой
Null];
IdJ1asa1,sb1 =
1
2
π
ArcTan
lsa1+δl[tsb1]
2
rmin
2
-
1
4 (lsa1+δl[tsb1])2
2
-
π
ArcTan
lsa1+δl[tsb1]
2
rmin
2
-
1
4 (lsa1+δl[tsb1])2
+ 2
ℏ2 c2
16
×
3
(lsa1 + δl[tsb1])2 + m02 c4 +
k7=1
sa
1
2
π
ArcTan
lsa1+δl[tsb1]
2
(rak7[lsa1,tsb1])2-
1
4 (lsa1+δl[tsb1])2
2
-
π
ArcTan
lsa1+δl[tsb1]
2
(rak7[lsa1,tsb1])2-
1
4 (lsa1+δl[tsb1])2
+ 2
ℏ2 c2
16
×
3
(lsa1 + δl[tsb1])2 + m02 c4
;
tsb1 =., {sb1, 0, MMAX};
lsa1 =., {sa1, 0, NMAX}
The Third Part of the Program New SCCPERVMUR.nb 5
sa1=0 lsa1=1.61626 × 10-15
sb1=0 tsb1=0.
sb1=1 tsb1=1.54803 × 10-24
sb1=104 tsb1=1.60995 × 10-22
sa1=1 lsa1=4.01584 × 10-13
sb1=0 tsb1=0.
sb1=1 tsb1=1.54803 × 10-24
sb1=104 tsb1=1.60995 × 10-22
sa1=10 lsa1=4.00129 × 10-12
sb1=0 tsb1=0.
sb1=1 tsb1=1.54803 × 10-24
sb1=104 tsb1=1.60995 × 10-22
sa1=20 lsa1=8.00097 × 10-12
sb1=0 tsb1=0.
sb1=1 tsb1=1.54803 × 10-24
sb1=104 tsb1=1.60995 × 10-22
sa1=30 lsa1=1.20006 × 10-11
sb1=0 tsb1=0.
sb1=1 tsb1=1.54803 × 10-24
sb1=104 tsb1=1.60995 × 10-22
sa1=40 lsa1=1.60003 × 10-11
sb1=0 tsb1=0.
sb1=1 tsb1=1.54803 × 10-24
sb1=104 tsb1=1.60995 × 10-22
sa1=50 lsa1=2. × 10-11
sb1=0 tsb1=0.
sb1=1 tsb1=1.54803 × 10-24
sb1=104 tsb1=1.60995 × 10-22
I n [] : = IdJ11a =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + sa11 d3, sb11 δ3}, IdJ1asa11,sb11},
{sa11, 0, NMAX}, {sb11, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];
IdJ11a[l, t]
6 The Third Part of the Program New SCCPERVMUR.nb
I n [] : =
оператор цикла
Dolaa1 = lmin + aa1 d3;
условный оператор
If[aa1 0 || aa1 1,
печатать
Print["aa1=", aa1, " ", "laa1=", laa1],
условный оператор
If[aa1 f1 || aa1 f2 ||
aa1 f3 || aa1 f4 || aa1 f5,
печатать
Print["aa1=", aa1, " ", "laa1=", laa1],
пустой
Null]];
оператор цикла
Doτbb1 = bb1 δ3;
условный оператор
If[(aa1 0 || aa1 1) && (bb1 0 || bb1 1 || bb1 MMAX),
печатать
Print["bb1=", bb1, " ",
"τbb1=", τbb1],
условный оператор
If[(aa1 f1 || aa1 f2 || aa1 f3 || aa1 f4 || aa1 f5) &&
(bb1 0 || bb1 1 || bb1 MMAX),
печатать
Print["bb1=", bb1, " ", "τbb1=", τbb1],
пустой
Null]];
J1aaa1,bb1 =
1
T[laa1] - τbb1
1
T[laa1] квадратурное интегрирование
NIntegrate[IdJ11a[laa1, t], {t, τbb1, T[laa1]}];
τbb1 =., {bb1, 0, MMAX};
laa1 =., {aa1, 0, NMAX}
The Third Part of the Program New SCCPERVMUR.nb 7
aa1=0 laa1=1.61626 × 10-15
bb1=0 τbb1=0.
bb1=1 τbb1=1.54803 × 10-24
bb1=104 τbb1=1.60995 × 10-22
aa1=1 laa1=4.01584 × 10-13
bb1=0 τbb1=0.
bb1=1 τbb1=1.54803 × 10-24
bb1=104 τbb1=1.60995 × 10-22
aa1=10 laa1=4.00129 × 10-12
bb1=0 τbb1=0.
bb1=1 τbb1=1.54803 × 10-24
bb1=104 τbb1=1.60995 × 10-22
aa1=20 laa1=8.00097 × 10-12
bb1=0 τbb1=0.
bb1=1 τbb1=1.54803 × 10-24
bb1=104 τbb1=1.60995 × 10-22
aa1=30 laa1=1.20006 × 10-11
bb1=0 τbb1=0.
bb1=1 τbb1=1.54803 × 10-24
bb1=104 τbb1=1.60995 × 10-22
aa1=40 laa1=1.60003 × 10-11
bb1=0 τbb1=0.
bb1=1 τbb1=1.54803 × 10-24
bb1=104 τbb1=1.60995 × 10-22
aa1=50 laa1=2. × 10-11
bb1=0 τbb1=0.
bb1=1 τbb1=1.54803 × 10-24
bb1=104 τbb1=1.60995 × 10-22
I n [] : = J11a =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + aa11 d3, bb11 δ3}, J1aaa11,bb11},
{aa11, 0, NMAX}, {bb11, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];
J11a[l, τ]
I n [] : =
⋯
Do[
условный оператор
If[(aa f1 || aa f2 || aa f3 || aa f4 || aa f5),
печатать
Print["aa=", aa],
пустой
Null];
laa = lmin + aa d3;
Int1aaa =
квадратурное интегрирование
NIntegrate[J11a[laa, τ], {τ, 0, T[laa] - ϵ3}];
laa =., {aa, 0, NMAX}]
8 The Third Part of the Program New SCCPERVMUR.nb
aa=10
aa=20
aa=30
aa=40
aa=50
I n [] : = F1a =
интерполировать
Interpolation[
таблица значений
Table[{lmin + a8 d3, Int1aa8}, {a8, 0, NMAX}],
порядок интерполяции
InterpolationOrder 5];
I n [] : = I1a =
квадратурное интегрирование
NIntegrate[F1a[l], {l, lmin, lmax3}]
O u t [] =
8.64476 × 10-12
I n [] : =
численное приближение
N
1
lP
Na I1a
O u t [] =
3.95264 × 10103
I n [] : =
график функции
Plot[F1a[l], {l, lmin, lmax3},
отображаемый диапазон графика
PlotRange {-1.5, 1.5}]
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-1.5
-1.0
-0.5
0.5
1.0
1.5
I n [] : =
график функции
PlotF1a[l], l, lmin, 2 × 10-12,
отображаемый диапазон графика
PlotRange {-1.5, 1.5}
O u t [] =
5.0×10-13
1.0×10-12
1.5×10-12
2.0×10-12
-1.5
-1.0
-0.5
0.5
1.0
1.5
The backward direction
The Third Part of the Program New SCCPERVMUR.nb 9
I n [] : = sam = s2 - 1;
I n [] : =
оператор цикла
Dolsa2 = lmin + sa2 d4;
условный оператор
If[sa2 0 || sa2 1,
печатать
Print["sa2=", sa2, " ", "lsa2=", lsa2],
условный оператор
If[sa2 f1 || sa2 f2 ||
sa2 f3 || sa2 f4 || sa2 f5,
печатать
Print["sa2=", sa2, " ", "lsa2=", lsa2],
пустой
Null],
пустой
Null];
оператор цикла
Dotsb2 = sb2 δ4;
условный оператор
If[(sb2 0 || sb2 1 || sb2 MMAX) && (sa2 0 || sa2 1 || sa2 f1 || sa2 f2 ||
sa2 f3 || sa2 f4 || sa2 f5),
печатать
Print["sb2=", sb2, " ", "tsb2=", tsb2],
пустой
Null];
IdJ2asa2,sb2 =
1
2
π
ArcTan
lsa2+δl
1
2 Δt[lsa2]+tsb2
2
rmin
2
-
1
4 lsa2+δl
1
2 Δt[lsa2]+tsb2
2
2
-
π
ArcTan
lsa2+δl
1
2 Δt[lsa2]+tsb2
2
rmin
2
-
1
4 lsa2+δl
1
2 Δt[lsa2]+tsb2
2
+ 2
ℏ2 c2
16
×
3
lsa2 + δl1
2 Δt[lsa2] + tsb2
2 + m02 c4 +
k8=1
sam
1
2
π
ArcTan
lsa2+δl
1
2 Δt[lsa2]+tsb2
2
(ramk8[lsa2,tsb2])2-
1
4 lsa2+δl
1
2 Δt[lsa2]+tsb2
2
2
-
π
ArcTan
lsa2+δl
1
2 Δt[lsa2]+tsb2
2
(ramk8[lsa2,tsb2])2-
1
4 lsa2+δl
1
2 Δt[lsa2]+tsb2
2
+ 2
ℏ2 c2
16
×
3
lsa2 + δl1
2 Δt[lsa2] + tsb2
2 + m02 c4
;
tsb2 =., {sb2, 0, MMAX};
lsa2 =., {sa2, 0, NMAX}
10 The Third Part of the Program New SCCPERVMUR.nb
sa2=0 lsa2=1.61626 × 10-15
sb2=0 tsb2=0.
sb2=1 tsb2=1.54803 × 10-24
sb2=104 tsb2=1.60995 × 10-22
sa2=1 lsa2=4.01584 × 10-13
sb2=0 tsb2=0.
sb2=1 tsb2=1.54803 × 10-24
sb2=104 tsb2=1.60995 × 10-22
sa2=10 lsa2=4.00129 × 10-12
sb2=0 tsb2=0.
sb2=1 tsb2=1.54803 × 10-24
sb2=104 tsb2=1.60995 × 10-22
sa2=20 lsa2=8.00097 × 10-12
sb2=0 tsb2=0.
sb2=1 tsb2=1.54803 × 10-24
sb2=104 tsb2=1.60995 × 10-22
sa2=30 lsa2=1.20006 × 10-11
sb2=0 tsb2=0.
sb2=1 tsb2=1.54803 × 10-24
sb2=104 tsb2=1.60995 × 10-22
sa2=40 lsa2=1.60003 × 10-11
sb2=0 tsb2=0.
sb2=1 tsb2=1.54803 × 10-24
sb2=104 tsb2=1.60995 × 10-22
sa2=50 lsa2=2. × 10-11
sb2=0 tsb2=0.
sb2=1 tsb2=1.54803 × 10-24
sb2=104 tsb2=1.60995 × 10-22
I n [] : = IdJ22a =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + sa22 d4, sb22 δ4}, IdJ2asa22,sb22},
{sa22, 0, NMAX}, {sb22, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];
IdJ22a[l, t]
The Third Part of the Program New SCCPERVMUR.nb 11
I n [] : =
оператор цикла
Dolaa2 = lmin + aa2 d4;
условный оператор
If[aa2 0 || aa2 1,
печатать
Print["aa2=", aa2, " ", "laa2=", laa2],
условный оператор
If[aa2 f1 || aa2 f2 ||
aa2 f3 || aa2 f4 || aa2 f5,
печатать
Print["aa2=", aa2, " ", "laa2=", laa2],
пустой
Null]];
оператор цикла
Doτbb2 = bb2 δ4;
условный оператор
If[(aa2 0 || aa2 1) && (bb2 0 || bb2 1 || bb2 MMAX),
печатать
Print["bb2=", bb2, " ",
"τbb2=", τbb2],
условный оператор
If[(aa2 f1 || aa2 f2 || aa2 f3 || aa2 f4 || aa2 f5) &&
(bb2 0 || bb2 1 || bb2 MMAX),
печатать
Print["bb2=", bb2, " ", "τbb2=", τbb2],
пустой
Null]];
J2aaa2,bb2 =
1
T[laa2] - τbb2
1
T[laa2] квадратурное интегрирование
NIntegrate[IdJ22a[laa2, t],
{t, T[laa2] + τbb2, 2 T[laa2]}];
τbb2 =., {bb2, 0, MMAX};
laa2 =., {aa2, 0, NMAX}
12 The Third Part of the Program New SCCPERVMUR.nb
aa2=0 laa2=1.61626 × 10-15
bb2=0 τbb2=0.
bb2=1 τbb2=1.54803 × 10-24
bb2=104 τbb2=1.60995 × 10-22
aa2=1 laa2=4.01584 × 10-13
bb2=0 τbb2=0.
bb2=1 τbb2=1.54803 × 10-24
bb2=104 τbb2=1.60995 × 10-22
aa2=10 laa2=4.00129 × 10-12
bb2=0 τbb2=0.
bb2=1 τbb2=1.54803 × 10-24
bb2=104 τbb2=1.60995 × 10-22
aa2=20 laa2=8.00097 × 10-12
bb2=0 τbb2=0.
bb2=1 τbb2=1.54803 × 10-24
bb2=104 τbb2=1.60995 × 10-22
aa2=30 laa2=1.20006 × 10-11
bb2=0 τbb2=0.
bb2=1 τbb2=1.54803 × 10-24
bb2=104 τbb2=1.60995 × 10-22
aa2=40 laa2=1.60003 × 10-11
bb2=0 τbb2=0.
bb2=1 τbb2=1.54803 × 10-24
bb2=104 τbb2=1.60995 × 10-22
aa2=50 laa2=2. × 10-11
bb2=0 τbb2=0.
bb2=1 τbb2=1.54803 × 10-24
bb2=104 τbb2=1.60995 × 10-22
I n [] : = J22a =
интерполировать
Interpolation[
уплостить
Flatten[
таблица значений
Table[{{lmin + aa22 d4, bb22 δ4}, J2aaa22,bb22},
{aa22, 0, NMAX}, {bb22, 0, MMAX}], 1],
порядок интерполяции
InterpolationOrder 5];
J22a[l, τ]
I n [] : =
⋯
Do[
условный оператор
If[(aam f1 || aam f2 || aam f3 || aam f4 || aam f5),
печатать
Print["aam=", aam],
пустой
Null];
laam = lmin + aam d4;
Int2aaam =
квадратурное интегрирование
NIntegrate[J22a[laam, τ], {τ, 0, T[laam] - ϵ4}];
laam =., {aam, 0, NMAX}]
The Third Part of the Program New SCCPERVMUR.nb 13
aam=10
aam=20
aam=30
aam=40
aam=50
I n [] : = F2a =
интерполировать
Interpolation[
таблица значений
Table[{lmin + a9 d4, Int2aa9}, {a9, 0, NMAX}],
порядок интерполяции
InterpolationOrder 5];
I n [] : = I2a =
квадратурное интегрирование
NIntegrate[F2a[l], {l, lmin, lmax4}]
O u t [] =
8.64432 × 10-12
I n [] : =
численное приближение
N
1
lP
Na I2a
O u t [] =
3.95244 × 10103
I n [] : = I0a = I1a + I2a
O u t [] =
1.72891 × 10-11
I n [] : =
численное приближение
N
1
lP
Δ1 Na I0a
O u t [] =
5.84186 × 10103
численное приближение
N
1
lP
Δ2 Na I0a
I n [] : =
график функции
Plot[F2a[l], {l, lmin, lmax4},
отображаемый диапазон графика
PlotRange {-1.5, 1.5}]
O u t [] =
5.0×10-12
1.0×10-11
1.5×10-11
2.0×10-11
-1.5
-1.0
-0.5
0.5
1.0
1.5
14 The Third Part of the Program New SCCPERVMUR.nb
I n [] : =
график функции
PlotF2a[l], l, lmin, 2 × 10-12,
отображаемый диапазон графика
PlotRange {-1.5, 1.5}
O u t [] =
5.0×10-13
1.0×10-12
1.5×10-12
2.0×10-12
-1.5
-1.0
-0.5
0.5
1.0
1.5
I n [] : = Tm2 =
абсолютное значение
AbsoluteTime[];
I n [] : = ComputationTime = Tm2 - Tm1
O u t [] =
17 558.4553638
The Third Part of the Program New SCCPERVMUR.nb 15
In[1]:= me = 9.11 × 10-31;
mμ = 1.88356 × 10-28;
mτ = 3.16751 × 10-27;
In[2]:= uplmνe = 3.6 × 10-36;
In[3]:= uplmνμ = 3.39 × 10-31;
In[4]:= uplmντ = 3.2445 × 10-29;
mu = 4.2785 × 10-30;
md = 8.4675 × 10-30;
ms = 1.78266 × 10-28;
mc = 2.22833 × 10-27;
mb = 8.46765 × 10-27;
mt = 3.052 × 10-25;
In[5]:= VU = 3.6 × 1080;
Na = 7.39 × 1079;
VA = 4
3
π rmax
3
- rmin
3
;
m = me;
c = 299 792 458;
h = 6.6260755 × 10-34;
ℏ=
h
2 π
;
In[12]:= rmin := 10-11
In[13]:= lmin := 1020 lP
In[14]:= lP := 1.616255 × 10-35
In[15]:= rmax := 2.5 × 10-11
In[93]:= rmax := 3.1 × 10-11
In[16]:= lmax := 2.00008 × 10-11
In[17]:= VAp1 = 6.12611 × 10-32;
VAp2 = 1.20599 × 10-31;
VAp3 = 9.0059 × 10-31;
VAp4 = 1.43257 × 10-30;
VAp5 = 2.25659 × 10-31;
VAp6 = 1.14899 × 10-29;
VAp7 = 1.14616 × 10-30;
VAp8 = 5.57109 × 10-30;
VAp9 = 1.4133 × 10-29;
VAp10 = 4.1846 × 10-30;
In[27]:= Δ1 = 0.739;
Δ2 = 0.24;
Δ3 = 0.0104;
Δ4 = 0.0046;
Δ5 = 0.0013;
Δ6 = 0.0011;
Δ7 = 0.00096;
Δ8 = 0.00065;
Δ9 = 0.00058;
Δ10 = 0.00044;
In[37]:= l = lmin;
In[38]:= R11 = 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin +
384 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+
36 ℏ4 rmin
2
+ 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22 rmin +
4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
4 16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin//
численное приближение
N
Out[38]= 1.00016 × 10-11
In[39]:= l =.
In[40]:= l = lmax;
In[41]:= R12 = 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin +
384 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+
36 ℏ4 rmin
2
+ 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22 rmin +
4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
4 16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin//
численное приближение
N
Out[41]= 3.00003 × 10-11
In[42]:= l =.
In[43]:= l = lmin;
In[44]:= R21 =
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 +
1 16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
256 c4 l5 m4
16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2 ℏ2 rmin +
36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2 Program SCCPERVMUR version for the comparison.nb
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22 rmin +
4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2+
1 16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
96 c2 l3 m2
ℏ2 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin +
384 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+
384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+ 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin
l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+ 2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+
4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2+
1 16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
9 l ℏ4
16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin +
384 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+
384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+ 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin
l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+ 2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+
4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2+
64 c4 l4 m4 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384
c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22
rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2+
24 c2 l2 m2 ℏ2 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2
l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22
rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2+
9 ℏ4 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2 ℏ2 rmin +
36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
Program SCCPERVMUR version for the comparison.nb 3
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22
rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
4 16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2+
1 2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
3l ℏ2 16
3 c2 l4 m2 ℏ2 -
3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin +
1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+ 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2
rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+ 2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3
ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
16
3 c2 l4 m2 ℏ2 + 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2
ℏ2 rmin + 64
3 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 12
3 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+
384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+ 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin
l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+ 2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3
ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
144 c2 l4 m2 ℏ2 + 27 l2 ℏ4 + 3
3 l2 ℏ4 + 2048
3 c4 l5 m4 rmin +
192 c2 l3 m2 ℏ2 rmin + 768
3 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 72
3 l ℏ4 rmin +
1024
3 c4 l4 m4 rmin
2
+ 384
3 c2 l2 m2 ℏ2 rmin
2
+ 36
3 ℏ4 rmin
2
+
3 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22
rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2+
16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 256 c4 l5 m4 16
3 c2 l4 m2 ℏ2 -
3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2 ℏ2 rmin +
36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l
16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+ 96 c2
l3 m2 ℏ2 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3
m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l
4 Program SCCPERVMUR version for the comparison.nb
16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
9 l ℏ4 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3
m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l
16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
64 c4
l4 m4 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2
ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2
rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+
4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2+
24 c2 l2
m2 ℏ2 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2
ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2
rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+
4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2+
9 ℏ4 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2
ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2
rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+
4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
4 16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2
2
2 16 c2 l2 m2 + 3 ℏ216
3 c2 l4 m2 ℏ2 + 3 l2 ℏ4 +
3
3
l2
ℏ4 + 1024
Program SCCPERVMUR version for the comparison.nb 5
c4
l5
m4
rmin + 384
c2
l3
m2
ℏ2
rmin + 64
3
c2
l3
m2
ℏ2
rmin + 36
l
ℏ4
rmin + 12
3
l
ℏ4
rmin + 1024
c4
l4
m4
rmin
2
+ 384
c2
l2
m2
ℏ2
rmin
2
+ 36
ℏ4
rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin
l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+ 2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22 rmin +
4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2//
численное приближение
N
Out[44]= 1.00032 × 10-11
In[45]:= l =.
In[46]:= l = lmax;
In[47]:= R22 =
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 +
1 16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
256 c4 l5 m4
16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2 ℏ2 rmin +
36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
6 Program SCCPERVMUR version for the comparison.nb
min
min
min
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22 rmin +
4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2+
1 16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
96 c2 l3 m2
ℏ2 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin +
384 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+
384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+ 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin
l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+ 2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+
4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2+
1 16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
9 l ℏ4
16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin +
384 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+
384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+ 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin
l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+ 2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+
4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2+
64 c4 l4 m4 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384
c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22
rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2+
24 c2 l2 m2 ℏ2 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2
l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22
rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2+
9 ℏ4 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2 ℏ2 rmin +
36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
Program SCCPERVMUR version for the comparison.nb 7
min
min
min
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22
rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
4 16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2+
1 2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
3l ℏ2 16
3 c2 l4 m2 ℏ2 -
3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin +
1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+ 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2
rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+ 2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3
ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
16
3 c2 l4 m2 ℏ2 + 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2
ℏ2 rmin + 64
3 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 12
3 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+
384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+ 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin
l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+ 2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3
ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
144 c2 l4 m2 ℏ2 + 27 l2 ℏ4 + 3
3 l2 ℏ4 + 2048
3 c4 l5 m4 rmin +
192 c2 l3 m2 ℏ2 rmin + 768
3 c2 l3 m2 ℏ2 rmin + 36 l ℏ4 rmin + 72
3 l ℏ4 rmin +
1024
3 c4 l4 m4 rmin
2
+ 384
3 c2 l2 m2 ℏ2 rmin
2
+ 36
3 ℏ4 rmin
2
+
3 8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22
rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2+
16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 256 c4 l5 m4 16
3 c2 l4 m2 ℏ2 -
3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2 ℏ2 rmin +
36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l
16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+ 96 c2
l3 m2 ℏ2 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3
m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
8 Program SCCPERVMUR version for the comparison.nb
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l
16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
9 l ℏ4 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3
m2 ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l
16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
16 c2 l2 m2 + 3 ℏ2
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
64 c4
l4 m4 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2
ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2
rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+
4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2+
24 c2 l2
m2 ℏ2 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2
ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2
rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+
4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2+
9 ℏ4 16
3 c2 l4 m2 ℏ2 - 3 l2 ℏ4 + 3
3 l2 ℏ4 + 1024 c4 l5 m4 rmin + 384 c2 l3 m2
ℏ2 rmin + 36 l ℏ4 rmin + 1024 c4 l4 m4 rmin
2
+ 384 c2 l2 m2 ℏ2 rmin
2
+ 36 ℏ4 rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+
2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2
rmin+ l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+
4 l 16 c2 l2 m2 + 3 ℏ22 rmin + 4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2
2
4 16 c2 l2 m2 + 3 ℏ22
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin
2
2
2 16 c2 l2 m2 + 3 ℏ216
3 c2 l4 m2 ℏ2 + 3 l2 ℏ4 +
3
Program SCCPERVMUR version for the comparison.nb 9
3
l2
ℏ4 + 1024
c4
l5
m4
rmin + 384
c2
l3
m2
ℏ2
rmin + 64
3
c2
l3
m2
ℏ2
rmin + 36
l
ℏ4
rmin + 12
3
l
ℏ4
rmin + 1024
c4
l4
m4
rmin
2
+ 384
c2
l2
m2
ℏ2
rmin
2
+ 36
ℏ4
rmin
2
+
8 l ℏ2 16 c2 l2 m2 + 3 ℏ2rmin
l 16
3 c2 l2 m2 + 3 1 +
3 ℏ2+ 2
3 16 c2 l2 m2 + 3 ℏ2rmin
3 l ℏ2 + 32 c2 l2 m2 + 6 ℏ2rmin+
l2 ℏ2 16
3 c2 l2 m2 + 3 -1 +
3 ℏ2+ 4 l 16 c2 l2 m2 + 3 ℏ22 rmin +
4 16 c2 l2 m2 + 3 ℏ22 rmin
2
2//
численное приближение
N
Out[47]= 5.00008 × 10-11
In[48]:= l =.
In[49]:= a11 := R11 - lmin
R12 - R11
lmax - lmin
In[50]:= k11 :=
R12 - R11
lmax - lmin
10 Program SCCPERVMUR version for the comparison.nb
In[51]:= a22 := R21 - lmin
R22 - R21
lmax - lmin
In[52]:= k22 :=
R22 - R21
lmax - lmin
In[53]:= a1 := a11
In[54]:= k1 := k11
In[55]:= a2 := a22
In[56]:= k2 := k22
In[57]:= r1 := a1 + k1 l
In[58]:= r2 := a2 + k2 l
In[59]:= Ρ2 := R21 + R22 - R21
l - lmin
lmax - lmin
In[60]:= u := lmax - lmin
10
In[61]:= l1 := lmin + u
In[62]:= l2 := lmin + 2 u
In[63]:= l3 := lmin + 3 u
In[64]:= l4 := lmin + 4 u
In[65]:= l5 := lmin + 5 u
In[66]:= l6 := lmin + 6 u
In[67]:= l7 := lmin + 7 u
In[68]:= l8 := lmin + 8 u
In[69]:= l9 := lmin + 9 u
Inoculating values:
In[70]:= a2 := a2
In[71]:= k2 := k2
In[72]:= Δlmin :=
3 l2
3 l + 6 rmin + 32 rmin l2 × m2 c2
ℏ2
In[73]:= Δl1 := Δlmin /. rmin →r1
In[74]:= Δl2 := Δl1 /. r1 →r2
In[75]:= m0 = m;
In[76]:=
условный оператор
Ifm != uplmνe || m != uplmνμ || m != uplmντ,
цикл ДЛЯ
Forn = 2;
l = lmin, Ρn ≤rmax, n++, l =.;
m =.;
c =.;
ℏ=.;
Program SCCPERVMUR version for the comparison.nb 11
a2 =.;
k2 =.;
an =.;
kn =.;
Δln =.;
Δln+1 =
3 l2
3 l + 6 Rn+1 + 32 Rn+1 l2 × m2 c2
ℏ2
;
Δln =
3 l2
3 l + 6 rn + 32 rn l2 × m2 c2
ℏ2
;
Ρn+1 = Rn+1 /.
уплостить
Flatten
решить уравнения
SolveRn+1 - 1
2
l - Δln+1⩵rn + 1
2
l - Δln, Rn+1[[1]];
l = lmin;
m = m0;
c = 299 792 458;
ℏ= 1.05361 × 10-34;
a2 = a22;
k2 = k22;
an = y1,n - (y2,n - y1,n) ×
lmin
l1 - lmin
;
kn =
y2,n - y1,n
l1 - lmin
;
условный оператор
IfΡn+1 > 0, l =.;
m =.;
c =.;
ℏ=.;
a2 =.;
k2 =.;
an =.;
kn =.;
rn+1 = Rn+1 /.
уплостить
Flatten
решить уравнения
SolveRn+1 - 1
2
l - Δln+1⩵rn + 1
2
l - Δln, Rn+1[[1]];
l = lmin;
m = m0;
c = 299 792 458;
ℏ= 1.05361 × 10-34;
a2 = a22;
k2 = k22;
an = y1,n - (y2,n - y1,n) ×
lmin
l1 - lmin
;
kn =
y2,n - y1,n
l1 - lmin
;
y1,n+1 = rn+1;
l =.;
l = l1;
y2,n+1 = rn+1;
dn+1 = y1,n+1 + (y2,n+1 - y1,n+1) × x - lmin
l1 - lmin
;
x = l2;
12 Program SCCPERVMUR version for the comparison.nb
l =.;
l = l2;
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-3, l =.;
x =.;
l = l3;
x = l3;
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l4;
x = l4;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l5;
x = l5;
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l6;
x = l6;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
x =.;
l = l7;
x = l7;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
x =.;
l = l8;
x = l8;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
x =.;
l = l9;
x = l9;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.6, l =.;
x =.;
l = lmax;
x = lmax;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.6 , l =.;
x =.;
rn+1 =.;
an =.;
kn =.;
rn+1 = an+1 + kn+1 l;
Ρn+1 =.;
Ρn+1 = an+1 + kn+1 l;
an+1 = y1,n+1 - (y2,n+1 - y1,n+1) ×
lmin
l1 - lmin
;
Program SCCPERVMUR version for the comparison.nb 13
kn+1 =
y2,n+1 - y1,n+1
l1 - lmin
;
l = lmin,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;, Ρn+1 =.;
l =.;
m =.;
c =.;
ℏ=.;
a2 =.;
k2 =.;
an =.;
kn =.;
Ρn+1 = Rn+1 /.
уплостить
Flatten
решить уравнения
SolveRn+1 - 1
2
l - Δln+1⩵rn + 1
2
l - Δln, Rn+1[[2]];
rn+1 = Rn+1 /.
уплостить
Flatten
решить уравнения
SolveRn+1 - 1
2
l - Δln+1⩵rn + 1
2
l - Δln, Rn+1[[2]];
l = lmin;
m = m0;
c = 299 792 458;
ℏ= 1.05361 × 10-34;
a2 = a22;
k2 = k22;
an = y1,n - (y2,n - y1,n) ×
lmin
l1 - lmin
;
kn =
y2,n - y1,n
l1 - lmin
;
y1,n+1 = rn+1;
l =.;
l = l1;
y2,n+1 = rn+1;
dn+1 = y1,n+1 + (y2,n+1 - y1,n+1) × x - lmin
l1 - lmin
;
x = l2;
l =.;
l = l2;
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-3, l =.;
x =.;
l = l3;
x = l3;
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l4;
x = l4;
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l5;
x = l5;
14 Program SCCPERVMUR version for the comparison.nb
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l6;
x = l6;
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
x =.;
l = l7;
x = l7;
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
x =.;
l = l8;
x = l8;
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
x =.;
l = l9;
x = l9;
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.6, l =.;
x =.;
l = lmax;
x = lmax;
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.6 , l =.;
x =.;
rn+1 =.;
an =.;
kn =.;
rn+1 = an+1 + kn+1 l;
Ρn+1 =.;
Ρn+1 = an+1 + kn+1 l;
an+1 = y1,n+1 - (y2,n+1 - y1,n+1) ×
lmin
l1 - lmin
;
kn+1 =
y2,n+1 - y1,n+1
l1 - lmin
;
l = lmin,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
Null;;,
пустой
Null//
затраченное время
Timing
Out[76]= {2674., Null}
In[77]:= n
Out[77]= 9281
In[78]:= l =.
In[79]:=
таблица значений
Tableak = y1,k - (y2,k - y1,k) ×
lmin
l1 - lmin
, {k, 3, n};
Program SCCPERVMUR version for the comparison.nb 15
In[80]:=
таблица значений
Tableki =
y2,i - y1,i
l1 - lmin
, {i, 3, n};
In[81]:=
таблица значений
Table[rj = aj + kj l, {j, 3, n}];
In[82]:= l = lmin;
In[83]:= rn
Out[83]= 2.5 × 10-11
In[84]:= rn-1
Out[84]= 2.49984 × 10-11
In[85]:= l =.
In[86]:=
таблица значений
TableΔlk =
3 l2
3 l + 6 rk + 32 rk l2 × m2 c2
ℏ2
, {k, 3, n};
In[87]:= N1min :=
π
ArcTan
l-Δlmin
2
rmin
2
- 1
4 (l-Δlmin)2
In[88]:=
таблица значений
TableN1i =
π
ArcTan
l-Δli
2
ri
2- 1
4 (l-Δli)2
, {i, 1, n};
In[89]:=
привести
Reducern ≥1
2
l -
3 l2
3 l + 6 rn + 32 rn l2 × m2 c2
ℏ2
, l
Reduce: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
Out[89]= -1.07745 × 10-15 ≤l ≤-1.07743 × 10-15 || l > -1.07739 × 10-15
In[90]:=
привести
Reducermin ≥1
2
l -
3 l2
3 l + 6 rmin + 32 rmin l2 × m2 c2
ℏ2
, l
Reduce: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
Out[90]= l ≤2.00008 × 10-11
In[91]:= l = lmax;
In[92]:= N1min
Out[92]= 2.00094
In[93]:= l =.
In[94]:= l = lmin;
16 Program SCCPERVMUR version for the comparison.nb
In[95]:= N1min
Out[95]= 38 876.8
In[96]:= l =.
In[97]:= Int1 =
квадратурное интегрирование
NIntegrateN1min
2
2
- N1min + 2
ℏ2 c2
16
×
3
l - Δlmin2 + m2 c4 +
j=1
n-1
N1j
2
2
- N1j + 2
ℏ2 c2
16
×
3
l - Δlj2 + m2 c4
, {l, lmin, lmax}
Out[97]= 9.1214 × 10-12
In[98]:=
численное приближение
N1
lP
Na Δ1 Int1
Out[98]= 3.08206 × 10103
численное приближение
N1
lP
Na Δ2 Int1
In[99]:=
квадратурное интегрирование
NIntegrateN1min
2
2
- N1min + 2
ℏ2 c2
16
×
3
l - Δlmin2 + m2 c4 , {l, lmin, lmax}
Out[99]= 5.1712 × 10-18
More precise value
In[100]:=
численное приближение
N
4.17191 × 10103
1
lP Na VA ∫lmin
lmax 1
l3
ℏ2 c2
16 × 3
l2 + m2 c4 ⅆl
Power: Infinite expression
1
0.
encountered.
Out[100]= ComplexInfinity
In[101]:= VA =.
In[102]:=
численное приближение
N4
3
π rmax
3
Out[102]= 6.54498 × 10-32
In[103]:= VA := 4
3
π rmax
3
Program SCCPERVMUR version for the comparison.nb 17
In[104]:=
численное приближение
N
4.17191 × 10103
1
lP Na VA ∫lmin
lmax 1
l3
ℏ2 c2
16 × 3
l2 + m2 c4 ⅆl
Power: Infinite expression
1
0.
encountered.
Out[104]= ComplexInfinity
In[105]:= r1 =.
In[106]:= n =.
In[107]:= r1 := l + rmin
In[108]:=
таблица значений
Table[rn =., {n, 2, 9281}];
In[109]:=
цикл ДЛЯ
Forn = 1; l = lmin, rn ≤rmax, n++, rn+1 = n + 1l + rmin//
затраченное время
Timing
Out[109]= {0.03125, Null}
In[110]:= n
Out[110]= 9281
In[111]:= rn
Out[111]= 2.50005 × 10-11
In[112]:= rn-1
Out[112]= 2.49988 × 10-11
In[113]:= l =.
In[114]:= N2min :=
π
ArcTan
l
2
rmin
2
- 1
4 l2
In[115]:=
таблица значений
TableN2i =
π
ArcTan
l
2
ri
2- 1
4 l2
, {i, 1, n};
In[116]:= Int2 =
квадратурное интегрирование
NIntegrateN2min
2
2
- N2min + 2
ℏ2 c2
16
× 3
l2 + m2 c4 +
j=1
n-1
N2j
2
2
- N2j + 2
ℏ2 c2
16
× 3
l2 + m2 c4
, {l, lmin, lmax}
Out[116]= 1.55936 × 10-13
18 Program SCCPERVMUR version for the comparison.nb
In[117]:=
численное приближение
N1
lP
Na Δ1 Int2
Out[117]= 5.26895 × 10101
численное приближение
N1
lP
Na Δ2 Int2
In[118]:=
квадратурное интегрирование
NIntegrateN2min
2
2
- N2min + 2
ℏ2 c2
16
× 3
l2 + m2 c4 , {l, lmin, lmax}
Out[118]= 5.16977 × 10-18
In[126]:= 3.08206 × 10103
5.26895 × 10101
Out[126]= 58.4948
In[120]:= VA =.
In[121]:= VA := 4
3
π rmax
3
- rmin
3
In[122]:= VM :=
i=1
10
(Na Δi VApi)
In[123]:= Vs := VU - VM
In[124]:= rmax =
3
4 π
Vs
1
3
Out[124]= 4.41304 × 1026
In[125]:=
предел
Limit
арктангенс
ArcTanl
x
, x →0
Out[125]=
l2 π
2 l
Program SCCPERVMUR version for the comparison.nb 19
88
Conclusion
In the paper an approach to estimation of vacuum energy in empty space and in the
limitation vacuum effect has been developed and improved. Two most common and widespread
elements – hydrogen and helium in the Universe were considered and for them the vacuum
energy at account the two fermions of SM – the electron and the u quark has been calculated
with the aid of the unique computer program, especially developed for this article. Also for
accurately the same case of the matter and the vacuum the calculation has been carried out on the
program which was used in the previous paper, and these two results were compared to find out
progress of the entire work.
The u quark free vacuum in the new approach with the linear dispersion has the large
enough lower limit of the energy in comparison with the free electron vacuum, and in the
previous approach without dispersion they have approximately the equal lower limits of the
energies with small enough difference. The difference between the two approaches of the
calculation of the free vacuum energy lower boundary for the u quark is therefore large enough
and it is not essential for the electron. In the previous approach the lower boundary of the u
quark free vacuum energy is a little great than the electron free vacuum energy. The lower limit
u quark vacuum energy in the presence of the heavy nuclear matter, i.e. the effect in the new
approach sufficiently severe differs from the value of the same kind of the energy lower
boundary for the electron vacuum, due to the u quark is enough heavy in comparison with the
electron. As the new computation and the computation on the previous scheme show, the
hydrogen atom severer wraps the vacuum, despite its smaller sizes than the helium atom has and
the half of the charge of the helium’s nucleus. The difference between the lower limits of the
vacuum energies near the matter for the same particles in the new approach, therefore, is greater
for the hydrogen and is smaller for the helium. The differences between the new approach and
the previous approach for the u quark are greater, than for the electron, due to the greater mass of
the quark.
The electron neutrino contribute greatly in comparison with the muon and the taon
neutrinos in the lower limit of the free vacuum energy in the new approach, and in this approach
the all neutrino vacuums have much more great the lower boundaries of the vacuum energies in
contrast those the same values in the previous approach. That is why the difference of the lower
limits between these two approaches for the neutrinos is enough great. In the previous
calculation the all values of the lower limits of the neutrinos vacuum energies differ a little, and
the highest energy has the heaviest neutrino – the taon neutrino. This is explainable. The lower
boundaries of the vacuum energies in presence of the effect in the previous approach have the
same order of the quantity that is in the new approach. That says the two approaches do not
contradict each other and the new one complements the old one. The difference between the u
quark vacuum energy lower boundary and the electron vacuum energy lower boundary near the
matter in the previous approach is less than the same one in the new approach. That is correct for
the both atoms. The difference between the lower boundaries of the vacuum energies without the
limitation vacuum effect, the difference of them between these two approaches behave the same
way, like the difference between the lower limits of the vacuum energies and the difference of
them between the two approaches with the effect.
The wrapping vacuum coefficients show that the heavy nuclear matter of the atoms really
wraps the vacuums, in the new approach mostly for the hydrogen, that we should wait exactly on
89
the numbers, and these numbers one can explain, probably, by the most abundance of hydrogen,
but for the electron, and not for the u quark, like for the lower limits of the vacuum energies (the
numbers) with the effect and without it. It is needed to say that this fact is not obvious; it
obviously seems correct that the helium’s nucleus must mostly wrap the vacuums, due to the
doubled charge of it and the more spatial sizes in comparison with the electron. For the helium
the logical dependence recovers in the both cases: the new one and the previous one. In the
previous approach the logical dependence restores, i.e. the most wrapping of the vacuum takes
place in the helium atom for the u quark and the least wrapping has the hydrogen atom for the
electron-positron vacuum. The uu vacuum in the hydrogen atom wraps smaller, than the
corresponding one in the helium atom in the previous approach. The non-integer value of the
electric charge of the u quark does not impact on non-prevailing of any values for this vacuum.
The exploring effect is very small for the new approach and it is much great for the previous
approach.
Appendix
Any approximate expression has its exact analog (from the right to the left)
max
max
min
min
0
1
l
n
P
n
P l
f l
nl
f l dl
l
,
where
max
min
max
Floor
P
l
l
n
l
,
but due to enormous number of summands cannot be calculated and, therefore, must not
consider. (Here the function ‘Floor’ means rounding down)
Acknowledgments
I am very grateful to Olga Volkova, EdS for the help in writing this paper. I especially
grateful to Yuri Rudenko, postgraduate for the help with the program applied to the article.
References
1. “This is why space needs to be continuous, not discrete | Forbes Media LLC” [Internet]
[cited
2024
May
08]
Available
from
https://www.forbes.com/sites/startswithabang/2020/04/17/this-is-why-space-needs-to-be-
continuous-not-discrete/?sh=3ff5302574ea
2. G. Nekrasov (2024). A statement of the Cosmological constant problem and an effect of
the reducing of vacuum by matter based on uncertainty relations. The Papers on the SIPS
2024 by Flogen Star Outreach Publishing [Internet] [cited 2025 November 10] Available
from
https://www.flogen.org/sips2024/articles/sips24_38_00411460f7c92d2124a67ea0f4cb5f8
5_FS.html
90
3. C. Amsler; et al. (Particle Data Group) (2008). "Review of Particle Physics: Leptons"
(PDF).
Physics
Letters
B.
667
(1–5):
1.
Bibcode:2008PhLB..667....1A.
doi:10.1016/j.physletb.2008.07.018. hdl:1854/LU-685594.
4. C. Amsler; et al. (Particle Data Group) (2008). "Reviews of Particle Physics: Quarks"
(PDF).
Physics
Letters
B.
667
(1–5):
1.
Bibcode:2008PhLB..667....1A.
doi:10.1016/j.physletb.2008.07.018. hdl:1854/LU-685594.
5. C. Amsler; et al. (Particle Data Group) (2008). "Review of Particle Physics: Neutrinos
Properties" (PDF). Physics Letters B. 667 (1–5): 1. Bibcode:2008PhLB..667....1A.
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(The
link:http://kencroswell.com/alchemy.html)
Archived
(The
link:https://web.archive.org/web/20110513233910/http://www.kencroswell.com/alchemy
.html) from the original on 2011-05-13.
8. Clementi, E.; Raimond, D. L.; Reinhardt, W. P. (1967). "Atomic Screening Constants
from SCF Functions. II. Atoms with 37 to 86 Electrons". Journal of Chemical Physics. 47
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10. Navas, S.; Amsler, C.; Gutsche, T.; Hanhart, C.; Hernández-Rey, J. J.; Lourenço, C.;
Masoni, A.; Mikhasenko, M.; Mitchell, R. E.; Patrignani, C.; Schwanda, C.; Spanier, S.;
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25.1.2
11. Linde, Andrei (2005). Particle Physics and Inflationary Cosmology. Contemporary
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12. Guth, Alan (1997). The Inflationary Universe: The quest for a new theory of cosmic
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(ed.).
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in
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(1
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doi:10.1093/oso/9780198786399.001.0001. ISBN 978-0-19-878639-9.
14. Malcolm S. Longair (2008). Galaxy Formation. Astronomy and Astrophysics Library.
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vixra | 2602.0152 | Astrophysics | A problem of the connection of cosmology with elementary particle physics is shown on thelevel of uncertainty relations. At the scales about 10^-2 m the contribution of one single typevirtual elementary particles in the lower boundary of vacuum energy is considered. The observedvalue of vacuum energy or energy density on the large scale of the Universe corresponds only tothis scale. This is the energy about 3.34 GeV per each one cubic meter. The minimal high energyphysics scale achieved by experiments at present is considered. The lower boundary of theenergy is generated by the quantum vacuum of empty space and the quantum vacuum limited bymatter in the Universe mainly at scales down to 10^-15 m and more much are not in agreementwith the observed value, as that is established. These lower limits for the energies of the vacuumare considered in the model of estimating where they generate by the presence of virtual particlesin free space and the virtual particles which are limited by matter and exist together with matterin the Universe. The numerical values of the boundary energies are obtained using the computeralgorithm. | Nekrasov Grigory Yu | Astrophysics | https://vixra.org/abs/2602.0152 | 1
A STATEMENT OF THE COSMOLOGICAL CONSTANT PROBLEM
AND AN EFFECT OF THE REDUCING OF VACUUM BY MATTER
BASED ON UNCERTAINTY RELATIONS
Grigory Yu. Nekrasov
Federal State University of Education 141014, Moscow region, Mytishchi, Russia
Keywords: Vacuum, Uncertainty relations, Virtual particles, Cosmological constant, Virtual
electrons and positrons, Vacuum energy, Matter, Interaction of vacuum with matter, Vacuum
energy density
ABSTRACT
A problem of the connection of cosmology with elementary particle physics is shown on the
level of uncertainty relations. At the scales about 10-2 m the contribution of one single type
virtual elementary particles in the lower boundary of vacuum energy is considered. The observed
value of vacuum energy or energy density on the large scale of the Universe corresponds only to
this scale. This is the energy about 3.34 GeV per each one cubic meter. The minimal high energy
physics scale achieved by experiments at present is considered. The lower boundary of the
energy is generated by the quantum vacuum of empty space and the quantum vacuum limited by
matter in the Universe mainly at scales down to 10-15 m and more much are not in agreement
with the observed value, as that is established. These lower limits for the energies of the vacuum
are considered in the model of estimating where they generate by the presence of virtual particles
in free space and the virtual particles which are limited by matter and exist together with matter
in the Universe. The numerical values of the boundary energies are obtained using the computer
algorithm.
INTRODUCTION
The Heisenberg uncertainty principle is the fundamental principle of the quantum
mechanics. Written down for any pair of conjugate quantities uncertainty relation follows from
the commutator of these quantities. Uncertainty relations can be used for the assessment of
quantities not having certain values in this quantum state via the determination of classical
quantities in this state. The description of the motion of quantum particles in such a way
becomes possible due to wave-particle duality or wave-corpuscle dualism. Such assessment has
been given in [1] for the energy of the particle in the first Bohr orbit and the radius of the first
Bohr orbit expressed via momentum of the particle which does not have certain value. This
approach has been applied to hydrogen atom, and the formula gained from the uncertainty
relations coincided with the formula for the energy of the particle in the Bohr atom theory.
In turn, the Bohr atom theory can exist due to the existence of wave-particle duality. The
assessment of some quantities on uncertainty relations has to give the correct order coinciding
with the conclusions of the theory of a quantum system, for example, Bohr atom theory or
Schrödinger theory. Consideration of fundamental laws in quantum theory on uncertainty
relations for specific quantities in the order of the magnitude of these quantities is justified. The
wave-particle duality allows an using the classical laws of motion for particles at the quantum
level. Quantum effects taking place at the quantum level remain at this level due to interaction is
at this level. This interaction leads to the spatial limitation of quantum particles happening,
usually, in the electric or/and the magnetic field(s) of other particles which make small in
2
amplitude finite motion in comparison
with
the
considered
particle.
The
described situation is implemented in an
atom where the considered particle is the
moving electron, and the second particle
is nucleus, in the field of which the
electron exists. Thus, its state in the
simplest
case
of
hydrogen
atom
implements as motion on the circular
stationary orbit with the value of the
main quantum number. Also, the wave-
particle duality gives the possibility for the existence of the description of particles’ scattering
known as Rutherford's theory, where quantum particles are considered as point-like particles
moving on classical trajectories.
1. Uncertainty Relations As An Approach To Assess The Observed Energy Of The
Universe And The Modern Achieved Threshold Of Energy
The represented approach is acceptable when quantum particles are considered, but
nevertheless the classical idea of particles is necessary. By means of such theory one can assess
some quantities appearing in "right theory". This theory containing at the same time classical and
quantum description is applicable like, for example, the Bohr's theory for hydrogen atom, and is
true due to the existing of the wave-particle duality. Consider the uncertainty relations for
momentum and coordinate
2
xp
x
.
(1.1)
The relation (1.1) is applicable for one particle so that the uncertainty of the coordinates of
particle is the width of the wave packet along the axis x , and the corresponding uncertainty of
momentum is the width of the wave packet in momentum space along the same axis [2]. This
relation can be applied for real particle or virtual particle. Virtual particles, according quantum
field theory (QFT), fill the vacuum everywhere and always. An approach considering in this
paper employs only the quantum vacuum of the quantum electrodynamics (QED), where it is
taken virtual electron-positron pairs and quantum photon vacuum is outside of the consideration.
The vacuums, generated by other matter particles and fields of other fundamental interactions are
to be the part of consequent work. If the vacuum is filled by virtual particles, one can imagine
one single virtual particle in the sizes of its, limited by interaction on the quantum level, wave
packet, then expanding the model, according the concept of vacuum, on all space of the Universe
by translation this cell for the all directions and the arbitrary distances of space, it can be gained
the simple approximate model of the fraction of real vacuum. Thus, this uncertainty relation as
correct for one particle will not lose force in the case of the large number of virtual particles in
the vacuum. So, one can use this relation for the vacuum, then, the first, it is needed to care that
the average value of every component of momentum (and energy) must be zero:
0
ip
,
(1.2)
0
E
,
(1.3)
that is justified for vacuum. This can be achieved by placing the middle of the corresponding
interval exactly at the point zero on its axis (see fig. 1). If that, the value
2
p for the momentum
can be used to express the uncertainty interval of the momentum through this value
fig. 1 – Uncertainty of the x component of the momentum
and the average value of the x component of the momentum
(median) for vacuum virtual particle
3
2
1
p
p
p ,
(1.4)
where
1
2
p
p and one can get
2
2
p
p .
(1.5)
The second, it is necessary to choose the detecting values of the uncertainty intervals, because
the interval of uncertainty does not imply the certain value of corresponding quantity. One needs
to give a value to the quantity which can give a value within the interval owing to the detecting
or the measurement in the general sense of word, according the wave-particle duality, applying
to this case and according the limit transition to the classical/quasiclassical physics. Let will be
chosen the maximum values of the uncertainty intervals in momentum space as detecting values
to get the maximal possible energy of the vacuum, and these intervals in such a way can be
expressed via this value (see fig. 1). This will show an upper limit for the energy exists. For
vacuum is essential energy but not momentum, therefore one can pass from the maximal value of
momentum according to (1.1) to the maximal value of energy corresponding this momentum on
the formula of special theory of relativity
2
2
2
2
4
2
2
E
p c
m c
(1.6)
in the inequality expressing the values
2
p from (1.1), according (1.5) and collecting them as it is
required in (1.6). At that as it is accepted in such assessments, one doesn’t need to use virtuality
for virtual particles to be taken into account, they are thought like real particles. It is also going
to use the equality condition of the momentum components. As a result of simple applying the
similar inequalities to each other one can receive the uncertainty relation for the energy and the
all location uncertainties along each of 3 space axes
2
2
2
2
4
2
2
2
1
1
1
16
c
E
m c
x
y
z
,
(1.7)
or
2
2
2
2
2
4
2
4
2
2
2
2
2
2
1
1
1
1
1
1
16
16
c
c
m c
E
m c
x
y
z
x
y
z
.
(1.8)
Passing from energy to energy density
2
2
1
1
2
4
2
2
2
1
1
1
16
c
x y z
m c
w
x y z
x
y
z
(1.9)
2
2
2
4
2
2
2
1
1
1
16
c
m c
x
y
z
.
At this stage it is needed to choose a form of the virtual particle wave packet. Now the
wave packet has the form of rectangular parallelepiped. As was considered above this
uncertainty relation can describe not only one particle and the vacuum as whole. This is actually
for the inequality for the energy density. From here it can be determined the scale of the
observed cosmological energy density. For this put
x
y
z
,
(1.10)
so it was chosen the cubic form of the wave packet or cell of the vacuum. As the vacuum energy
according observation is positive, one should take the inequality with positive energy
2
2
3
2
4
2
3
16
c
w
x
m c
x
.
(1.11)
The energy density here is the observed cosmological energy density, which follows from the
cosmological constant [3] on the formula [4] and has the value in SI
w
Then, it can be gained the foll
(for one single type of particl
chosen acceptable solution (re
.
Then the minimal energy of
uncertainty one can gain havin
It is well-known that t
[5] (for instance, one can take
(‘HEP’ is ‘High Energy Phys
single type of particles, a num
It leads to the conclusion th
energy is insignificant. One c
energy density of real electron
be on the confirmed in
theory. This fact as was
manner of this article is yet kn
cosmological constant proble
understanding for a resoluti
problem might express in
cosmological energy is not th
only ordinal matter and intera
already found in nature and
undiscovered essence.
2. Uncertainty Relation
Quantum Vacuum A
2.1.
The General Co
Considering structure
provided by P. Dirac, the prob
(not vice versa in this contex
particles. The problem can be
of the real electron in an atom
in an atom, i.e. in the electric
them location uncertainties, th
vacuum limitation by matter
4
4
10
3
3
J
GeV
5.34 10
3.34
8
m
m
c
G
.
lowing inequality of eighth order
8
2
2
4
2
2
2
2
3
0
16
x
w m c
x
c w
,
les) which must be reduced to the number in
eal, positive). Further it is laid x
l
. Thus, o
0.054m
l
.
such virtual particle, i.e. particle having thi
ng multiplied the volume of this scale by the
3
14
8.41 10
J
0.53MeV
E
l w
.
the modern high energy physics is gone dow
18
6.18 10
m
), which according the assessm
2
2
3
2
4
2
3
16
HEP
HEP
HEP
c
w
l
m c
l
sics’) gives the energy density only at this s
mber
43
3
J
10
m
HEP
w
.
hat the contribution of the quantum vacuum
can get relative error between these numbers
n-positron vacuum on the 53 orders of magnit
experiment
shown in
nown as the
em [6]. An
ion of this
n that the
e energy of
actions have
also other
ns As An Approach To Assess The Energy O
nd The Energy Of The Vacuum In Presenc
onsideration
of the vacuum, the mechanism of the pro
blem of description an interaction any type o
xt) comes down to the problem of the limitat
e addressed, as the author offers, like the pro
m, by consideration the virtual electron of the
field of atomic nucleus. Atomic nuclei are en
herefore, must be small; as the vacuum pola
effect must act simultaneously. The vacuum
fig. 2 – Uncertainty of the energy and
energy (median) for vacuum
(1.12)
(1.13)
nequality, and must be
one gains this scale
(1.14)
s lower limit of linear
energy density (1.12):
(1.15)
wn to 10-18 meters scale
ment
(1.16)
scale and only for one
(1.17)
m in the cosmological
s and calculate that the
tude lower then it must
Of The Free
ce Of Matter
ocesses in which was
of vacuum with matter
tion of vacuum virtual
oblem of the limitation
e electron-positron pair
nough massive, so that
arization effect and the
m limitation by matter
d the average value of the
m virtual particle
5
effect or simply the vacuum limitation effect is the name for a new effect of the reducing of sizes
of wave packets vacuum virtual particles creating near strong condensed matter, and can be
found as yet one phenomenon of QFT from the qualitative analysis. This effect is explained by
the quantum nature of the virtual particles of vacuum. The virtual particle is described by wave
function, maximum(s) of which can be localized, like in the case of the electron in atom. Such
matter field or wave function has much strength with non-zero strength gradient.
The elementary particles of matter are also offered as such natural limiting structures,
when they are localized. For example, the electron can limit vacuum energy density by its own
electric field. If to consider the naked electron (i.e. the point without the field), according the
formula for the strength of the electric field of the electron in its coordinates:
2
ke
E
r
,
0
r
, the
strength of the field will be equal to infinity E . Thus, the strength of the field in the
coordinates of the electron limits the vacuum, so that the energy density of the vacuum is equal
to infinity as well as the energy density of the electric field of the electron:
2
2
2
0
0
4
2
2
E
k e
w
r
,
0
r
and w .
The electron in its coordinates limits the wave function of the virtual particle of the
vacuum to the zero sizes, so that the vacuum energy density is equal in this point to infinity. And
it is not surprising, the electron in the modern physics is considered as the point-like particle, is
the singularity with infinite space-time curvature and the characteristics of density (the energy of
field, mass). Also and that is important, the electron limits the wave functions of the virtual
particles of the vacuum at any distance from its coordinates. But, apparently, the greatest
contribution in the energy density is made by the vacuum particles limited near the electron.
Any particle, for example, proton or atomic nucleus can limit vacuum particles, and not
only electrons.
The calculation of the scale for the observed energy density of the vacuum which is
performed above is not accurate, as matter, which limits the vacuum everywhere, is not
considered. It is necessary for the correct calculation to consider the density (and quantity) of
matter in the Universe which limits the vacuum. The formula (1.8) or equivalently (1.9) does not
change, but on one’s representations, gained from the qualitative analysis of the problem, matter
will lead to the effective reducing of the considered scale on the value
l
making up the new
scale on which the energy/energy density is set without matter. Though matter has small location
uncertainties, it is not localized with absolute accuracy, singularities in the coordinates of
particles do not exist, they are cut out from the picture of matter, in which they can
asymptotically be for the simplicity of the model.
So, the lower assessment of the energy density existing due to the virtual particles on the
scale l in the free vacuum gives by the formula
2
2
3
2
4
2
3
16
v
c
w
l
l
m c
l
,
(2.1.1)
‘wv’ has brackets to mark the dependence of the assessment. And in the vacuum containing
matter the scale l must be effectively reduced, so that the assessment will give by the formula
2
2
3
2
4
2
3
16
v
v m
c
w
l
l
w
l
l
l
m c
l
l
,
(2.1.2)
‘v’ means ‘vacuum’ and ‘m’ means ‘matter’.
There exists in the quantum mechanics the uncertainty relation for energy and time,
which is suitable for this reasoning. It has the view
here
E
is the uncertainty o
lifetime of virtual particle-ant
view the pair can have any en
the temporal interval is enoug
decreasing the lifetime and v
particles, as it is written for e
particles.
The uncertainty interv
intervals (see fig. 2), therefore
and it is implied that the valu
is the detecting value.
As was mentioned abo
act simultaneously, but both
electron-positron creates exac
the virtual electron is closer t
energy
2
E in (2.1.4) and in (
(2.1.3) for the theory.
If t
is the lifetime of
where v is the velocity of the
formula is approximate unless
that its motion is accelerated
electron moves in average un
scale, caused by presence of
first, it is needed to place refe
have the small sizes of t
packets in location space, in th
of this phenomenon (see fig.
the nucleus the pair electron
creates exactly on the axis o
connecting the proton, the ele
the positron. The electron i
near the proton and the posit
from the proton. This creates
which is justified due to th
directions, i.e. along any axes,
which are located on the axe
other directions are not instan
the rays going from the center
And for the vacuum limitati
polarization effect, the first, t
that electrons should be creat
proton. Otherwise the pair is g
orientation of the pair. The se
going to be rotated from this
6
2
E t
,
of the energy, which a quantum system ca
tiparticle pair before annihilation and vanishi
nergy which hits the arbitrary uncertainty int
gh short. Concretely, one can arbitrary increa
vice versa. Note, that the relation (2.1.1) is c
energy density, and the relation (1.1) is corre
val for energy according (1.3) is arranged
e it is correct
2
2
E
E
,
ue of the energy
2
E , the maximum value of t
ove the vacuum polarization effect and the va
these effects act much more considerably
ctly on the axis, on which the nucleus and this
to the nucleus and the virtual positron is far
(1.6), (1.7) and (1.8) is the same energy. H
f the pair, it must approximately be correct
2
v t
s
,
e electron, s is the half of the passed distanc
s to think that the velocity of the electron mus
d. However, in this approximation one can
niformly. A challenge is determination an eff
matter in space. This can be done using sim
erence matter – the proton or another nucleus,
the wave
he picture
3). Near
n-positron
or the line
ectron and
is located
tron is far
s a model
hat in real electron-positron vacuum virtu
, but for the quantum polarization effect are s
es or the rays going from the charged center
ntly contributed in the effect. The directions
r are not equal to zero contribute as dipoles o
ion effect everything is effectively the sam
the rays directions are preferable for the eff
ted near the proton or nucleus and virtual po
going to annihilate much more quickly than
econd, if the direction has zeroth projection o
orientation owing to the electrical force, ac
fig. 3 – Geometrical scheme of the
(2.1.3)
an have and
t
is the
ing. From this point of
terval of the energy, if
ase the energy interval
correct only for virtual
ect for virtual and real
d like the momentum
(2.1.4)
the uncertainty interval
acuum limitation effect
when the virtual pair
s pair are lying so, that
from the nucleus. The
Hence, one can employ
(2.1.5)
ce by the electron. This
st really be variable, so
think that the virtual
fective reducing of the
mple 2D geometry. The
, as they are heavy and
ual pairs create in all
significant only dipoles
r, i.e. the particle, and
s whose projections on
of least dipole moment.
me as for the vacuum
fect with the condition
ositrons – far from the
in the case of the right
on the ray, such pair is
ting on the electron to
vacuum limitation effect
7
attract it and on the positron to push away it from the particle. Thus, as it is seen, this case is
going to be transformed to the intermediate case with nonzero projections on the rays or even to
the case which is implied the pair, lying exactly on the ray. These resulted cases are justified
only on the enough distance from the proton, for the pair must be able to rotate, it must have
enough free space before the pair vanishes. The third, if the pairs create not exactly on the rays,
this is intermediate case, they contribute as dipoles of least dipole moment in dependence on the
angle between the corresponding ray and the axis on which the pair lies. Also in this case they
are going to rotate to the rays directions as far as they could do it for them lifetime. In any case a
model which the author offers is the first approximation and it is justified.
When the electron lying on the axis approaches to the proton in the time of scattering at
once after the creation, the wave packet of the electron begins to move between the strength lines
of the static electrical field of the proton coming nearer to the charge center. Now the dispersion
of the wave packets of the virtual particles is not taken into account. In the process of that motion
the electron’s wave packet is reducing and it moves only between the same electrical field
strength lines (see fig. 3). At that even if one takes into account the dispersion of the virtual
matter waves, the positron moving away from the proton can increase the sizes of its wave
packet and by that could decrease the vacuum energy. And the electron considering at an account
of the dispersion nevertheless is going to reduce the sizes of its wave packet, so the dispersion
will be suppressed by the tightening of the electron into the considering singularity between the
same strength lines, i.e. actually by the vacuum limitation effect. The electron will move in such
a way up to the certain limit of distance and it will turn back to the positron to annihilate. This
limit from the center of the electron’s wave packet to the center of the proton is designated as r .
The electron goes the full way 2s , i.e. in the one direction from the positron to the proton and in
the opposite direction to the positron and from the proton for its lifetime
t
. In the initial state
the electron has the linear sizes of its wave packet l , in the point of the maximum removal from
the creation point it has linear sizes l
l
, the corresponding angles are seen on the Figure 3,
and the reducing of the linear sizes of the electron’s wave packet is designated as l
, and this is
the required effective reducing of the scale generated by matter.
So, basing on the picture and using (2.1.5) one can gain the formula
tan
l
v t
.
(2.1.6)
Also the following formula can be gained using Figure 3
tan
2
l
l
r
.
(2.1.7)
Combining these two formulae, one gets the formula
2
1
l
l
r
v t
.
(2.1.8)
Expressing the lifetime from (2.1.3) in the form of inequality and using (2.1.4), simultaneously
expressing the lifetime from (2.1.8), and collecting this all in the one single inequality, one can
gain the inequality
2
8
1
E r
v
l
l
.
(2.1.9)
Further it is laid
2
E
E
,
(2.1.10)
for the energy in (2.1.4) and in (1.6); energy entering in (1.7) and in (1.8) is the same. The
following formula of special theory of relativity will be needed
8
2
E
c
v
p
.
(2.1.11)
Multiplying the inequality (2.1.9) by the energy E and dividing it into the squared speed of light
one can gain the inequality
2
2
2
8
1
vE
E r
l
c
c
l
,
(2.1.12)
where it is thought
0
E
. Using (1.5) without index 2 in (1.1) and the similar inequalities for the
other space components, expressing the momentum from these inequalities, equating the right
hand side of (2.1.11) and the gained vector expression, one can take normalization operation in
these inequalities and can gain the inequality
2
3
4
vE
c
x
.
(2.1.13)
Coupling (2.1.12) and (2.1.13) one gets
2
2
2
3
8
4
1
vE
E r
l
l
c
c
l
.
(2.1.14)
Expressing t
from (2.1.8)
2
1
r
t
l
v
l
,
(2.1.15)
it is seen, for it must be
0
t
, the following inequality must be executed
l
l
.
(2.1.16)
Taking the positive part of (1.8) and taking into account (1.10) it can be gained the
assessment
2
2
2
4
2
3
16
c
E
m c
l
,
(2.1.17)
also using (2.1.15) in (2.1.3) at account (2.1.4) it can be gained the assessment
1
8
v
l
E
r
l
.
(2.1.18)
It is needed to suppose the equality between (2.1.17) and (2.1.18). Equating them, one can solve
this algebraic equation for the velocity of the particle
1
2
2
2
4
2
8
3
1
16
r
l
c
v
m c
l
l
.
(2.1.19)
Now the velocity v is expressed via l ,
l
and r variables. The using (2.1.19) in (2.1.13) can
give yet one assessment for the energy
1
2
2
2
2
2
2
4
2
3
3
1
32
16
c
l
c
E
m c
rl
l
l
.
(2.1.20)
Also one should suppose the equality between (2.1.17) and (2.1.20). Then, solving the gained
equation for the scale reducing, one gains
2
2
2
2
2
3
3
6
32
l
l
m c
l
r
rl
.
(2.1.21)
9
Having gained the result (2.1.21) for the effective vacuum scale reducing by matter one can
check the inequality (2.1.16), and it is seen from here that this inequality-condition is executed.
The result (2.1.21) must be substituted in (2.1.2) for one single scale l and distance r to
the proton or nucleus or much localized heavy particle. It is obvious that in reality all the scales
exist and all they contribute to the vacuum energy/energy density. If to take into account this fact
and that as was described above about any orientation of the axis of the pair’s scattering, one
needs, the first, to sum each the right hand side of (2.1.2) inequality for its own scale or linear
size of the electron’s wave packet l , to take into account all the scales. Further it is needed to
insert some important remarks. One considers only the ‘right’ orientation of the electron-positron
pairs, namely, negative charge must be in the hemisphere turned to the positive-charged matter
considered. Also one considers only an increasing of the vacuum energy, which happens when
the virtual electron attracts to the proton, decreasing sizes of its wave packet, and the moving
away positron, as was said above, decreases or at least does not change the vacuum energy, when
it increases sizes of its wave packet. So, the moving away positron once after the creation of the
pair might have the negligible addition to the vacuum energy (it is possible) like the coming (to
the electron) positron for annihilation. The coming positron decreases sizes of its wave packet
only to its former sizes, when it was just created, and the electron, which has been so in detail
considering in this paper, when it goes back to the annihilation point, is increasing the sizes of its
wave packet that might slightly increase of the vacuum energy (it is possible). This is because
the electron yet had the reduced sizes of its wave packet when it was scattering together with the
positron from the creation point. Such increasing is smaller than the former sizes of the
electron’s wave packet. Of course, this all must be taken into account in the accurate calculation,
which will be done in consequent work. However, the positron in this situation like the electron
in empty space (without matter) contributes to the vacuum energy that was seen from the Section
1. Further the previous will be discussed. The second, because the axes can have arbitrary
orientation remaining in the right hemisphere, they have any projection on the corresponding ray,
so that the projection can vary from zero to maximum value 2s lying exactly on the ray, and this
is the dipole without taking into account the rotation of real dipoles in the static electrical field of
the nucleus. Thus, this dipole-projection has, like the ray dipoles, the distance to the nucleus, i.e.
it is on the distance from the nucleus to the center masses of the dipole. So, as in this model all
the dipoles lay exactly on the rays, to gain the full energy density one should sum every
component of the sum in the first step varying the distance r from the least value at which the
considering model is applicable to the maximum value at which the model is still working.
As well-known 3D space in the usual physics is continuous, therefore the scale l varies
continuously, and the sum, about which was said in previous subparagraph actually must be
replaced with an integral. But if one does that, a problem is appeared, because the integral is
actually a sum with differential that changes the dimensionality of the quantity and adds this
unnecessary difference. It is needed only the sum, but the sum must be continuous. This problem
can be resolved by consideration of the definition of integral
1
lim
b
N
i
i
N
i
a
f
x dx
f
x
x
,
(2.1.22)
l
must go to zero, so that
vacuum energy. Thus, on this
space with minimal in avera
implies anisotropy on fundam
Planck length is determined by
and has the value
Thus, the continuous sum ca
approximately to the continuo
can be restricted as
or if integrand decreases rapid
that is not meaning the Univer
for a while. As it is seen one g
Above it was describe
found lower estimation (2.1.2
one single wave packet or all
space. It is clear that this can
location or spatial packaging o
and well-localized particle wi
which virtual particles are. T
completely acceptable. Actua
particles packaging is a conce
it for each layer of arbitrary
particles’ wave packets is thou
one flat layer with the balls of
fig. 4 – Meridian slash of the o
p
10
where
has
dimensi
differen
ix
can
brackets
limit
therefor
the integ
restore
dimensi
sum to
problem
in cont
one gains the infinite expression and corres
s stage it is needed to suppose not continuous
age in all the directions (because the discr
mental scale) length, which one chose the P
y fundamental physical constants
3
P
G
l
c
35
1.62 10
m
Pl
.
an no more be continuous, and represents t
ous integral divided into the Planck length. T
min
max
,
l
l
l
dly enough
min,
l
l
,
rse must surely be infinite. And one has the re
min
max
,
r
r
r
gets an integral with at least one infinite limit.
ed one virtual pair and, correspondingly, on
2) is assessment for the energy density
v m
w
l a universe consisting of such wave packet
n’t describe the real Universe. At this stage
of the virtual particles. One must consider on
ith certain vicinity. Within this atom one co
The cubic structure of the vacuum virtual pa
ally, the most convenient for the representat
entric ball layers. So that, particles can be in t
thickness and distance from the center of sp
ught a spherical or almost spherical. In the Fi
f the particles’ wave packets on the zeroth me
one layer of the concentric ball layers
ackaging
i
b
a
x
N
. The last
the
additional
ionality x due to the
nce b
a
. The segment
n be taken out from the
s of the sum and it in
is
infinite
small,
re one should divide
gral into the step l
to
the
right
ionality and only the
o be left. But a new
m is appeared, namely,
inuous space the step
spondingly the infinite
s, and discrete physical
rete structure of space
Planck length
Pl . The
(2.1.23)
(2.1.24)
the discrete sum equal
The integral on scale l
(2.1.25)
(2.1.26)
estriction
(2.1.27)
.
e virtual electron. The
l which can describe
s in each cubic cell of
one needs to describe
ne single atom or heavy
onsiders spatial cells in
articles in space is not
tion and description of
this layer up to full fill
pheres. A shape of the
igure 4 is pictured such
eridian of the ball. One
needs to package all the balls
as they are gone in each such
side of regular tangential poly
As, the perimeter of such poly
Also according the Figure 4, t
Equating (2.1.29) and (2.1.30)
where the radius of the inscri
Figure 4
Thus, N is the number of th
number of the balls, because
polygon. It is gained the equat
From here
On the Figure 4
pictured 2D projection on t
meridian
plane
of
t
considering real 3D model
the virtual particles packagin
One needs to understand ho
many balls are going to go
one single arbitrary concentr
ball layer. To understand this,
is needed and sufficient
consider an equatorial slash
the 3D picture, pictured on t
Figure 5. The meridian sectio
is included, then, at changin
the azimuthal angle all t
sections, the flat circles a
summed without the polar bal
which were considered in th
zeroth meridian flat circle. T
11
of the wave packets in each one and all one
h layer. This must be calculated geometrical
ygon is given by the formula
2
tan
N
b
R
N
.
ygon
2
tan
N
L
Nb
NR
N
.
the perimeter is equal
L
N l
l
.
), one gains
2
tan
l
l
R
N
,
ibed circle can be calculated from the Pytha
2
2
1
4
R
r
l
l
.
he sides or the number of the vertices of the
e the center of each ball is located exactly
tion for N
2
2
1
2tan
4
l
l
r
l
l
N
.
2
2
arctan
2
0.25
N
l
l
r
l
l
.
is
the
the
of
ng.
ow
in
ric
, it
to
of
the
on
ng
the
are
lls
he
The balls on the equator go in twice, that
fig. 5 – Equatorial slash of the one layer of the co
around a particle is parted
es these layers as much
ly. As well-known the
(2.1.28)
(2.1.29)
(2.1.30)
(2.1.31)
agoras’ theorem, using
(2.1.32)
e polygon and it is the
y on the vertex of the
(2.1.33)
(2.1.34)
is why one needs the
oncentric ball layers space
d on
12
multiplier ½. So, it is needed to sum
2
N
balls as many times without considering in projection
on the equator two meridian balls and without duplication, i.e. using the one-half multiplier as
they go in. Therefore, one gains the full number of the balls (wave packets) going in the one
single arbitrary ball layer
2
2
2
N
A
N
.
(2.1.35)
It is useful to simplify this result
2
2
2
N
A
N
.
(2.1.36)
At that the thickness of the ball layer is
l
l
,
(2.1.37)
the radius of the most sphere and the radius of the least sphere correspondingly
2
1
2
r
l
l
,
(2.1.38)
1
1
2
r
l
l
.
(2.1.39)
2.2.
The Theoretical Vacuum Energy In Presence Of Matter, The Theoretical Free
Vacuum Energy And The Observed Vacuum Energy
If
v m
w
l
is multiplied by
3
l
l
, it can be gained the energy
v m
E in this cell of space
with the side l
l
, i.e.
3
v m
v m
E
w
l
l
.
(2.2.1)
Let it be designated
:
cell
v m
v m
E
E
.
(2.2.2)
If l is fixed and r is arbitrary, then, according the considered geometrical model of the vacuum
near matter, its energy will be equal
cell
v m
a
v m
E
N AE
,
(2.2.3)
where
a
N is the number of atoms in the Universe, and here each ball is now thought being in a
cube with side l
l
. The energy (2.2.2) like the energy satisfied positive part of (1.8) at
condition (1.10) not necessarily can relate to the cubic wave packet of the virtual electron, it can
correspond to a ball-shaped wave packet as well. In the model described in this work it is thought
that the wave functions of virtual particles at the constant scale l are not overlapped. Therefore
the distance r must change discretely, to do not allow the wave functions are overlapped. As
was noted yet, radii the most and the least spheres are set by formulae (2.1.38) and (2.1.39); with
the substitution
min
r
r
and holding l
const
that is related to the first ball layer, where
min
r
const
and this radius is the introduced value. The next distances can be calculated by
consistent solving of the system of the algebraic equations
1
1
2
min
,
,
r l
r
l
,
2
1,
r l
,
(2.2.4)
1
2
2
1
,
,
r l
r l
,
2
2,
r l
,
(2.2.5)
1
3
2
2
,
,
r l
r l
,
2
3,
r l
,
(2.2.6)
and so on, for
1r ,
2r ,
3r and so on. Now l can be variable. Range of l is (2.1.25) or (2.1.26),
where
min
0
l
means the ultraviolet cutoff. The full vacuum energy near matter for one single
13
type of particles (particles with them own antiparticles, the pairs) according to the all said above
now takes the form
min
0
0
1
,
v m
v m
n
n
P l
E
E
l r dl
l
,
(2.2.7)
where 0
min
r
r
. Then, the estimation for the full vacuum energy near matter takes the form
min
2
2
2
0
2
4
2
0
,
1
3
,
2
2
16
,
n
v m
a
n
n
P l
n
N l r
c
E
N
N l r
m c dl
l
l
l l r
,
(2.2.8)
where
, n
N l r
is defined by (2.1.34),
, n
l l r
is defined by (2.1.21) and discretization of
nr is
defined by the system of equations like (2.2.4) – (2.2.6). The infinite sum in (2.2.8) can be
reduced to an approximate finite sum if one takes into account the sizes of real atoms. The
experimentally determined sizes (radii) of atoms have values in the range 30 pm – 300 pm ±5
pm [7], where 1 pm is the picometer, as
12
1pm
1 10
m
, the minimal value has hydrogen atom
25 pm and the maximum value has cesium atom 260 pm, and the most heavy atoms existing in
the Universe such as uranium and plutonium have a value 175 pm with accuracy ±5 pm [7] (1 Å
= 100 pm).
Thus, basing on the Section 1 and the current section of this paper, the full vacuum
energy in the Universe with one single type of virtual particles (with them own antiparticles)
taking into account matter existing in the Universe and empty space without matter takes the
form
0
0
0
v
v m
E
E
E
,
(2.2.9)
where
0
v
E
is the full vacuum energy in empty space of the Universe. The energy
0
v m
E is already
found, to find the energy
0
v
E
it is needed to multiply the minimal value of (2.1.1) by the volume
of one single cell, in which the energy is; and one finds the energy in one single cell of empty
space at fixed l
3
v
v
E
l w l
.
(2.2.10)
Analogously to (2.2.2), let it be designated
:
cell
v
v
E
E
.
(2.2.11)
The energy density
v
w l in the inequality (2.1.1) describes one single cell of space or all empty
universe at fixed scale l . To take into account all the scales, one needs, like it was done in
(2.2.7), to integrate the energy density entering in (2.2.11) over all the scales, and, as it was
discussed above in the relation of the definition (2.1.22), one needs to divide this integral into the
Planck length Pl to gain the approximate result (see the text above related to (2.1.22)), i.e.
min
0
1
v
v
P l
w
w dl
l
,
(2.2.12)
the range of l can be taken from (2.1.25) or, as it was done here, from (2.1.26); and the spatial
ultraviolet cutoff
min
l
was used. It is needed to note that here the space is thought discrete as
well. Thus, now it is thought that the energy densities
v
w and
0
v
w
describe all empty space of a
universe. To gain the full vacuum energy of empty space of the Universe (the part of all the
space), one should multiply the energy density (2.2.12) by the volume of empty space of the
Universe
0
0
v
v
s
E
w V
.
(2.2.13)
14
Then, the assessment (2.1.1) for the full vacuum energy of empty space of the Universe will be
min
2
2
0
3
2
4
2
1
3
16
v
s
P
l
c
E
V
l
m c dl
l
l
.
(2.2.14)
Basing on (2.2.9) the estimation for the full vacuum energy can be gained and takes the form
min
2
2
2
2
4
0
2
0
,
1
3
,
2
2
16
,
n
a
n
n
P l
n
N l r
c
E
N
N l r
m c dl
l
l
l l r
(2.2.15)
min
2
2
3
2
4
2
1
3
16
s
P
l
c
V
l
m c dl
l
l
.
Now in the formulae above the energy density of the space cell relates to the cubic wave packet
of the virtual particle that is not natural enough. Not only energy can describe the both types of
wave packets, are cubic and ball-shaped, energy density can also be transformed allowing it to
correspond to ball-shaped wave packets as well. Any energy density in this work consists of
energy divided into volume in which this energy is distributed. As already was said, energy is
invariant to changing between cubic and ball wave packets, this means that it is needed to
transform only volume into which energy is divided. Now, one has the volume of a cube with
side l or l
l
, to consider the volume of a ball one should multiply the volume of a cube by
6
in all the formulae for energy density.
In this paper the vacuum energy distributed only in average on chemical elements
existing in the Universe has been estimating. The accurate calculation considering the vacuum
energy distributed on the quantities of chemical elements existing in the Universe will be the part
of consequent work.
One defines the average number of particles in the following way
av
m
U
a
V
N
m
,
(2.2.16)
where
m
is the average density of matter in the Universe;
U
V is the volume of the Universe;
av
m is the mass of one average particle in the Universe. Each particle gives rise the spherical
vicinity around itself. The volume of vicinity is the volume of atom, thus, the volume of all
matter in the Universe can be calculated on the formula
A
a
M
V N
V
.
(2.2.17)
As well-known the average density of matter in the Universe is equal to the critical
density (the error makes up about 1%) [3, 4]
m
c
,
(2.2.18)
2
3
8
c
H
G
,
(2.2.19)
where H is the Hubble constant [8]. It is required to calculate the mass of the average particle. If
that how many each sort of atoms are in the Universe is known, then the number of all heavy
well-localized particles and atoms is known as well, to find the mass of the average particle, one
must sum the multiplications of the masses of the particles of each sort by the numbers of the
particles of these sorts to find the full mass of the Universe (the mass of all matter in the
Universe) and one must divide this total mass into the number of all particles, as a result the
mass of the average particle will be found; for atoms it will have the view
15
93
1
av
93
1
k
k
k
k
k
m N
m
N
,
(2.2.20)
where
k
m is the mass of k-th nucleus of this chemical element,
k
N are the numbers of atoms in
the Universe numbered by index k, 93 is the number of the natural chemical elements.
One receives the expression for the estimation (2.2.14) of the vacuum energy density
related to the part of the Universe without matter
2
2
0
2
2
2 2
2
2 2
min
min
2 2
2
2 2
min
min
1
3
3
16
16
64
36
v
s
P
mc
E
V
m c l
m c l
l
l
m c l
.
(2.2.21)
It is possible to suppose
min
P
l
l
,
(2.2.22)
if one does that, the assessment (2.2.21) takes the form
2
2
0
2
2
2 2
2
2 2
2 3
2
2 2
3
3
16
16
64
36
v
s
P
P
P
P
mc
E
V
m c l
m c l
l
m c l
.
(2.2.23)
Actually, the volume of the Universe is
U
s
M
V
V
V
.
(2.2.24)
Hence, if
s
V is not known (not observed), the volume of all empty space can be expressed from
here
s
U
M
V
V
V
,
(2.2.25)
where
U
V is the volume of the observed part of the Universe, which can be assessed from
observations. The volume of matter also is known due to that the full number of atoms in the
Universe has been assessed as well. If one accepts hypothesis (2.2.22), substituting (2.2.25) in
(2.2.23), considering the values SI for m , c , ;
Pl is (2.1.24),
80
3
3.5 10 m
U
V
[9],
1
80
1H
10
a
N
[10] (if to think that the all atoms are hydrogen atoms),
1
32
3
1H
6.55 10
m
A
V
(diameter
1
11
1H
2 25pm
5 10
m
d
), the lower number estimation
0
193
2.32 10
J
v
E
.
(2.2.26)
This value is absolutely far from the observable value
0 obs
71
1.87 10 J
v
E
,
(2.2.27)
calculated on the value (1.12) on the formula
0 obs
v
U
M
E
V
V
w
.
(2.2.28)
It is seen that only in free space the divergence is 122 orders of magnitude, this is the
cosmological constant problem, with the hypothesis of discrete space.
2.3.
Features Related To The Model
In the concept of discrete space is not well-understood how matter moves through this
space. If one considers one point particle, that it can move by jumps from one point to another
divided by the Planck length in the all three directions. At that the fundamental quanta of space is
the cube with the Planck length side. For point particle time interval between jumps does not
exist, and particle can be in this point, the vertex of the cube or be in other vertex, that particle
can do jumps instantaneously. This may not be said for macroscopic bodies, they can move
16
through that discrete space with varies velocities. Every point in such body (the point of the
wave functions) does jump from one to the next point of space but they delay in each point of
space for the period t
(here it is not the lifetime of the particle) of infinitely divisible time. It is
alike the moving between two points, as that if the particle-point started from this point at the
moment of time t , in the next point of space it must delay for the period
1t
t
t
, having
jumped between these two points instantaneously. Thus, doing such jumps with delays, this from
large, macroscopic scale is going to see as a motion with seeming arbitrary velocity. But on the
fundamental level for each quanta of space this ‘motion’ is alike the continuous real motion
between each two points of space, i.e. along the Planck length
Pl and the delay is going to seem
the period of time for which the real continuous motion would happen. Thus, the instant velocity
of motion in discrete non-infinite divisible space (constrained by the fundamental scale which is
the Planck length) happening for infinite divisible and continuous time is to be
P
f
l
v
t
(2.3.1)
for each linear quanta of space
Pl . Indeed in such a way the motion in discrete space must be
justified. This article presumes the discrete nature of real physical space. Also in the model
described in the Section 2 of this paper it is neglected with anisotropy on the fundamental level,
which occurs due to the discreteness of space. One can easily imagine what really is a
fundamental wave packet or a wave packet reduced down to the fundamental level: due to the
discreteness of space the Planck-sized wave packets of particles are eight-point, on the number
of the vertices – the points of a cube with the side Pl .
The following consideration relates to the Planck scale and the vacuum near matter. It can
be for the free vacuum
min
P
l
l
,
(2.3.2)
that means the minimal length is the Planck length. The vacuum near matter can be reduced, in
such case (2.3.2) is justified at maximum degree of reducing. Or (2.3.2) is not right for the free
vacuum and it is right in such case
min
P
l
l
l
,
0
l
,
(2.3.3)
so that for the vacuum near matter (2.3.2) is right at maximum limiting, i.e. the reducing of
overplanck scale to the Planck scale happens. The author states that in free space (2.3.2) is
justified as it is the most possible, and near matter does happen the reducing of the Planck scale.
2.4.
The Final Results Of The Calculation
The estimation (2.2.8) can be calculated on the computer in the first place for a universe
consisting only of hydrogen atoms. It is needed to say that the Universe often considers as such
universe [10] on the reason that in the Universe the hydrogen is prevailed. For hydrogen it is
important to note the existence of the values of the reducing of the Planck scale. These values
describe the Planck-sized vacuum wave packets for the hydrogen atom and can be obtained on
(2.1.21) by substituting
P
l
l
,
min
r
r
and
max
r
r
. Here
min
r
is the minimal distance to the
proton which can still think a point and
max
r
is the radius of the atom. One obtains
2
60
min
2
2
2
min
min
2
3
,
7.58 10
m
3
6
32
P
P
P
P
l
l l
r
m c
l
r
r
l
,
(2.4.1)
17
2
60
max
2
2
2
max
max
2
3
,
2.53 10
m
3
6
32
P
P
P
P
l
l l
r
m c
l
r
r
l
.
(2.4.2)
The Planck-sized wave packets of the vacuum virtual particles do not actually contribute in the
theory because if the Planck length is small, the reducings (2.4.1) and (2.4.2) are yet smaller, so
that one can neglect them in calculation. For example, this relates to the number of particles on
the zeroth meridian (2.1.34) and the energy density (2.1.2). Thus, these formulae take the forms
2
2
arctan
2
0.25
N
l
r
l
,
(2.4.3)
2
2
3
2
4
2
3
16
v m
c
w
l
l
m c
l
(2.4.4)
for the Planck-sized wave packets. Also for the computation it is needed to have in the view the
constraint following from (2.1.34), namely
2
2
1
0
4
r
l
l
,
(2.4.5)
which thanks to the known conditions
0
r
and l
l
can be led to the form
1
2
n
n
r
l
l
for all n.
(2.4.6)
That the range (2.1.26) for l is not applicable but the range (2.1.25) is correct follows from
(2.4.6) and from that the origin of
nr is in the system of algebraic equations (2.2.4) – (2.2.6) and
so on. In such a way one at once gains the limitation for scales in the task of the computation of
the estimation for the vacuum energy near matter which follows from (2.4.6), (2.1.21) and the
dependence
nr l as was said above. In nature this limitation can be caused by the spatial
limitation of the acting of the interaction fields which matter itself has.
Also it must be understandable that in the real practical calculation one should consider
limited atom, an object having finite sizes in space, therefore the number n of the radii
nr must
be limited by the maximum value. Thus, it must be computed
max
min
2
2
2
0
2
4
2
0
,
1
3
,
2
2
16
,
l
n
k
v m
a
k
k
P l
k
N l r
c
E
N
N l r
m c dl
l
l
l l r
.
(2.4.7)
It was accepted in the computation that hydrogen atom has values:
11
min
10
m
r
,
20
min
10
P
l
l
,
11
max
3 10
m
r
. It can be obtained from (2.4.6) that
11
max
2.00008 10
m
l
by substituting
min
nr
r
, and all other values
nr including
max
r
give true conditions if the previous conditions
are accepted. Solving the system of the algebraic equations (2.2.4) – (2.2.6) analytically, one can
quickly understand that the volume of the expressions increases very rapidly. In fact the second
substitution for number
2
n
already gives a large final result. However, one could notice that
the first, the second and the third results can be calculated analytically on the computer and have
approximately straight lines plots. This does give a resolution of the problem. As one, n -th result
must be substituted in the next having number n on unit more, the author came to the conclusion
that all
nr l are approximately straight lines. However, this conclusion must be verified on each
step of the computation. For this all interval from min
l
to max
l
was divided into ten segments with
eleven points including the start and the end points and in each point values of the function
gained from exact solution for the first step were compared with the points calculated from
18
approximate linear representation of this exact solution. And this comparison was done for each
value of n. This has been done by the written program by the author, the listing of this program
written in the “Wolfram language®” is applied below the main text. Thus, the computed value at
the all applied above conditions in the right hand side of the estimation (2.2.8) allows one to
write down the numerical assessment
0
104
1.31797 10
J
v m
E
.
(2.4.8)
This final result has been gained by the computer program which is done
12347
n
iterations
for 9300 seconds with the rest. The general assessment (2.2.15) for the full vacuum energy, i.e.
the vacuum energy of empty space of the Universe and the vacuum energy in the space near
matter takes the exact quantitative form
193
104
0
2.32 10
1.32 10
J
E
.
(2.4.9)
The gained data testifies in sake of the cosmological constant problem, it really takes place and
the lower boundary for the full vacuum energy exceeds the observed value over 122 orders of
magnitude.
At the next stage it is needed to introduce a wrapping vacuum coefficient to show that
matter really reduces vacuum leading to the increasing of vacuum energy for the entire Universe.
If to assess the energy of an empty space in each hydrogen atom taking into account admissible
for the atom scale interval provided in the previous computation, also taking into account that
there is no reducing of the wave packets of the virtual particles in empty space, one should
compute the fraction
max
min
104
2
2
2
4
3
2
1.31797 10
1
1
3
16
l
a
A
P
l
c
N V
m c dl
l
l
l
,
(2.4.10)
where
1
is the wrapping vacuum coefficient; with more the wrapping of the vacuum by
matter in the entire Universe is more. The approximate value of the working volume of the atom
3
31
3
max
4
1.13 10
m
3
A
V
r
(2.4.11)
at the all discussed above conditions and the values of the quantities leads to the numerical
inequality for the coefficient
176.055
.
(2.4.12)
The more accurate estimation of the working volume of the hydrogen atom, i.e. the volume
occupying the reduced/not reduced wave packets of the virtual particles according this model,
namely
3
3
31
3
max
min
4
4
1.09 10
m
3
3
A
V
r
r
(2.4.13)
leads to the lower limitation for the wrapping vacuum coefficient
182.515
.
(2.4.14)
As become obvious for the reader the lower edge of the vacuum energy in all the hydrogen
atoms volumes at the absence of them nuclei, the protons that means there is no the reducing of
the vacuum virtual particles can be calculated by assuming
,
0
n
l r l
for all n. That is in this
method of calculation one should compute
max
min
2
2
2
1
2
4
2
0
,
1
3
,
2
2
16
l
n
k
a
k
k
P l
N l r
c
N
N l r
m c dl
l
l
,
(2.4.15)
where for N must be used (2.4.3) and at
0
k
must be
min
r
r
. For this case the scale l is no
more restricted with the maximum value, because the inequality (2.4.6) now gets
19
1
2
nr
l
.
(2.4.16)
Also assuming
,
0
n
l r l
, the system of the equations (2.2.4) – (2.2.6) can be solved by simple
way, as, one obtains
min
nr
nl
r
.
(2.4.17)
Substituting (2.4.17) in (2.4.16), it can obtain
min
0.5
r
l
n
,
(2.4.18)
but
0
l
, hence, (2.4.18) is always true. Thus, in this case l must be restricted on top by the
value max
l
from the previous calculation. The number
1
n up to which the sum in (2.4.15) must
be summed one can explain that solving the algebraic equations up to the condition
max
nr
r
, the
value
nr , when n is maximal, is a little exceeded the value
max
r
. So, computing (2.4.15) and
making up the fraction like (2.4.10) with (2.4.15) in the denominator and the value from (2.4.9)
in the numerator, one could gain the numerical condition
77.5007
.
(2.4.19)
The conclusion is that the matter in the Universe really limits or wraps (the author introduces
such term) the vacuum, at that the vacuum energy increases.
And finally, purely qualitative it can be shown that the lower assessment for the vacuum
energy is able to be calculated without any treatment to the concept of individual atoms. If one
would consider a one single atom with the volume of all empty space of the entire Universe in
the task of calculation of the vacuum energy being in empty space or the free vacuum energy,
this method has never yet been considered, then it must be applied
min
1
2
r
l
,
(2.4.20)
the maximum value of the radius of such imagined atom must be calculated from the condition
3
max
4
3
s
V
r
,
(2.4.21)
it takes the form
1
3
max
3
4
s
r
V
.
(2.4.22)
The condition on the range for scales is to be (2.1.26). The minimal number of balls on the
zeroth meridian at the condition (2.4.20) can be found out by substituting (2.4.20) in (2.4.3) and
taking the limit
0
lim arctan
2
x
l
x
,
(2.4.23)
thus,
0
min
2
N
N
. This is the formal number. The substitution (2.4.17) in (2.4.3) and taking
into account (2.4.20) leads to
1
1
arctan 2
1
N
k k
,
(2.4.24)
where
1,
k
n
and n is to be set from iterating (2.4.17) at the condition (2.4.20) up to that the
condition
max
nr
r
fails to give true. Thus, the inequality for the free vacuum energy calculated
by this method takes the qualitative form
20
min
2
2
2
1
0
2
4
2
0
1
3
2
2
16
n
v
k
P l
N k
c
E
N k
m c dl
l
l
.
(2.4.25)
This method has a feature in packaging of the wave packets vacuum virtual particles. The
inequality (2.4.25) has the qualitative role and cannot be computed because the value (2.4.22) is
too much, namely
26
max
4.37 10 m
r
,
(2.4.26)
so that, it must be done too much number of the iterations to find all
nr and the final n.
VU = 3.5 × 1080;
Na = 1080;
VA = 8.18 × 10-33;
m = 9.1 × 10-31;
c = 3 × 108;
h = 6.62 × 10-34;
ℏ=
h
2 π
;
rmin := 10-11
lmin := 1020 lP
lP := 1.62 × 10-35
rmax := 3 × 10-11
lmax := 2.00008 × 10-11
a1 := 9.99998 × 10-12
k1 := 0.999976
a2 := 9.99996 × 10-12
k2 := 1.99996
r1 := a1 + k1 l
r2 := a2 + k2 l
Ρ2 := 1.00032 × 10-11 + 3.99976 × 10-11
l - 1.62 × 10-15
2.00008 × 10-11 - 1.62 × 10-15
l1 := 2.00154 × 10-12
l2 := 4.00146 × 10-12
l3 := 6.00138 × 10-12
l4 := 8.0013 × 10-12
l5 := 1.00012 × 10-11
l6 := 1.20011 × 10-11
l7 := 1.40011 × 10-11
l8 := 1.6001 × 10-11
l9 := 1.80009 × 10-11
Inoculating values:
a2 := 9.99996 × 10-12
k2 := 1.99996
Δlmin :=
3 l2
3 l + 6 rmin + 32 rmin l2 × m2 c2
ℏ2
Δl1 := Δlmin /. rmin r1
Δl2 := Δl1 /. r1 r2
цикл ДЛЯ
Forn = 2;
l = lmin, Ρn ≤rmax, n++, l =.;
m =.;
c =.;
ℏ=.;
a2 =.;
k2 =.;
an =.;
kn =.;
Δln =.;
Δln+1 =
3 l2
3 l + 6 Rn+1 + 32 Rn+1 l2 × m2 c2
ℏ2
;
Δln =
3 l2
3 l + 6 rn + 32 rn l2 × m2 c2
ℏ2
;
Ρn+1 = Rn+1 /.
уплостить
Flatten
решить уравнения
SolveRn+1 -
1
2
(l - Δln+1) rn +
1
2
(l - Δln), Rn+1〚1〛;
l = lmin;
m = 9.1 × 10-31;
c = 3 × 108;
ℏ= 1.05361 × 10-34;
a2 = 9.99996 × 10-12;
k2 = 1.99998;
an = y1,n - (y2,n - y1,n) ×
lmin
l1 - lmin
;
kn =
y2,n - y1,n
l1 - lmin
;
условный оператор
IfΡn+1 > 0, l =.;
m =.;
c =.;
ℏ=.;
a2 =.;
k2 =.;
an =.;
kn =.;
2 Program SCCPERVMUR.nb
rn+1 = Rn+1 /.
уплостить
Flatten
решить уравнения
SolveRn+1 -
1
2
(l - Δln+1) rn +
1
2
(l - Δln), Rn+1〚1〛;
l = lmin;
m = 9.1 × 10-31;
c = 3 × 108;
ℏ= 1.05361 × 10-34;
a2 = 9.99996 × 10-12;
k2 = 1.99998;
an = y1,n - (y2,n - y1,n) ×
lmin
l1 - lmin
;
kn =
y2,n - y1,n
l1 - lmin
;
y1,n+1 = rn+1;
l =.;
l = l1;
y2,n+1 = rn+1;
dn+1 = y1,n+1 + (y2,n+1 - y1,n+1) ×
x - lmin
l1 - lmin
;
x = l2;
l =.;
l = l2;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-3, l =.;
x =.;
l = l3;
x = l3;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l4;
x = l4;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l5;
x = l5;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l6;
x = l6;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
x =.;
l = l7;
x = l7;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
Program SCCPERVMUR.nb 3
x =.;
l = l8;
x = l8;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
x =.;
l = l9;
x = l9;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.6, l =.;
x =.;
l = lmax;
x = lmax;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.6 , l =.;
x =.;
rn+1 =.;
an =.;
kn =.;
rn+1 = an+1 + kn+1 l;
Ρn+1 =.;
Ρn+1 = an+1 + kn+1 l;
an+1 = y1,n+1 - (y2,n+1 - y1,n+1) ×
lmin
l1 - lmin
;
kn+1 =
y2,n+1 - y1,n+1
l1 - lmin
;
l = lmin,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;, Ρn+1 =.;
l =.;
m =.;
c =.;
ℏ=.;
a2 =.;
k2 =.;
an =.;
kn =.;
Ρn+1 = Rn+1 /.
уплостить
Flatten
решить уравнения
SolveRn+1 -
1
2
(l - Δln+1) rn +
1
2
(l - Δln), Rn+1〚2〛;
rn+1 = Rn+1 /.
уплостить
Flatten
решить уравнения
SolveRn+1 -
1
2
(l - Δln+1) rn +
1
2
(l - Δln), Rn+1〚2〛;
l = lmin;
m = 9.1 × 10-31;
c = 3 × 108;
ℏ= 1.05361 × 10-34;
4 Program SCCPERVMUR.nb
a2 = 9.99996 × 10-12;
k2 = 1.99998;
an = y1,n - (y2,n - y1,n) ×
lmin
l1 - lmin
;
kn =
y2,n - y1,n
l1 - lmin
;
y1,n+1 = rn+1;
l =.;
l = l1;
y2,n+1 = rn+1;
dn+1 = y1,n+1 + (y2,n+1 - y1,n+1) ×
x - lmin
l1 - lmin
;
x = l2;
l =.;
l = l2;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-3, l =.;
x =.;
l = l3;
x = l3;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l4;
x = l4;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l5;
x = l5;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.8, l =.;
x =.;
l = l6;
x = l6;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
x =.;
l = l7;
x = l7;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
x =.;
l = l8;
x = l8;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.7, l =.;
x =.;
Program SCCPERVMUR.nb 5
l = l9;
x = l9;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.6, l =.;
x =.;
l = lmax;
x = lmax;
⋯
If
абсолютное значение
Abs[rn+1 - dn+1] < 10-2.6 , l =.;
x =.;
rn+1 =.;
an =.;
kn =.;
rn+1 = an+1 + kn+1 l;
Ρn+1 =.;
Ρn+1 = an+1 + kn+1 l;
an+1 = y1,n+1 - (y2,n+1 - y1,n+1) ×
lmin
l1 - lmin
;
kn+1 =
y2,n+1 - y1,n+1
l1 - lmin
;
l = lmin,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;,
пустой
Null;;//
затраченное время
Timing
{9300.937500, Null}
n
12 347
l =.
таблица значений
Tableak = y1,k - (y2,k - y1,k) ×
lmin
l1 - lmin
, {k, 3, n};
таблица значений
Tableki =
y2,i - y1,i
l1 - lmin
, {i, 3, n};
таблица значений
Table[rj = aj + kj l, {j, 3, n}];
таблица значений
TableΔlk =
3 l2
3 l + 6 rk + 32 rk l2 × m2 c2
ℏ2
, {k, 3, n};
N1min :=
π
ArcTan
l-Δlmin
2
rmin
2
-
1
4 (l-Δlmin)2
6 Program SCCPERVMUR.nb
таблица значений
TableN1i =
π
ArcTan
l-Δli
2
ri
2-
1
4 (l-Δli)2
, {i, 1, n};
привести
Reducern ≥
1
2
l -
3 l2
3 l + 6 rn + 32 rn l2 × m2 c2
ℏ2
, l
Reduce::ratnz : Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
-8.09885 × 10-16 ≤l ≤-8.09871 × 10-16 || l > -8.09852 × 10-16
привести
Reducermin ≥
1
2
l -
3 l2
3 l + 6 rmin + 32 rmin l2 × m2 c2
ℏ2
, l
Reduce::ratnz : Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a
corresponding exact system and numericizing the result.
l ≤2.00008 × 10-11
l = lmax;
N1min
2.001
l =.
l = lmin;
N1min
38 786.9
l =.
Int =
квадратурное интегрирование
NIntegrate
N1min
2
2
- N1min + 2
ℏ2 c2
16
×
3
(l - Δlmin)2 + m2 c4 +
j=1
n
N1j
2
2
- N1j + 2
ℏ2 c2
16
×
3
(l - Δlj)2 + m2 c4
, {l, lmin, lmax}
2.13511 × 10-11
численное приближение
N
1
lP
Na Int
1.31797 × 10104
квадратурное интегрирование
NIntegrate
N1min
2
2
- N1min + 2
ℏ2 c2
16
×
3
(l - Δlmin)2 + m2 c4 , {l, lmin, lmax}
5.15089 × 10-18
Program SCCPERVMUR.nb 7
VA =.
численное приближение
N
4
3
π rmax
3
1.13097 × 10-31
VA := 1.13 × 10-31
численное приближение
N
1.31797 × 10104
1
lP Na VA ∫lmin
lmax 1
l3
ℏ2 c2
16 × 3
l2 + m2 c4 l
176.055
VA =.
численное приближение
N
4
3
π rmax
3
-
4
3
π rmin
3
1.08909 × 10-31
VA := 1.09 × 10-31
More precise value
численное приближение
N
1.31797 × 10104
1
lP Na VA ∫lmin
lmax 1
l3
ℏ2 c2
16 × 3
l2 + m2 c4 l
182.515
r1 =.
n =.
r1 := l + rmin
таблица значений
Table[rn =., {n, 2, 12 347}];
цикл ДЛЯ
For[n = 1; l = lmin, rn ≤rmax, n++, rn+1 = (n + 1) l + rmin]
n
12 346
rn
3.00005 × 10-11
rn-1
2.99989 × 10-11
l =.
8 Program SCCPERVMUR.nb
N2min :=
π
ArcTan
l
2
rmin
2
-
1
4 l2
таблица значений
TableN2i =
π
ArcTan
l
2
ri
2-
1
4 l2
, {i, 1, n};
Int =.
Int =
квадратурное интегрирование
NIntegrate
N2min
2
2
- N2min + 2
ℏ2 c2
16
×
3
l2 + m2 c4 +
j=1
n-1
N2j
2
2
- N2j + 2
ℏ2 c2
16
×
3
l2 + m2 c4
, {l, lmin, lmax}
2.75495 × 10-13
численное приближение
N
1
lP
Na Int
1.70059 × 10102
квадратурное интегрирование
NIntegrate
N2min
2
2
- N2min + 2
ℏ2 c2
16
×
3
l2 + m2 c4 , {l, lmin, lmax}
5.14944 × 10-18
1.31797 × 10104
1.70059 × 10102
77.5007
r1 =.
rmax =.
r1 :=
3
2
l
VM := Na VA
Vs := VU - VM
rmax =
3
4 π
Vs
1
3
4.3718 × 1026
таблица значений
Table[rk =., {k, 2, n}];
n =.
Program SCCPERVMUR.nb 9
цикл ДЛЯ
Forn = 1; l = lmin, rn ≤rmax, n++, rn+1 = (n + 1) l +
1
2
l
$Aborted
n
321 572 413
rn
5.20947 × 10-7
предел
Limit
арктангенс
ArcTan
l
x
, x 0
l2 π
2 l
N3min := 2
таблица значений
TableN3i =
π
ArcTan
1
2
i (i+1)
, {i, 1, n};
Int =.
Int =
квадратурное интегрирование
NIntegrate
N3min
2
2
- N3min + 2
ℏ2 c2
16
×
3
l2 + m2 c4 +
j=1
n-1
N3j
2
2
- N3j + 2
ℏ2 c2
16
×
3
l2 + m2 c4
, {l, lmin, ∞}
численное приближение
N
1
lP
Int
10 Program SCCPERVMUR.nb
31
CONCLUSION
In this work the observed value for the vacuum energy density in the Universe was
compared with the lower boundary for the vacuum energy density corresponding to the verified
high energy physics scale that is the least space scale and the divergence in the 53 orders of
magnitude was found. In the work the cosmological constant problem was stated on the data of
the calculated observed full vacuum energy of the entire Universe and has been calculated the
lower value for the free vacuum energy, i.e. the vacuum energy of all empty space of the
Universe. The carried out research testifies that the cosmological constant problem really takes
place. It was considered the part of the quantum electrodynamics vacuum, namely, the electron-
positron quantum vacuum. Also the estimation for the energy of the vacuum near the matter has
been calculated. The lower value of the energy has been found below the lower boundary of the
free vacuum energy on the 89 orders of magnitude. The main purpose of this work was to
consider the interaction of the vacuum with matter. In the result, the minimal energy of the
vacuum near matter has been obtained. The total lower assessment for the full vacuum energy
including the free vacuum and the vacuum interacting with matter confirmed the approximate
known value for the vacuum energy, and the found in this work value exceeds the observed
value mainly on the 122 orders of magnitude. This result confirmed the previous number for the
free vacuum as the approximate estimation for the minimal value of the vacuum energy. Note,
that the known excess of the observed value is the 120 orders of magnitude. Thus, the found
value is approximately relevant to the well-known and accepted value on the order of magnitude.
These assessments for the all Universe have been done in the model where all atoms of the
Universe are thought with hydrogen atoms. In consequent work it is supposed the considering all
types of the atoms existing in the Universe. In this work has been considered only the motion of
the virtual particles that leads to essential increasing of the vacuum energy. That is why also in
consequent work the consideration of the full motion of virtual particles in the pair and all the
types of vacuums respective to all the types of the particles of matter is supposed. The
aforementioned calculations except the calculation of the energy density on the achieved by the
high energy physics scale have been done at the assumption of discrete space. Actually, these
calculations can approximately be executed only in the hypothesis of discrete space. According
the approach of this work space must be discrete; otherwise any vacuum energy would be
infinite.
ACKNOWLEDGEMENTS
I am very grateful to Olga Volkova, EdS for the help in writing of this manuscript.
REFERENCES
1. Savelyev I.V. Physics a general course Volume 3 Quantum optics. Atomic physics. Solid
state physics. Physics of the atomic nucleus and elementary particles 3rd printing edition.
Moscow: Mir Publishers; 1980. p. 70-74.
2. Landau L.D., Lifshits E.M. Quantum mechanics (non-relativistic theory), Volume 3. 3rd
edition revised and enlarged. New York (NY): Pergamon Press Inc.; 1977. p. 45-49.
3. D.N. Spergel, R. Bean, O. Dore et al. Wilkinson Microwave Anisotropy Probe (WMAP)
of Three Year Results: Implications for Cosmology [Internet]. Princeton University; 2007
Feb 27 [cited 2024 June 20] Available from https://arxiv.org/pdf/astro-ph/0603449
32
4. S. Weinberg Gravitation and cosmology, Principles and applications of the general theory
of relativity. New York – London – Sydney – Toronto: John Wiley and Sons, Inc.; 1972.
p. 613.
5. P. Langacker The standard model and beyond, Series in high energy physics, cosmology,
and gravitation, 2nd edition. | Boca Raton, FL: CRC Press, Taylor& Francis Group; 2017.
p. 12.
6. S. Weinberg 1989. The cosmological constant problem. In: Review of Modern Physics.
Volume 61. p. 1-23.
7. Slater J. C. 1964. Atomic Radii in Crystals. In: Journal of Chemical Physics.
Volume 41 (10). p. 3199–3205.
8. Landau L.D., Lifshitz E.M. The Classical Theory of Fields, 3rd edition, revised. Oxford:
Pergamon Press; 1971. p. 346.
9. "Universe volume Wolfram|Alpha" [Internet]. [cited 2024 June 20] Available from
https://www.wolframalpha.com/input?i=Universe+volume
10. H. Kragh 2003. Magic Number: A Partial History of the Fine-Structure Constant. In:
Archive for History of Exact Sciences. Volume 57 (5). p. 395–431. |
vixra | 2602.0146 | Astrophysics | It may seem odd to talk about glowing dark matter. Something dark is typically hypothesized to be hiding in the darkness of space, along with dark energy. All favored dark models are simply outdated. This essay will explain how the brilliance of so-called dark matter has always been there. Dark phenomena clearly belong within any Theory of Everything. | Clark M. Thomas | Astrophysics | https://vixra.org/abs/2602.0146 | WIMPs and Glowing Dark Matter
By: Clark M. Thomas
© February 23, 2026
Abstract
It may seem odd to talk about glowing dark matter. Something dark is
typically hypothesized to be hiding in the darkness of space, along with
dark energy. All favored dark models are simply outdated. This essay will
explain how the brilliance of so-called dark matter has always been there.
Dark phenomena clearly belong within any Theory of Everything.
What Space Observatories Recently Did and Did Not See
Recent high-energy James Webb Space Telescope (JWST) data from
electromagnetic (EM) waves in the Milky Way’s galactic plane has been
modeled to suggest degrading dark matter (DDM), to where some of our
galaxy’s vast energy is now visible as gamma rays from WIMPs (weakly
interacting massive particles).[1] This tidy hypothesis is a favored model.
However, there are challenges to this idea.[2],[3]
of
1
5
Another recent JWST image below reveals intergalactic dark matter
gravitational effects on multiple galaxy clusters, along with vast strings of
magnetism.[4] There is much to unpack in “glowing dark matter” images.
Because “dark matter quantum clouds” interface by push/shadow gravity
with human-visible baryonic matter, we cannot elegantly model cosmic dark
matter without mass and EM charges, or only with Relativity spacetime. In
no way does all this seemingly quantum-particulate “massive energy” equate
quantum field theories with spacetime General Relativity (GR). Cosmic data
reveals the continuing need for causative 4D physics in a 21st-century
Theory of Everything (TOE) within and among all multiversal linear scales.
In the JWST image below the rendered “blue” intergalactic dark regions
are more dense than rendered “pink” regions of mass. Thus “blue” here is
more strongly net-blocking, equipotent, omnidirectional, 4D-multiversal
“quantum sea” flows. In no way can myriad local spacetime gravity sheets
clearly explain the elegant net gravity relationships as shown on the right
side of the image immediately below.
Yielding similar and very important results, another relatively smaller
cosmic scale inside our local 1.74 billion cubic light years (with humans at
the geophysical center) involves the Dipole Repeller: [5]
of
2
5
Why WIMPs Are Galactic Gamma Ray Candidates
Even though WIMPs have never been observed, they correlate well within
certain math paradigms of what astrophysics “should” be. Many millions of
dollars have been spent chasing this massive, but ephemeral, theorized
ghost particle. Because we verified the Higgs Boson a few years ago, such
success does not indicate that WIMPs will be likewise found.[6] If WIMPs
aren’t soon experimentally captured, then it would be wise to also seriously
entertain very different causative physics models that could more elegantly
lead to unifying physics and astrophysics paradigms.
The idea that the newly discovered bright X-rays and gamma rays layer
(see above) inside our Milky Way (MW) plane is the product of decomposing
WIMPs is vague and tantalizing. A better idea is that these visible gamma
rays are the routine product not of decomposing ghost WIMPs, but of
dialectically lengthening other physics primary elements, such as beaded
Coulombic strings and primary clusters.
The tools for physics verification that we now lack, and may never have,
cannot directly measure precise Coulombic forces inside individual yin/yang
spheres around 10 to the negative 38 meters (10E-38 m) smaller than our
human linear dimension, which is our point of reference. That’s a thousand
times smaller on the linear scale than even the upper limits of the Planck
dimension, which is negative 35 from our human size. (No physics object
has zero 3D dimensions; nor are there any with only one or two dimensions.
Math goes to pure zero, as in Zeno’s Achilles paradox, but physics does not.)
Individual 3D y/y spheres simultaneously contain both plus and minus
electromagnetism, which can express at their surface as neutral “primary
EM” and magnetism. Note that push/shadow gravity and dipolar EM are not
the same; but magnetism can be a bridge, especially with the emergent
strong and weak nuclear forces. Transverse waves of y/y Coulombic beaded
strings appear to be without dipolar EM only because they cohere with
neutral primary EM at their own juxtaposed ends.
What dialectically emerges from different y/y beaded strings, either
dipolar or primary, helps shape all larger structures. Emergent clusters, and
so-called primary particles such as quarks, all trace their origins to much
smaller Coulombic spheres, and to their beaded strings of different lengths
with different transverse wave frequencies. Composite single quarks, for
example, are twenty linear metric dimensions larger than fundamental single
y/y spheres. Single quark size is also similar to the dimensional difference
between individual humans and individual composite quarks.[7]
of
3
5
Simultaneity of cause and effects is philosophically expressed
by the Nichiren Buddhist conceptual word, renge (pronounced
“ren’-gay”). A basic and popular, two-dimensional illustration of
foundational cosmic reality is shown here to the right:
Four-dimensional, multiversal components all have the same renge power
existing within each of their Coulombic spheres. Everything in linear-
dimensional reality expresses the granular dialectical differences within
renge that we now call physics, astrophysics, and chemistry. There is no
need to invoke absurd pure-math models with 10E500 2D universes.[8]
What is most important for the transverse-waves frequency phenomenon
is how many initially very short photonic EM strings link up into longer
strings with less individual electron volts (eV) wave energy.
Primary
frequency strings (looking from left to right) in this illustration below
continue to get longer, first revealing themselves to our space instruments
as gamma rays. Next, some lengthen to glow within human visible light
frequencies. Finally, long transverse waves can vanish from our unaided
vision into the red side of the full EM spectrum. We observe some of this
transformation in distant so-called “dark energy” red shifting: [9]
Lengthening photon strings can also express themselves beyond visible
frequencies as very long “quantum sea” transverse waves, even up to the
very long “gravity waves” first caught by the LIGO arrays.[10]
Many wavy strings combine into formerly-called primary particles. So
bound, they may no longer exhibit light within visible wave lengths, nor in
fact light as we commonly recognize it. Nevertheless, wavy vibrations within
complex objects can escape and emit radiant energy, as with U235.
of
4
5
All that we have just described is another way of envisioning how the
smallest components of real physics can transform and link to become the
largest components of our 4D multiverse. The dialectically largest is
composed of the smallest — but the smallest is not composed of the largest.
The question of gravity is simple too: Instead of modeling with causative
net push/shadow gravity, Einstein’s spacetime math reversely correlates with
increasingly precise GPS satellite orbital data. Nevertheless, GPS is likewise
congruent with correctly explained causative, net push-shadow theory.[11]
Within some proximal linear dimensions, GR spacetime seems elegant,
but not in others near and far. It is only when we dialectically trace real
physics starting from the super-tiny, yin/yang Coulombic spheres — with
their initially short beaded-string structures — that the true multiversal
harmony of large and small emerges like a beautiful renge flower embracing
all “bubble” universes such as our own.
References
[1] https://phys.org/news/2025-12-decaying-dark-unidentified-ray-
emission.html
[2] https://www.astronomy-links.net/wimps.html
[3] https://astronomy-links.net/lambda.dark.matter.pdf
[4] https://apnews.com/article/dark-matter-galaxies-map-james-webb-
telescope-150691a1349cd39961ca24ab0e87c688
[5] https://astronomy-links.net/DipoleRepellerExplained.pdf
[6] https://en.wikipedia.org/wiki/Higgs_boson
[7] https://astronomy-links.net/String.Types.pdf
[8] https://astronomy-links.net/Hawking.legacy.pdf
[9] https://www.vecteezy.com/free-vector/electromagnetic-spectrum
[10] https://astronomy-links.net/LIGO.and.GR.pdf
[11] https://astronomy-links.net/LightSpeed.pdf
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5 |
vixra | 2602.0143 | Astrophysics | The alternative mechanism of the emergence of cosmic microwave background radiation (CMB), associated with the thermal radiation of primordial gas-dust clouds in the early Universe, is considered. The emergence of such clouds in the theory of infinite hierarchical nesting of matter is a natural stage in matter evolution. The mass, radius, and spatial concentration of typical primordial gas-dust clouds, the distance between neighboring clouds, and the power of CMB energy generation per unit volume and per nucleon of the early Universe were calculated. The masses and radii of these clouds correspond to the masses and radii of the observed Bok globules. The presented mechanism is consistent with the cluster model describing the appearance of angular multipoles in the CMB power spectrum. In addition to CMB radiation, cosmic infrared background (CIB) radiation and cosmic optical background (COB) radiation are also considered. According to the presented model, the sources of CIB are primordial protoplanetary clouds. As for the COB radiation, it is associated with the radiation of the first protostars. During evolution, each primordial cloud, with a mass of about 31 solar masses, first generates CMB radiation, and then CIB and COB radiations. Since protostars give rise to neutron stars, the concentration of primordial gas-dust clouds is also the concentration of observed neutron stars. In the course of the calculations, a new definition of the radiation intensity is used, which is based on the vector of the surface energy flux density and accounts for the angles of incidence of radiation on a flat receiver from all sides of the hemisphere. According to Poynting's theorem, the relationship between the intensity and energy density of black body radiation is derived from the concept of photons. | Sergey G. Fedosin | Astrophysics | https://vixra.org/abs/2602.0143 | 1
Jordan Journal of Physics, Vol. 18, No 4, pp. 529-549 (2025).
https://jjp.yu.edu.jo/index.php/jjp/article/view/323. https://doi.org/10.47011/18.4.10
On the origin of cosmic microwave background radiation
Sergey G. Fedosin
PO box 614088, Sviazeva str. 22-79, Perm, Perm Krai, Russia
ORCID 0000-0003-3627-2369, E-mail: sergey.fedosin@gmail.com
The alternative mechanism of the emergence of cosmic microwave background radiation
(CMB), associated with the thermal radiation of primordial gas-dust clouds in the early
Universe, is considered. The emergence of such clouds in the theory of infinite hierarchical
nesting of matter is a natural stage in matter evolution. The mass, radius, and spatial
concentration of typical primordial gas-dust clouds, the distance between neighboring clouds,
and the power of CMB energy generation per unit volume and per nucleon of the early Universe
were calculated. The masses and radii of these clouds correspond to the masses and radii of the
observed Bok globules. The presented mechanism is consistent with the cluster model
describing the appearance of angular multipoles in the CMB power spectrum. In addition to
CMB radiation, cosmic infrared background (CIB) radiation and cosmic optical background
(COB) radiation are also considered. According to the presented model, the sources of CIB are
primordial protoplanetary clouds. As for the COB radiation, it is associated with the radiation
of the first protostars. During evolution, each primordial cloud, with a mass of about 31 solar
masses, first generates CMB radiation, and then CIB and COB radiations. Since protostars give
rise to neutron stars, the concentration of primordial gas-dust clouds is also the concentration
of observed neutron stars. In the course of the calculations, a new definition of the radiation
intensity is used, which is based on the vector of the surface energy flux density and accounts
for the angles of incidence of radiation on a flat receiver from all sides of the hemisphere.
According to Poynting's theorem, the relationship between the intensity and energy density of
black body radiation is derived from the concept of photons.
Keywords: cosmic microwave background; infinite hierarchical nesting of matter; early
Universe; cosmology: theory; matter evolution.
PACS: 98.70.Vc
1. Introduction
2
The cosmic microwave background radiation (CMB) in the wavelength range of 0.3-30 mm
contributes most to the total energy of cosmic background radiation. The standard explanation
for the origin of CMB based on the Big Bang concept, in which CMB appeared in the early
Universe. However, the idea of the Big Bang still has drawbacks [1]; therefore, other alternative
cosmological theories continue to appear. For example, in the quasi-steady-state cosmological
model, it is assumed that CMB could be the result of processing stellar radiation by cosmic
dust [2].
However, even in this case, there are difficulties associated with the fact that the CMB is
too homogeneous and isotropic and has a spectrum of ideal black body. According to the
dynamic Universe model [3] and the hierarchical Universe model [4], the stellar radiation in
the early Universe could be sufficient for CMB to have the observed energy density and be
isotropic, so that the Big Bang is not needed. According to [5], the models based on a Universe
in dynamical equilibrium without expansion predicted the 2.7 K temperature prior to and better
than models based on the Big Bang. In addition, it was shown in [6] that isotopes of all the
observed chemical elements can be formed from hydrogen in stars during the time of the order
of 100 billion years, which makes it possible to do without the Big Bang.
It is known from measurements [7] that the CMB temperature corresponds to the blackbody
temperature
2.7255
T =
К. If the CMB is in equilibrium with respect to some global blackbody
of the Universe, it would have a volumetric energy density equal to
4
14
4
4.17 10
T
u
c
−
=
=
J/m3, where is the Stefan–Boltzmann constant, and c is the speed of light. This relation
characterizes, for example, the state of a hollow black body, which is in thermal equilibrium
with radiation in the inner cavity. The surface of such a cavity emits and absorbs radiation
energy with an intensity of
4
6
3.13 10
I
T
−
=
=
W/m2. This means that ideal receivers, close
in their properties to a blackbody, will measure in the cavity the CMB intensity on the order of
I . In this case, the contribution to the intensity I will be made by photons incident on the
receiver at various angles.
In measurements, the angular intensity
dI
J
d
=
is often used, where denotes the solid
angle in steradians, from which the radiation arrives at the receiver. In accordance with the
Stefan–Boltzmann law, for CMB radiation, if it were in thermal equilibrium with matter as in
a hollow black body, the following relation would be true:
4
7
9.96 10
I
T
J
−
=
=
=
W/(sr·m2). This value is in accordance with the results in [8].
3
When plotting the radiation spectrum of a blackbody, the dependence of the spectral angular
intensity d J
d
on the radiation frequency is usually plotted. This value reaches a maximum
when a small frequency range d is selected near the frequency
160.23
m
=
GHz,
corresponding to the maximum in CMB radiation. In accordance with Wien’s law of
displacement for the frequency there is
10
5.879 10
m
kT
T
h
=
, where the constant
2.821439...
, h is the Planck constant, k is the Boltzmann constant, and the radiation
temperature T is measured in Kelvin.
The purpose of this work is to explain the origin of background radiation in the model of a
hierarchical Universe. Based on the thermal equilibrium of radiation and matter of radiation
sources in the early Universe, we will find the sizes of these sources, their masses and
concentration in space. As will be shown below, in the presented approach the formula
4
4 T
u
c
=
for the energy density of background microwave radiation in the Universe can no
longer be valid, as well as for background infrared radiation and background optical radiation.
2. Definition of intensity
The standard definition considers intensity as the amount of energy passing per unit of time
through a unit area oriented perpendicular to the direction of energy propagation. To take into
account various orientation angles of the receiver with respect to the incident radiation, a
different definition should be applied. In the concept of photons, we can assume that intensity
is the magnitude of a certain vector, namely the vector of the surface density of the radiation
energy flux.
Let us assume that the radiation receiver responds only to the radiation component, which
is perpendicular to the receiver plane. This can happen, for example, when the receiver is
sensitive to the momenta of the photons falling on the receiver from all sides. Then the
momenta components of the set of photons, which are parallel to the receiver plane, mutually
cancel each other, and the sum of the perpendicular momenta components of the photons is
considered. In this case, we can assume that the energy flux surface density vector I is
determined by the amount of incident radiation energy on the flat radiation receiver from all
sides of the hemisphere per second per unit area S of the receiver, taking into account the
angular dependence:
4
(
)
(
)
(
)
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ˆ
ˆ
ˆ
ˆ.
i
i
i
i
i
in
i
in
n
i
i
i
d E
d
d
d
c
dS
dt
dS
dt
d F
d
c
c
c
P
cP
dS
dS
=
=
=
=
=
=
=
p n
p n
I
n
n
F n n
n
n
n
(1)
where
i
i
E
c p
=
denotes the energy of the photon with momentum amplitude
ip ; the index
i specifies the photon number during summation in (1); ˆ ip is a unit vector directed along the
photon momentum
ip , such that
ˆ
i
i
i
p
=
p
p ; ˆn is a unit normal vector directed to the receiver
plane from the hemisphere;
i
i
d
dt
=
p
F
defines the force acting from the photon on the receiver;
(
)
ˆ
in
i
F =
F n is the force projection on the normal;
in
in
d F
P
dS
=
denotes the pressure from the
photon’s side, perpendicular to the receiver surface; and
n
in
i
P
P
=
is the total perpendicular
pressure from all photons.
If we take into account that the electromagnetic radiation pressure is
4
3
3
u
I
P
c
=
=
, then
according to (1), we have
3
4
n
P
P =
,
3
ˆ
ˆ
4
n
cP
cP
=
=
I
n
n . (2)
In (2), the vector I is directed in the same way as the unit normal vector ˆn and is
proportional to the speed of light and the electromagnetic pressure on the receiver.
The difference between
nP and P in (2) can be attributed to the well-known 4/3 problem,
according to which the mass-energy
P
m of the electromagnetic field of a charged body moving
at a very low velocity, derived from the Poynting vector and proportional to the field
momentum density, is 4/3 times greater than the mass-energy
E
m , corresponding to the field
energy density. The relation
3
4
P
E
m
m =
corresponds to the equality
3
4
n
P
P =
, so that the
radiation pressure P is related to the Poynting vector, and the pressure
nP exerted on the
receiver is related to the field energy density. According to [9], the 4/3 problem occurs because
5
the electromagnetic field energy density and the field momentum density are not the four-
momentum components but rather the field stress-energy tensor components.
To check formula (1), first place the receiver on the surface of the sphere of radius r , at the
center of which there is a certain source with the power of isotropic radiation W . In this
situation, the radiation falls on the receiver at a right angle. In (1) the scalar product (
)
ˆ
ˆ
1
i
=
p n
is obtained, and the energy flux density recorded in the receiver is then equal to:
2
4
W
I
r
=
. (3)
Let us consider another situation, when one of the many existing radiation sources is the
volume element dV . We can assume that
u dV
dW
t
=
is the differential of the radiation
energy, leaving the volume element dV per unit time t
. Let us choose the hemisphere radius
R
c t
= , where c is the speed of light, and place the receiver at the coordinate origin on the
plane ZOX at the center of the hemisphere. This situation is shown in Figure 1.
Z
X
Y
D
dV
O
R
r
6
Now, all the radiation energy contained in the hemisphere with a radius R falls into the
receiver at time
t
. Then, in spherical coordinates for the intensity differential of the volume
element dV located inside the hemisphere at a distance r from the center, in accordance with
(1) and (3), we have:
2
sin
sin
4
u
Q
dV
dI
r
t
=
, (4)
where
2
sin
dV
r dr
QdQd
=
is the volume element considered as a source of isotropic
radiation; the product sin
sin
Q
defines the angular dependence of the radiation intensity so
that at
0
Q =
photons move along the OZ axis and do not enter the receiver at all; and at
0
=
, photons move in the ZOX plane and do not enter the receiver either. If
2
Q
=
, and
2
=
, photons fall on the receiver at a right angle to its surface and make the maximum contribution
to the intensity. The appearance of sin
sin
Q
in (4) follows from the fact that, according to
Figure 1,
(
)
ˆ
0,1,0
= −
n
,
(
)
ˆ
sin
cos ,sin
sin ,cos
i
Q
Q
Q
= −
p
, and in definition (1), we have
(
)
ˆ
ˆ
sin
sin
i
Q
=
p n
.
Integration over the hemisphere’s volume replaces the summation in (1) and gives the
following:
2
0
0
0
sin
sin
4
4
R c t
u
uc
I
dr
Q dQ
d
t
=
=
=
. (5)
Fig. 1. The receiver D (in this case rectangular in shape) is located at
the coordinate origin on the plane ZOX at the center of the hemisphere
of radius R, the current radius r and the angles Q and specify the
position of the radiating volume element dV in the spherical coordinate
7
The calculation in (5) shows how, in the case of equilibrium blackbody radiation, we can
understand the relation between the intensity I and the energy density u of radiation that
enters the receiver from different directions. Hence, we can see that I is actually related to the
mass-energy of the field energy density, and not to the mass-energy of the momentum density,
which is found using the Poynting vector.
3. CMB energy production
It is known that CMB radiation originates from large distances; therefore, in one way or
another, it is generated by many sources. Let us assume that, on average, each cubic meter of
the early Universe was a source of CMB and produced L joules of CMB energy per second;
then, the volumetric power L of energy generation is measured in W/m3. Next, we proceed as
in [4].
Let us suppose that radiation sources uniformly fill the hemisphere, while radiation from
some sources does not fall on the receiver at a right angle. This means that to determine the
intensity I , we should integrate the entire hemisphere’s volume and take into account the
angles of incidence of radiation on the receiver, similar to (4) and the definition of intensity in
(1). We will place the receiver at the origin of the coordinate system and position it in the ZOX
plane to measure the CMB energy.
If some radiating volume is located at a distance r from the origin of coordinates, then the
effective amount of energy dI , incident per unit time on the unit area of the receiver, will be
equal to:
(
)
2
exp
exp
sin
sin
4
Hr
L
snr
Q
dV
c
dI
r
−
−
=
, (6)
where H is the Hubble constant.
The first exponent exp
Hr
c
−
in (6) describes the exponential decrease in the energy of
CMB photons as they travel a distance r . As a result, the wavelength of the photons is shifted,
which is known as the cosmological redshift of the spectra of distant radiation sources.
8
The second exponent
(
)
exp
snr
−
sets the degree of scattering of photons on their way to
the receiver, reducing the number of arriving photons. This exponent corresponds to the Beer–
Lambert law for light scattering, and s is the scattering cross section, n is the concentration
of objects scattering light.
Let us take the integral in (6) over the volume of the hemisphere of infinite radius:
(
)
2
0
0
0
exp
exp
sin
sin
4
4
L
Hr
L
I
snr dr
Q dQ
d
H
c
sn
c
=
−
−
=
+
. (7)
In (7), the relation between the measured intensity I and the volumetric power L of CMB
energy generation in cosmic space is presented.
Considering that the exponent exp
Hr
c
−
in (7) describes the exponential decrease in
energy, as well as in the frequency of CMB photons as they travel a distance r , the following
is obtained for the photon wavelength and redshift:
0 exp Hr
c
=
,
0
0
0
1
exp
1
Hr
z
c
−
=
=
−=
−
,
(
)
ln
1
c
r
z
H
=
+
.
(8)
If in (8) z is small compared to unity, then
(
)
ln
1
z
z
+
, which leads to the Hubble law in
the form
cz
r
H
.
Due to the decrease in energy and scattering of photons in (6), some blurring of images of
distant galaxies should be observed, since photons change their motion direction as a result of
scattering. In fact, the observed blurring is insignificant, which can be explained by the small
size of the electrogravitational vacuum particles described in [4] and [10], which are unable to
significantly change the direction of the photons’ momenta. We can also refer to more recent
works [11-12], in which, in light of new data, the observed dependence of the duration of
supernova explosions on the distance to them, the dependence of the surface brightness of
galaxies on the redshift, the relationship between redshift, relict radiation, and the blackbody
spectrum were analyzed.
9
In addition, observations of the angular radii and surface brightness of galaxies at a given
luminosity do not correspond to the expanding Universe hypothesis in the
CDM
−
model,
but are in good agreement with relation (8), which describes the relationship between distance
and redshift, as well as with the static Universe model [13-14], in which the surface brightness
does not depend on the redshift z . With relation (8), the supernovae type Ia data give almost
the same result as the
CDM
−
model.
4. The sources of CMB
In the theory of infinite hierarchical nesting of matter [4], [10], [15], it is assumed that the
substance of a certain level of matter arises in the course of evolution of the substance of lower
levels of matter. Therefore, stars as objects of the stellar level of matter appear after the
compression of large gas clouds. The main objects of these clouds are nucleons belonging to
the nucleon level of matter. In turn, the appearance of gas clouds is a consequence of evolution
of the substance of the praon level of matter, and praons can form the substance of nucleons in
the same way as nucleons can form the substance of stars.
Based on this, suppose that in the early Universe, the entire volume was more or less
uniformly filled with CMB sources at concentration n , the average radius of these sources was
equal to a , and the effective temperature of particles on the surface of these sources was equal
to the temperature
sT . In this case, we can write:
4
2
4
s
L
T a n
=
. (9)
In (9), the CMB generation power per unit volume L is expressed through the surface area
of a typical CMB source, equal to
2
4 a
, through the intensity
4
sT
of radiation from this
surface, and through the concentration n of CMB sources in Universe.
Substituting L from (7) in (9) and taking into account that
4
I
T
=
, we find:
4
2
4
sT a n
T
H
sn
c
=
+
. (10)
10
We assume that during the time needed for CMB photons to reach the Earth from distant
regions of the Universe, the number of baryons and their concentration in cosmic space did not
change significantly. Then, in the first approximation, we can use the conclusions of the
CDM
−
model (Lambda-cold dark matter model), where the critical mass density reaches
the value
2
27
3
9.2 10
8
cr
H
G
−
=
=
kg/m3, if we assume that the Hubble constant H equals 70
km/(s·Mpc) or
18
2.268 10−
s-1 [16].
The
physical
density
of
the
visible
baryonic
matter
in
this
case
is
28
0.0227
2.1 10
b
cr
−
=
=
kg/m3. This value is chosen in such a way that, among other things,
it best fits observations of the amount of visible matter in galaxies. This approach will be
sufficient for us since we will further derive various relationships, the physical meaning of
which does not depend on the specific value of
b
.
Let us now take into account that the CMB sources, that is, the primordial gas-dust clouds,
were located discretely in space with a concentration
b
n
m
=
, where m is the mass of a typical
CMB source. Each source has a cross section equal to
2
s
a
=
. At very large distances, all
sources begin to overlap each other, which makes it difficult to see the most distant CMB
sources and weakens the intensity of the radiation that could be at the radiation receiver. This
leads to the appearance of the exponent
(
)
exp
snr
−
in (6).
The mass of a typical source is expressed by the formula
3
4
3
s
b
a
m
n
=
=
, where
s
is
the mass density of the source substance. Expressing n from here and substituting into (10),
taking into account the equality
2
s
a
=
, we have:
3
3
4
b
b
s
n
m
a
=
=
,
4
4
4
1
3
s
s
b
T
T
Ha
c
=
+
. (11)
The last formula in (11) relates the radius a , mass density
s
of CMB sources and the
effective temperature
sT of the surface particles of these sources.
5. The origin of energy in CMB sources
11
For the particles of numerous CMB sources to have a kinetic temperature on the order of
sT
and to be able to subsequently radiate at this temperature, it is necessary that the particles of
these sources somehow acquire the corresponding thermal energy as the energy of proper
motion.
Let us turn to the theory of infinite hierarchical nesting of matter, according to which
different matter levels are found in the Universe, and the main objects of these levels have
significantly different masses and sizes. In particular, there are metagalactic, stellar, nucleon,
praon, and graon levels of matter [4], [10], [15], [17-21].
All matter levels structured by gravitational clustering. This process is accompanied by the
opposite process of fragmentation when particles collide with each other and with radiation
quanta. In large gas clouds, under appropriate conditions, atoms and molecules can combine
under the action of gravitational forces first into molecular complexes and then into more
massive dust particles, until planets, stars and their clusters are formed. Dust particles of micron
size have a fairly dense core surrounded by a layer of loose matter. The minimum time required
for the formation of such particles can be estimated by the approximate formula for the radial
fall of matter to the accretion center under the action of gravitation [22]:
3
2
t
G
, (12)
where G is the gravitational constant, and is the mass density of matter at the initial
moment of fall. For example, with a density of
100
=
kg/m3 in (12) we obtain a duration of
approximately 2.3 hours. The lower the initial mass density of an object is, the longer it takes
for such an object to be formed. For a gas cloud with an initial density of
20
10
−
=
kg/m3, the
time t will be approximately
7
2.7 10
years. If we substitute in (12) the current density of
baryonic matter
28
2.1 10
b
−
=
=
kg/m3, the corresponding duration of metagalaxy
formation will be on the order of 180 billion years.
A more accurate calculation accepted in astrophysics takes into account the time required
for a gas cloud to increase its density with decreasing radius instead of taking into account the
time of fall into the accretion center. Let us assume that the evolution of matter in the
hierarchically structured Universe leads over time to the formation of baryonic matter with
average mass density
b
. This process cannot be uniform everywhere, and in those places,
12
where it goes faster, the matter can compress under the action of gravitation, regardless of the
surrounding volumes of space with lower density. For acceleration the particles’ motion in the
gravitational field outside the gas cloud with the mass M , we have:
2
2
2
d r
GM
dt
r
= −
. (13)
This equation (13) is also suitable for describing the motion of the gas cloud’s outer shell.
The solution of (13) should be sought in the form dr
A
B
dt
r
=
+
,
2
2
2
2
d r
A
dt
r
= −
.
Hence, it follows that
2
A
GM
=
, and if
2
b
GM
B
r
= −
, where
br is the initial radius of the
cloud, we obtain the relation for the velocity of the shell motion, which is associated with the
law of conservation of energy:
2
2
2
b
dr
GM
GM
dt
r
r
=
−
. (14)
For the case of cloud compression, the coordinate r decreases over time t , and therefore,
we use the following equation:
2
2
b
dr
GM
GM
dt
r
r
= −
−
. (15)
The solution of the differential equation (15), in the case of compression from radius
br to
radius sr , is as follows:
3
3
1
arctg
1
1
arctg
1
2
2
2
2
s
b
b
b
s
b
b
b
b
b
b
s
s
r
r
r r
r r
r
r
r
r
r
r
t
r
GM
r
r
GM
r
GM
GM
=
−+
−
=
−+
−
.
(16)
13
The maximum time is reached when the matter falls onto the point center with the radius
0
sr =
. Assuming that
1 3
3
4
b
b
M
r
=
, in the case in (16),
3
max
3
2
2
32
b
b
r
t
GM
G
=
=
. This
time depends only on the initial mass density
b
of the cloud and is estimated, since the
solution does not consider the pressure forces in the gas cloud, which rapidly increase as the
radius decreases.
Considering this approach, two scenarios are possible. In the first of them, the matter of the
observable Universe with an average density
b
arises from praons, the smallest particles of
the lowest level of matter, in a period of time determined by the physical conditions of this
process. To understand how a nucleon can be formed from a set of praons, it is enough to
imagine a similar process, in which a set of nucleons in a large gas cloud is compressed to the
maximum extent under the action of gravitation. If the mass of the emerging star is large
enough, then the result of its evolution would be a supernova and the birth of a neutron star.
Praons, nucleons and neutron stars are similar because they have the highest possible mass
densities and the strongest electromagnetic fields at their levels of matter. In this case, it is
assumed that at the level of nucleons the particles’ matter is held together not by ordinary
gravitation but by strong gravitation [10], [15].
In the second case, baryonic matter is first created; this matter is distributed in space with a
certain mass density
m
and subsequently compressed to density
b
. Let
b
significantly
exceed
m
. By substituting
1 3
3
4
b
b
M
r
=
instead of sr and
1 3
3
4
m
m
M
r
=
instead of br , and
neglecting the first term in (16), we obtain
m
3
32
m
t
G
. As an estimate, we will substitute
here
28
2.1 10
b
−
=
kg/m3 instead of
m
and will obtain the corresponding minimum
compression time, if it actually took place: min
3
145
32
b
t
G
=
billion years.
For comparison, in the standard cosmological
CDM
−
model, based on the general theory
of relativity, the age of the Metagalaxy is estimated to be approximately 13.8 billion years.
Moreover, in order to explain the spatial flatness, homogeneity, isotropy and large-scale
structure of Metagalaxy, the model includes the hypothesis of cosmological inflation at the
early stages of the Big Bang. The exotic character of such inflation is associated with the fact
that during a period of time from 10−42 sec to 10−36 sec after the start of the Big Bang at the
14
initial Planck matter density of approximately 1096 kg/m3 the radius of the Metagalaxy should
have increased by a factor of 1026 [23]. As can be seen from the estimates made above, if the
hypothesis of cosmological inflation is not used, the minimum age of the observable Universe
should be an order of magnitude greater than in the
CDM
−
model.
We can consider a typical CMB source as a relativistic uniform system and estimate its
internal thermal energy using the virial theorem [24-25]. Considering the contributions of
gravitational energy and pressure field energy to the system’s potential energy, according to
[26], the following relation is obtained for the kinetic energy
k
E :
2
81
100 14
k
Gm
E
a
. (17)
However, the energy
k
E can be approximately expressed in terms of the average
temperature
sT of the source:
3
2
s
k
p
mkT
E
m
, (18)
where the ratio of the source mass to the nucleon mass in the form
p
m
N
m =
specifies the
total number of nucleons N as an estimate of the total number of atoms, and k is the
Boltzmann constant.
Comparing expressions (17) and (18) for
k
E in view of the relation
3
4
3
s
a
m
=
gives us
the following:
25 14
18
s
p
s
kT
a
Gm
=
. (19)
A primordial gas-dust cloud with mass m , which is the source of CMB, can be considered
as a blackbody, in which matter is in thermal equilibrium with CMB radiation. The radiation
15
energy density inside the cloud should be equal to
4
4
s
s
T
u
c
=
. When a typical CMB source is
formed in the form of a gas-dust cloud, the binding energy
E
should be released, which is
equal in order of magnitude to the total kinetic energy
k
E of the cloud particles. A more precise
estimate in [26] gives
5 14 14
1
1.57
3
27
k
k
E
E
E
−
. We can assume that the binding
energy E
is radiated from the cloud by means of CMB radiation. In this case, the following
equality must be satisfied:
4
3
4
4.71
4
s
k
s
s
T
E
E
u
c
V
a
=
=
=
, (20)
where
3
4
3
s
a
V
=
is the cloud’s volume.
Substituting
k
E from (18) in (20) and considering the relation
s
s
m
V
=
, we find:
3
8
4.71
s
p
s
T m
ck
=
. (21)
6. Parameters of CMB sources
Relations (11), (19) and (21) can be considered as a system of three equations to determine
unknown quantities a ,
sT and
s
. Substituting
s
(21) into (19), we get:
5
4.71 14
12
s
p
k
c
a
T m
G
=
. (22)
Multiplying a (22) by
s
(21), we find:
2
10
14
3
4.71
s
s
T
a
cG
=
. (23)
16
Substituting (23) into (11) leads to a quadratic equation for
2
sT :
4
4
2
4
3
40
14
0
9
4.71
s
s
b
HT
T
T
T
c G
−
−
=
. (24)
Solving equation (24) gives the surface temperature of a typical CMB source:
(
)
2
2
4
3
3
2
20
14
400 14
1
3.472
9
4.71
81 4.71
s
b
b
HT
H T
T
T
c G
c G
=
+
+
=
K. (25)
Substituting
sT (25) into (22) and into (21), we obtain the radius a of the CMB source in
the form of a gas-dust cloud and the density
s
of the substance of the cloud:
16
2.088 10
a
=
m.
18
1.629 10
s
−
=
kg/m3. (26)
The radius of the cloud in (26) reaches the value
0.68
a =
pc.
Next, taking into account (26), we find the mass of the source:
3
31
4
6.21 10
3
s
a
m
=
=
kg or 31.2
c
M , (27)
where
c
M is the mass of the Sun.
Parameters (26-27) of a typical CMB source correspond to a rather large gas-dust cloud, the
particles of which acquire their kinetic energy due to gravitational work to compress matter.
When the particles collide, the energy of motion is converted into heat and can then be radiated
in the form of CMB quanta.
If we take into account that the obtained in (26-27) parameters of the sources belong to gas
clouds in the early Universe, then we can expect that the first stars appeared precisely in such
clouds. Later, similar clouds could give rise to the first open star clusters, which became the
main elements of emerging galaxies. For comparison, the number of stars in currently observed
open star clusters can be more than one hundred, the typical masses of clusters can exceed
17
50
c
M , the core’s radius can reach approximately 0.6 pc, and the radius of the corona in a
typical cluster can reach 6 pc.
The concentration of CMB sources in the early Universe is found through the density of
baryonic matter
b
and the mass m of a typical source according to the first relation in (11):
60
3.4 10
b
n
m
−
=
=
m–3. (28)
If we assume that each source is located in a certain cubic volume in a cubic lattice, then
the shortest distance between the nearest sources will equal
19
3
1
6.7 10
s
R
n
=
m. This means
that the distance between the centers of the nearest sources is
3
3.2 10
s
R
a
times greater than
the radius a of a typical source and is equal to the value on the order of
2.16
s
R
kpc at
0.68
a =
pc according to (26).
The value of L , that is CMB generation power per unit volume of the Universe, is found
from (9) taking into account
sT (25), a (26) and n (28), or from (7) taking into account the
relations
4
I
T
=
,
2
s
a
=
:
31
4
2
4
2
4
4
1.5
0
3 1
s
H
L
T a n
T
a n
c
−
=
=
+
=
W/m3. (29)
The values of L may differ slightly in different directions, reflecting the variability of the
Hubble parameter and the spatial matter distribution.
The average concentration of nucleons in the Universe is
0.125
b
b
p
n
m
=
=
m–3. Taking this
into account, from (29) the power of CMB energy generation per nucleon of the Universe is
determined:
30
1.22 10
b
L
n
−
=
W/ nucleon. (30)
18
Dividing the binding energy
1.57
k
E
E
of one SMB source by the number
p
m
N
m
=
of
nucleons in this source, taking into account expressions
k
E (18) and
sT (25), we find the
binding energy per nucleon:
22
4.71
2
1.1 10
s
kT
E
N
−
=
=
J/ nucleon. (31)
On the other hand, an estimate of the photon’s concentration in the volume of a CMB source
in a state of temperature equilibrium between radiation and matter at the temperature
3.472
sT =
K is obtained as
8
8 4
1
. 9
0
f
f
s
n
k
=
=
m-3, where
3
14
16
4
4.22 10
3
3
s
s
f
s
T
u
s
c
T
−
=
=
=
J/(K·m3)
is
the
volumetric
density
of
the
CMB
entropy
and
the
coefficient
2
4
0
45
0.2776...
4
1
d
e
=
−
Dividing the photon energy density
4
4
s
s
T
u
c
=
(20) by the photon concentration
fn , we
obtain the average energy per photon:
22
3
1.29 10
4
s
s
f
u
kT
n
−
=
=
J/photon. (32)
Note that the binding energy per nucleon
E
N
in (31) and the energy per photon
s
f
u
n in (32)
are close to each other in magnitude. After the photons leave the CMB sources, fill outer space
and reach the Earth, their average temperature will decrease from value
3.472
sT =
K to value
2.7255
T =
K. In this case, the energy of a photon with a frequency of
160.23
m
=
GHz
corresponding to the maximum in CMB radiation at temperature
2.7255
T =
K is equal to
22
1.06 10
m
h
−
=
J.
19
Hence, the ratio of the number of CMB photons in cosmic space to the number of matter
nucleons present in this space should be on the order of unity. On average, we can assume that
each nucleon of the observable Universe produces only one CMB photon.
In the course of our calculations, we assumed that all the nucleons present in the Universe
with an average density of
28
2.1 10
b
−
=
kg/m3 were compressed by gravitation into
primordial gas-dust clouds with an average density of
18
1.629 10
s
−
=
kg/m3 (26). These
clouds play the role of typical CMB sources. Let the volume of a typical source be denoted by
3
4
3
s
s
m
a
V
=
=
and the volume of the same, but not compressed matter in the homogeneous
Universe with density
b
and with the same mass m (27) be denoted by V . Then, the ratio of
the volumes equals
9
7,
1
6
0
7
s
s
b
V
V
=
=
. Now, in view of
k
E (18), the binding energy
1.57
k
E
E
of one source, and the number of nucleons
p
m
N
m
=
in the source, we can estimate
the average CMB energy density in the Universe in the following form:
23
1.57
1.57
1.57
3
4.71
2
2
1.4 10
k
b
k
b
b
s
b
s
s
s
p
E
E
kT N
kT
E
u
V
V
m
m
m
−
=
=
=
=
=
=
J/m3. (33)
When deriving relation (20), we use the expression
s
s
E
u
V
=
. Combined with (33) and the
relation
9
7.
1
6
0
7
s
s
b
V
V
=
=
, this gives the following:
9
7.76 10
s
s
s
b
u
V
u
V
=
=
=
. (34)
In (34), a significant difference is obtained between the averaged CMB energy density u
in (33), and between the photon energy density
su in the case of thermal equilibrium of photons
with the matter of CMB sources. The difference in magnitudes of
su and u results from the
fact that the CMB generated in the volume
s
V of each source was in equilibrium with the matter
only in this volume. When the CMB from each source fills the volume V and starts mixing
20
with the radiation from nearby sources, the CMB energy density decreases from
su to u . As a
result, the SMB observed on Earth is thermal radiation, the energy density of which in the
Universe. is u .
In
CDM
−
model, the difference in quantities of
su and u is not taken into account, and
it is assumed that the energy density of the СМВ is equal to
4
14
4
4.17 10
T
u
c
−
=
=
J/m3 in
the entire space of the Universe at temperature
2.7255
T =
K of СМВ.
In this case, the estimate of the CMB photon concentration in the form
3
8
16
4
4.1 10
3
3
f
f
s
T
u
n
k
ck
kT
=
=
=
=
m-3 applies to the entire Universe. Then the ratio of
the number of CMB photons to the number of nucleons will equal
9
3.3 10
f
f
p
b
b
n
n m
n
=
=
,
which is close in magnitude to the ratio of the volumes
s
V
V
in (34).
Hence, the following question arises: why is the number of photons so much greater than
the number of nucleons? This problem, known in cosmology as the entropy problem, is usually
solved via the concept of the hot Universe based on the assumption of adiabatic space
expansion from the initial state of equilibrium of radiation and matter.
In contrast, in our approach the numbers of СМВ photons and nucleons are approximately
the same, and there is no need for the hot Universe. Since the SMB is currently not in
equilibrium with matter, the formula for energy density
4
4 T
u
c
=
cannot be applied to the
entire Universe. Indeed, the primordial gas-dust clouds, which were the sources of CMB in the
early Universe, were located in a discrete way, occupied a small volume and therefore could
not play the role of a global black body, limiting the volume of the Universe.
7. The angular harmonics of CMB
Using the Fourier transform, the observed CMB power spectrum can be expanded to
include spherical harmonics [27-28].
The spherical harmonic
0
=
in the angular power spectrum corresponds to the average
value of the CMB temperature. The dipole anisotropy on the CMB temperature distribution
map has an amplitude of approximately 0.1% and is related to the spherical harmonic
1
=
[29]. The dipole anisotropy is well explained by the Doppler effect and by the motion of the
21
Earth together with the Sun relative to the reference frame, in which the CMB intensity is the
same in all directions. This motion changes the CMB radiation wavelength measured on Earth,
depending on the angle between the Earth’s total velocity in cosmic space and the direction of
the sky region from which the CMB originates.
The spherical harmonics
2
are related to CMB temperature fluctuations, the root-mean-
square deviation of which reaches several tens of μK relative to the average temperature. The
CDM
−
model assumes that such temperature fluctuations could be caused by fluctuations
in the density of matter in the early Universe, which had the state of very dense hot plasma of
electrons and baryons.
One alternative explanation is that the angular power spectrum of the CMB can be obtained
if CMB photons, upon their appearance, interact with matter, which was structured into some
objects, clusters and particles [30]. The average distance between the centers of the objects in
the case of their cubic arrangement is approximately 108 m, the mass of one object is
17
8.8 10−
kg, and the mass density of the object is
23
9 10−
kg/m3. Similarly, there are clusters inside the
objects, the distance between the centers of which is about 12 cm. If their mass density is
23
9 10−
kg/m3, then the cluster mass is
25
1.2 10−
kg. The position and amplitude of the main
peak of CMB power spectrum at
360
and of the subsequent peaks depend mainly on the
mutual distances between the mentioned objects, and on their mass density and internal
structure.
It is assumed that the cluster contains 40 to 100 particles, such as protons, helium nuclei
and electrons; therefore, it represents an atomic-molecular complex, containing hydrogen and
helium. On average, an object contains approximately
9
10 clusters. The results obtained for the
structure of objects, clusters and particles were found under the condition of using a radiation
wavelength equal to 1.9 mm. This wavelength corresponds to the maximum blackbody
spectrum distribution at the temperature
2.7255
T =
K and most exactly reflects the properties
of CMB from the standpoint of structural analysis. Moreover, the angular power spectrum of
CMB radiation is the same for all wavelengths of the CMB.
The presented parameters can be combined with the described scheme of CMB emergence
in the early Universe, in which matter evolution first leads to formation of nucleons and
electrons. Then gravitation compresses the matter into gas-dust clouds and sets the matter
particles in motion. When the particles collide, the kinetic energy is converted into thermal
energy and is radiated in the form of CMB photons. The mass density of objects in [30] is less
than the mass density of gas clouds
18
1.629 10
s
−
=
kg/m3, as found in (26). We can assume
22
that the objects in [30] were located in the less dense part of the shell of gas clouds. Then, CMB
photons, passing through these objects, clusters and particles inside them, can form the
currently observed angular radiation power spectrum.
The possibility that these objects, clusters and particles could appear in primordial gas
clouds follows from the value of the Jeans mass [31], which can be simplified as follows:
1 2
3 2
9
10
26
10
J
c
T
M
M
n
. (35)
The temperature T in (35) must be specified in K, and the particle concentration n must
be specified in
3
m−.
Let us substitute in (35) the cloud surface temperature
3.472
sT =
K (25) instead of T , and
take into account
18
1.629 10
s
−
=
kg/m3 (26) and the concentration of cloud particles
8
9.7 10
s
s
p
n
m
=
=
m-3 instead of n . This gives
5.4
J
c
M
M
=
. The Jeans mass is less than the
gas cloud mass
31.2
c
m
M
=
(27), which allows fragmentation of the cloud into smaller
structural components.
The nonuniform distribution of matter in cosmic space also contributes to the small-scale
fluctuations in the CMB temperature. Thus, correlations between the optical radiation of
galaxies and CMB fluctuations are described in [32], and for radio sources, such correlations
are presented in [33]. The cold anomalies in CMB temperature are mysterious and cannot be
explained from the standpoint of the
CDM
−
model, the most famous of which is the WMAP
cold spot discovered by the WMAP space observatory in the Eridanus constellation [34-35].
The cold spot is approximately
73
dT =
µK colder than the CMB temperature,
2.7255
T =
K.
These anomalies can be explained as follows. Since
4
I
T
=
for CMB, in the first
approximation, we have
3
4
dI
T dT
=
. If we direct the radiation receiver exactly to the
anomalous spot, then in (6), we can set
2
Q
=
,
2
=
, and we can write the following:
(
)
3
2
exp
exp
4
4
Hr
L
snr
dV
c
dI
T dT
r
−
−
=
=
. (36)
23
According to (7) and considering the relation
4
I
T
=
, we find:
4
4
L
I
T
H
sn
c
=
=
+
,
4
4
H
L
T
sn
c
=
+
. (37)
Substituting L (37) into the expression for dI (36), we find the relationship between the
temperature difference dT , the distance r and the volume dV generating CMB radiation and
leading to the contribution dT to the CMB temperature T :
(
)
2
exp
exp
4
H
Hr
T
sn
snr
dV
c
c
dT
r
+
−
−
=
. (38)
Equation (38) assumes that the volume dV contains primordial gas-dust clouds functioning
as CMB sources, which, on average, have the mass density
18
1.629 10
s
−
=
kg/m3 (26) and
the concentration
60
3.4 10
b
n
m
−
=
=
m–3 (28) in the early Universe. In this case, the
concentration n is present in (38) in two terms. An increase n in the term H
sn
c
+
leads to
an increase of dT , but the term
(
)
exp
snr
−
acts in the opposite way, reducing dT . Thus, if
in some direction in the volume dV the concentration of sources differs from the value n due
to some anomaly, then this will lead to a change of dT in (38).
We can see from the relation
b
n
m
=
that if the mass m of a typical CMB source is constant,
then the change in the concentration of sources n can be associated with a local change in the
mass density of baryons
b
in the volume dV under consideration. Thus, fluctuations in the
mass density
b
in cosmic space can influence fluctuations in the measured CMB temperature
in various directions.
According to (8), the redshift z in the model under consideration can be related to the
distance r by the formula:
(
)
ln
1
c
r
z
H
=
+
. As an example, let us place a certain volume dV
24
at a distance corresponding to
2
z =
, that is, at
ln3
4.7
c
r
H
=
=
Gpc. Let us assume that dT
equal to 20 μK, which is close to the value of the root-mean-square small-scale CMB
fluctuations. Using further the relation
2
s
a
=
, the values of
16
2.088 10
a
=
m (26) and
60
3.4 10
n
−
=
m–3 (28), from (38) we find
74
9.35 10
dV =
m3, which for a spherical volume
gives the radius of this volume of the order of
197
v
R =
Mpc. Such the radius is close to the
radius of the known Giant Void in the constellation Canes Venatici [36]. Consequently, the
presence of voids in accordance with (38) leads to the observed CMB fluctuations.
It is known that, for sufficiently large spherical harmonics the relation
2
= holds true,
where is the effective sky viewing angle [28]. The main peak in the CMB angular power
spectrum occurs at
360
, which corresponds to the angle
1
. Moreover, if the harmonic
number
360
and decreases, and the angle
1
and increases, the power in the
angular spectrum decreases gradually without any particular peaks. Why do harmonics with
small manifest in the spectrum in a different way than harmonics with large
360
, which
form the spectrum in the form of sinusoid damping in amplitude?
For the above example with the volume dV , let us calculate the angle in radians using
the formula:
2
0,084
v
R
r
=
=
, and the angle in degrees
180
4 ,8
=
=
. The angle in
radians corresponds to the spherical harmonic
2
75
=
=
in the CMB angular power
spectrum. It turns out that supervoids with a low density of matter or, on the contrary, denser
regions of space can make a significant contribution to CMB temperature fluctuations only at
small . As for the appearance of spherical harmonics with large values
360
in the power
spectrum, in the model presented above, according to [30], they are explained by the fact that
CMB radiation interacts with matter, which is structured into some objects, clusters, and
particles. Thus, different mechanisms lead to different forms of CMB power spectrum for small
and large spherical harmonics. In contrast, the
CDM
−
model has difficulties explaining the
low power and form of the angular spectrum for harmonics associated with large angles and
with small [37].
8. Infrared and optical background radiation
The dependence of the angular intensity J on the cosmic background radiation frequency
in [9] shows that there are other angular intensity peaks near the CMB, including those of the
25
cosmic infrared background (CIB) and the cosmic optical background (COB). The total
intensity of CIB and COB radiation is almost 10 times less than the intensity of cosmic
microwave background radiation (CMB).
It is believed that the main contribution to CIB and COB comes from the processing of
radiation from protostars and young stars by cosmic dust. We can make this assumption more
concrete. Let us apply the obtained results with respect to the CMB to estimate the parameters
of space objects, which could produce CIB and COB radiation.
We can assume that the maximum J for the CIB is obtained at a radiation frequency of
12
2 10
CIB
Hz, and for the COB the maximum occurs at a frequency of
14
3 10
COB
Hz.
As a first approximation, let us assume that the Wien displacement law for radiation from a
black body is valid for the radiation frequency. This gives the corresponding radiation
temperatures
34
CIB
T
=
К and
5103
COB
T
=
К.
Substituting temperatures
CIB
T
and
COB
T
in (25) instead of temperature T makes it possible
to estimate the surface temperatures of CIB and COB sources in the early Universe:
426
s CIB
T
=
K.
6
9.6 10
s COB
T
=
K. (39)
Substituting temperatures (39) into (21-22) instead of
sT , we obtain the corresponding mass
densities and radii of CIB and COB sources. We can also estimate the masses of sources by
multiplying the mass density and the volume of the corresponding source:
12
3.01 10
s CIB
−
=
kg/m3.
14
1.
10
7
CIB
a
=
m.
31
6.19 10
CIB
m
=
kg. (40)
34
s COB
=
kg/m3.
9
7.5 10
COB
a
=
m.
31
6.008 10
COB
m
=
kg. (41)
According to (40), the sources of CIB radiation are gas-dust clouds with a radius of the
order of
1136
CIB
a
=
AU. and with mass
31.1
c
CIB
m
M
=
. For comparison, in the Solar System
the dwarf planet Sedna at aphelion moves away from the Sun at a distance of 937 AU. From
(41) it follows that COB radiation sources are objects with mass
30.2
c
COB
m
M
=
and with a
radius of the order of
10.8
c
COB
a
R
=
, where
c
R is the radius of the Sun.
26
In (26) and in (27) it was found that a typical CMB radiation source has a radius of 0.68
pc and a mass of the order of. 31.2
c
M . The sources of radiation CIB and COB in (40-41) also
have masses of the order of 31
c
M . It turns out that during cosmological evolution, CMB
microwave radiation sources first become isolated in the form of gas clouds with a radius of
the order of 0.68 pc. When these clouds are subsequently compressed by gravitational forces
to a radius of the order of
1136
CIB
a
=
AU, protoplanetary systems containing gas and dust arise,
leading to infrared background radiation CIB. The compression of clouds slows down due to
the appearance of pressure in the gas, and the process of planet formation begins in the clouds.
At the same time, the bulk of gas and dust in the center of each cloud continues to compress.
As a result, primordial stars emerge, producing optical background radiation COB. Nuclear
reactions begin in the depths of these stars, preventing the gravitational compression of matter.
Thus, the CMB, CIB and COB emissions are associated with the most long-term and
equilibrium phases in the evolution of primordial gas-dust clouds.
Since the masses of CIB and COB sources approximately coincide with the mass
31.2
c
m
M
=
(27) of CMB sources, the concentration of CIB and COB sources in the early
Universe approximately coincides with the concentration
60
3.4 10
b
n
m
−
=
=
m–3 (28) of CMB
sources. Due to their large masses, primordial stars that are the sources of COB should
transform into neutron stars. Thus, the concentration n can be considered as the concentration
of primordial neutron stars. The average distance between such stars can be estimated using
the formula
19
3
1
6.7 10
s
R
n
=
m or
2.16
s
R
kpc. On the other hand, the observable
Universe has a volume of the order of
80
3.6 10
m3, which contains about
24
10 stars [38]. The
concentration of stars in the Universe is on the order of
56
3.6 10−
m–3. Comparing the
concentration of all stars with the concentration of primordial neutron stars leads to the fact
that there are about
4
10 ordinary stars per a neutron star. This ratio of stars is indeed confirmed
by observations.
Modern instruments allow us to measure angular power spectra not only for CMB, but also
for CIB radiation [39]. Thus, the methods for analyzing the structure of radiating objects,
developed in [30] for CMB, can also be applied to CIB radiation. According to [30], the mass
density of the medium, which contains objects, clusters, and particles and is responsible for the
appearance of CMB angular harmonics, equals
23
9 10−
kg/m3. This mass density does not
exceed the mass density in (40-41) of the objects, which can be the sources of CIB and COB
27
radiation. This implies the possibility that the cause of harmonics in the CMB and CIB power
spectra may be the same objects, clusters and particles located in the shells of the corresponding
gas-dust clouds at different stages of compression of these clouds.
In (34) it was shown that the formula for the energy density of СМВ
4
13
4
1.1 10
s
s
T
u
c
−
=
=
J/m3 (20), where
3.472
sT =
K is the temperature of СМВ sources in the early Universe, cannot
be applied to the entire Universe. Instead of
su , a significantly lower average volumetric energy
density of the CMB was calculated, equal to
23
1.4 10
u
−
=
J/m3 according to (33). This was
because the gas mass m occupying the volume V in (34) in the initially homogeneous
Universe with the mass density
b
, only after being compressed into a gas cloud with a volume
s
V and density
s
. would start radiating like a black body with the temperature
3.472
sT =
K.
As СМВ photons move through space, their energy decreases due to cosmological redshift. In
addition, photons interact with the matter of many СМВ sources and are partially scattered. As
a result, the intensity of the СМВ radiation decreases, and the spectrum of СМВ photons
becomes close to the observed spectrum of the radiation of a black body with temperature
2.7255
T =
K.
For CIB and COB radiation the situation is largely similar. As in the case of CMB radiation,
CIB and COB radiation are nonequilibrium and for them there is no global blackbody
consisting of matter and bounding the entire Universe. This means that in fact we always
observe discrete sources of radiation, which at large distances merge into an almost uniform
background.
9. Discussion of results
A well-known problem of the Big Bang theory in cosmology is the complete lack of
understanding of the nature of such an explosion and of the origin of matter as such. The
subsequent use of the general theory of relativity in the
CDM
−
model adds new problems,
such as singularities and metric space expansion, which are incomprehensible from the
perspective of physics, the appearance of unidentified dark matter and mystical dark energy.
The use of multiple fitting parameters in the
CDM
−
model further undermines the
credibility of the modern version of the Big Bang theory.
In [40], six fitting parameters are listed, and it is concluded that despite the accuracy of the
results’ fitting, it is still not enough to consider the
CDM
−
model correct. In [13], the
following conclusion was reached: ”However, in cosmology, it has unfortunately been the case
28
that even a long series of failed predictions has not generally led to the rejection of theories,
but rather to their unlimited modification with ad hoc hypotheses, such as inflation, non-
baryonic matter, and dark energy.”
It is noted in [41] that it could be possible to improve the situation with predictions in the
CDM
−
model if, in addition to the seven fitting parameters of the model, we would assume,
for example, the existence of early or dynamical dark energy, neutrino interactions,
cosmological models with additional interactions, primordial magnetic fields, modified
theories of gravitation, etc.
On the other hand, cosmology in the theory of infinite hierarchical nesting of matter finds
the source and cause of origin of matter and the forms of its existence in the uniform
evolutionary process of transformation of the main carriers at all matter levels [15]. This means,
for example, that the evolution of the matter of planets and stars at the stellar level of matter is
due to the evolution and action of carriers belonging to lower matter levels. Each matter level
has its own main carrier as the most stable and balanced object, such as a neutron star, nucleon,
praon, graon, etc., respectively. The similarity principle assumes the existence of the same
coefficients of similarity in mass, size and speed of processes between the respective objects
of the adjacent matter levels, which allows us to find the physical parameters of the main
carriers of matter. As a consequence, a neutron star contains as many nucleons as each nucleon
contains praons, and as each praon contains graons.
The main driving forces for matter evolution are electromagnetic and gravitational forces,
which can be reduced to the action of carriers of the lowest matter levels, moving at relativistic
speeds [10], [42-46]. This point of view is supported in [47] by the fact that the evolution of
matter in the early Universe turns out to be little dependent on external factors and is
determined mainly by internal factors.
It follows from the principle of similarity of matter levels that the analogs of a neutron star
at the nucleon level of matter are nucleons, and the analogs of white dwarfs are the so-called
nuons [4], which have the same mass range as nucleons. Nuons are similar in their properties
to muons, but the origins of these particles are different: nuons appear similar to white dwarfs
in the course of long-term evolution of matter, and muons appear mainly in the rapid decay of
pions, while pions are assumed to be the analogs of low-mass and therefore unstable in the
decay of neutron stars. Moreover, neutral nuons play the role of dark matter, which manifests
itself through gravitational effects both on the motion of stars and gas clouds inside galaxies,
29
and on the motion of galaxies themselves during their interaction [48]. In contrast, the
CDM
−
model has difficulties explaining dark matter, describing its evolution and origin.
The CMB generation process continues to occur today, although on a smaller scale. The
properties of the coldest dark nebulae with masses up to 100
c
M are quite close to those of
primordial gas-dust clouds. Thus, practically opaque Bock globules, which are distinguished
by their black color, have a temperature in the range from several degrees to 30 K and a typical
mass of up to 30
c
M . From the standpoint of thermal radiation, such objects can model the
properties of a blackbody quite well. Here, we provide as examples references to the spectra of
the infrared sources IRS 1 and IRS 2 in [49-50] and to the spectrum of the Bock globule B335
in [51].
One review [52] described the properties of 248 small molecular clouds, most of which are
Bok globules. It is assumed that in our Galaxy system the average distance between such
globules is 600 pc, and the average mass of a globule is approximately 11
c
M . This distance
can be compared with the value
2.16
s
R
kpc found for the distance between the primordial
gas-dust clouds of the early Universe through the concentration n of CMB sources in (28).
According to [52], for most globules the radiation temperature of gas does not exceed 4.5
K, and the kinetic temperature does not exceed 8.5 K. Since globules are heated by radiation
from the surrounding star background, the dust temperature in globules turns out to be higher
than the gas temperature and depends on the measured frequency band and on the size of the
dust particles. On average, the dust temperature is close to 25 K. Under such conditions, the
spectrum of some globules appears to be not the blackbody spectrum at one fixed temperature,
but rather the sum of the spectra of individual components consisting of gas and dust.
In [53], by analyzing the absorption lines of water molecules in a large cloud of water vapor
near the HFLS3 galaxy, it was found that the temperature required for this exciting radiation
ranges from 16.4 to 30.2 K. Since the HFLS3 galaxy has a redshift of the order of magnitude
6.34
z =
and is located sufficiently far away, this temperature is considered from the point of
view of the
CDM
−
model as the CMB temperature at an earlier time. Moreover, due to the
space expansion since then, the CMB temperature should have decreased to the current value
of
2.7255
T =
K.
On the other hand, radiation at a temperature ranging from 16.4 to 30.2 K is quite typical
for gas clouds and Bok globules under the action of radiation from surrounding stars, including
those in the most distant galaxies. In the model we are considering, the CMB temperature at
the moment of emission coincides with the temperature
3.472
sT =
K (25) of gas-dust clouds
30
with masses on the order of 31
c
M , so distant from us that their redshift is much greater than
the redshift of the observed galaxies.
How does the CMB radiation spectrum almost exactly correspond to the blackbody
spectrum? Here, the following circumstances can be taken into account. First, we assume that
there were no stars around the primordial gas-dust clouds of the early Universe that could
noticeably heat the clouds and influence the form of the clouds’ spectrum. Then, the spectrum
of each cloud could be sufficiently close to the spectrum of a blackbody with the temperature
3.472
sT =
K.
Another circumstance is associated with the size of the visible Universe. Substituting in (8)
the maximum measured value of redshift
1089
z =
for the CMB according to [54] at
70
H =
km/(s·Mpc), we find the radius of the visible Universe
26
9.24 10
r =
m or 30 Gpc. Light can
travel this distance in
98
r
t
c
=
=
billion years. This time is less than the minimum compression
time
min
3
145
32
b
t
G
=
billion years, which was found according to (16) for the
compression of all baryonic matter during the formation of the visible Universe.
It should be noted that when measuring CMB intensity, it is necessary to exclude radiation
from bright point sources such as clusters of stars and galaxies from the obtained data to
determine the background radiation precisely. However, the redshift of the most distant
observable galaxies does not exceed
12
z =
. For example, the redshift of the galaxy GN-z11
equals
11.09
z =
according to [55]. This redshift is significantly less than the redshift of the
CMB, which reaches
1089
z =
. Thus, CMB radiation travels the main part of its way in
unexplored distant regions of cosmic space.
In (6-7) the fact was used that the cross sections of CMB sources in the early Universe and
the concentration of sources are such that at very large distances, these cross-sections begin to
overlap each other; and, therefore, the Beer–Lambert law becomes valid. As a result, the CMB
radiation coming to the Earth from distant sources has enough time to interact with the matter
of multiple closer sources and additionally thermalize. This approach inevitably turns the CMB
spectrum into an averaged spectrum that is close to the equilibrium spectrum of a blackbody.
The presented model is consistent with the results in [56], where the so-called virial gas
clouds located in the halo of galaxies make it possible to explain the rotational anisotropy
observed in the CMB. In [57], within the framework of the standard model of the expanding
Universe, the evolution of virial clouds from the surface of the last scattering to the formation
31
of primordial stars of population III was considered. These virial clouds, which are also in
thermal equilibrium with the CMB, as in our approach, have almost the same density as the
primordial gas-dust clouds in our model. Thus, the conclusions in [56-57] concerning
primordial gas-dust clouds prove our calculations.
The Earth and the Sun are known to move relative to the reference frame, in which the
CMB is isotropic, at a speed of approximately 370 km/s. If we take into account the motion of
the Sun in our galaxy and its motion in the Local Group of Galaxies, then the speed of the
Local Group of Galaxies relative to the CMB’s isotropic reference frame will be about
627
g
V =
km/s [58]. In the
CDM
−
model, the cosmological redshift is interpreted as a result
of the Universe expansion, which has the mathematical meaning of a change in the spacetime
metric, caused by an unknown factor. The physical meaning of this space expansion is an
obvious subject for discussion regarding the justifiability of the use of mathematical hypotheses
in the real physics of phenomena. It is assumed that at large distances from the Earth, galaxies
and other objects, located there, are moving away from each other at tremendous speeds due to
space expansion. These speeds can significantly exceed the speed
g
V of motion of the Local
Group of Galaxies relative to the isotropic reference frame of the CMB. So why from the entire
speed spectrum do we observe a relatively small speed
g
V , is it by chance?
From the viewpoint of the theory of infinite hierarchical nesting of matter, the answer lies
in the fact that, for matter evolution and for the emergence of a new matter level with more
massive objects, neither the Big Bang, nor the metric expansion of spacetime or high speeds of
motion are needed. The deviation of the speed of galaxies and star clusters from the speed of
the CMB’s isotropic reference frame can be caused only by the gravitational action of galaxies
on each other. When averaging the matter’s speeds over the volume of the visible Universe,
the obtained average speed must coincide with the speed of the CMB’s reference frame because
CMB occurs in the early Universe and, on average, is stationary relative to the global
distribution of matter.
Taking this into account, the problem of space flatness on cosmological scales becomes
understandable when even at very large distances, spacetime is practically not curved.
Therefore, there is no great need to calculate any curved metric, which is always needed in the
general theory of relativity, even in flat Minkowski spacetime. In this case, instead of the
general theory of relativity, it is more convenient to use the covariant theory of gravitation [59-
60], in which the metric effects are separated from the gravitational effects. This means that
gravitation does not depend on a metric; rather, it is a real physical force, such as the
32
electromagnetic force, which exists even in Minkowski spacetime, when there is no spacetime
curvature.
Paradoxes arising from the concept of space expansion were analyzed in [61], such as the
violation of the law of conservation of energy for local comoving volumes, the Newtonian form
of Friedmann's equations, the superluminal velocities of distant galaxies as a result of space
expansion, and Hubble's law in inhomogeneous distributions of galaxies, etc. The main reason
for the appearance of such paradoxes is the general theory of relativity due to the absence of
the energy-momentum tensor of the gravitational field in this theory, which casts doubt on the
possibility of using this theory in cosmology.
It was noted in [62] that the time scale in the
CDM
−
model does not correspond to the
time, which was required for the formation of large galactic clusters and voids in the early
Universe. In this regard, it is assumed that this discrepancy may be due to the use of the general
theory of relativity, which should be replaced with another gravitation theory, for example,
modified Newtonian dynamics (MOND). The authors of [62] draw the following conclusion
from their article: at the present moment, we understand neither the distribution of matter and
energy in the Universe nor the law of gravitation, which governs this.
It was found in [63] that a supercluster of galaxies with a radius of about 6 Mpc rotated at
an angular velocity equal to 2 .9
degrees per 10 billion years, or
19
1.6 10
−
=
s-1.
According to [64], large galactic clusters with sizes on the order of
800
c
R
Mpc can
experience general motion at velocities up to
1000
c
V
km/s. Assuming that this motion arises
from rotation, for the angular velocity in the first approximation, we have
20
4 10
c
c
c
V
R
−
s-1. The large-scale structure of the Universe has the form of a cosmic web and consists of
individual filaments that contain galactic clusters. The difference
V
in the linear velocity of
rotation of different points in the filaments can reach 100 km/s at a distance
R
between the
points equal to 1 Mpc, which gives an estimate of the angular velocity of rotation
18
3 10
f
V
R
−
s-1 according to [65].
If we assume that the entire visible Universe also rotates, then its limiting rotation can be
estimated by the formula for the first cosmic velocity, assuming that the matter at the edge of
the Universe is in equilibrium between the gravitational force and the centripetal force:
33
3
4
3
U
U
U
U
GM
G
R
=
=
.
where
U
M
,
U
R and
U
denote the mass, radius and mass density of the Universe,
respectively. Substituting here the mass density
28
2.1 10
b
−
=
kg/m3 instead of
U
, we find
19
2.4 10
U
−
=
s-1, which has the same order of magnitude with respect to the rotation of large
galactic structures. This rotation corresponds to a period of approximately 830 billion years.
It is obvious that any general rotation of the observable Universe contradicts the Big Bang
since, due to rotation the Universe would have a nonzero angular momentum. In the scenario
of the Big Bang and subsequent inflation up to the state of the observable Universe, it turns out
that, taking into account the law of conservation of angular momentum, the object that gave
rise to the Universe should have had an enormous angular momentum for its small size, which
seems completely improbable. At the same time, in the hierarchical model, the object, from
which the early Universe was formed could consist of a huge cloud of praons. If this cloud with
the sizes of the order of the observable Universe had any general rotation, then the emerging
Universe would have the same rotation after the evolution of praonic matter and its
transformation into nucleons and nuons.
We believe that other, less complex cosmological problems can also find their solution
within the framework of the theory of infinite hierarchical nesting of matter and a stationary
Universe. For example, in [66] it is indicated that expansion of space is not required to explain
the change in luminosity in the spectra of supernovae. Thus, cosmology can be constructed
with a minimum of assumptions and paradoxical conclusions that contradict the traditional
logic of physics.
10. Conclusions
Cosmic microwave background radiation (CMB) along with the effect of the cosmological
redshift of radiation spectra are usually considered phenomena that find an acceptable
explanation within the framework of the Big Bang theory. Indeed, these phenomena are rather
difficult to explain, which eventually led to the idea of the Big Bang. However, due to the
significant drawbacks of this theory, which are described above, we consider this theory to be
too exotic and radical and suggest another explanation for emergence of CMB.
In our approach, the necessary source of CMB energy turns out to be gravitational energy,
which, under matter clustering in primordial gas-dust clouds in the early Universe, is released
34
in the form of the kinetic energy of motion of matter particles, according to the virial theorem.
The subsequent collisions of particles convert kinetic energy into thermal energy, heating the
particles, so that gas-dust clouds can radiate as black bodies at the temperature
3.472
s
T =
K
(25). During the time until CMB radiation from distant regions of the Universe reaches the
Earth, the temperature of this radiation decreases to the value
2.7255
T =
K.
Based on this approach, we first find formula (7) for the volumetric power L of CMB
energy generation in cosmic space, and then we find the main characteristics of primordial gas-
dust clouds, including their radius
0.68
a
pc (26), mass
31
c
m
M
(27), volume
concentration of clouds
60
3.4 10
n
−
m–3 (28) and distance between neighboring clouds
2.2
s
R
kpc. For the volumetric power L of CMB energy generation, the value
31
1.5 10
L
−
W/m3 (29) is obtained, while the generation power per nucleon of a typical gas-dust cloud, as
well as per nucleon in the early Universe, is equal to
30
1.2 10
b
L
n
−
W/nucleon (30).
We can conclude that the CMB originated as thermal radiation from primordial gas-dust
clouds. This thermal radiation could interact with a large number of particles in each cloud,
which, in view of the slowly changing equilibrium state of the clouds and their opacity and
weak reflection, provides the CMB radiation spectrum close to that of blackbody radiation. In
addition, CMB radiation coming from very large distances passes through many separate CMB
sources in the form of gas clouds on its way to Earth. This further contributes to the
transformation of the CMB spectrum into a blackbody spectrum.
As indicated in Section 6 with reference to [30], the angular harmonics in the CMB power
spectrum can be explained if we take into account matter clustering near the surface of
primordial gas-dust clouds. This possibility is supported by the presence of both small and
large dust particles observed in Bok globules, leading to significant polarization of radiation in
the millimeter range [67-69] up to values on the order of 10 % or more. In this regard, the
degree of polarization of the CMB is also approximately 10 %. Thus, the primordial gas-dust
clouds generating CMB during their evolution could take the form of globules.
In addition to CMB, the approach under consideration can be applied to infrared
background radiation (CIB) and optical background radiation (COB). The estimates made in
Section 7 give reason to believe that the CIB emission could have originated in protoplanetary
clouds, while the contribution to the COB emission was made by primordial stars.
Although the blackbody CMB radiation coming to the Earth from all directions has an
intensity
4
I
T
=
, this does not mean that the ubiquitous occurrence of CMB energy density
35
in cosmic space is equal to
4
14
4
4.17 10
T
u
c
−
=
=
J/m3, similar to blackbody radiation. As
shown in (33), the average CMB energy density in the Universe should equal
23
1.4 10
u
−
=
J/m3. This happens because the CMB is radiation that is not in equilibrium with the global
blackbody. Indeed, all CMB sources cannot form a closed surface entirely surrounding the
radiation of the Universe, which is a necessary condition for the validity of the formula
4
4 T
u
c
=
. The same applies to CIB radiation and COB radiation, which are also
nonequilibrium.
Due to the difference between the energy density u and the energy density
s
u (20) of
blackbody radiation, it becomes possible to move away from the model of the hot expanding
Universe. It follows from this model that the CMB energy density is equal to u everywhere, so
that the ratio of the number of photons to the number of nucleons is equal to
9
3.3 10
f
f
p
b
b
n
n m
n
=
=
, which is close in magnitude to the ratio of volumes
s
V
V in (34). In our
approach, the energy density u corresponds to the condition of practically the same number
of photons and nucleons, which does not lead to the problem of excess photons over nucleons
or to the need to introduce the hot Universe model.
Statements and declarations
The authors have no relevant financial or non-financial interests to disclose.
Data availability
The data underlying this article are available in the article and in its online supplementary
material.
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vixra | 2602.0111 | Astrophysics | We investigate whether longu2011term atmospheric pressure measurements contain a coherent, frequencyu2011stable signal consistent with the expected gravitationalu2011wave (GW) emission from the eclipsing binary Zeta Phoenicis. The system's orbital period of 1.6697739 days implies a GW frequency of 13.863 $mu$Hz, a regime inaccessible to conventional interferometric detectors. Using 20 years of hourly pressure data from more than 100 stations, we apply a communicationsu2011engineering approach combining coherent integration, superheterodyne frequency shifting, and iterative phaseu2011demodulation to isolate weak, structured oscillations. After compensating for frequency drift and multiple phase modulations, we recover a narrow, persistent spectral feature at the predicted frequency. One modulation matches the annual Doppler signature expected from Earth’s orbital motion, while additional lowu2011frequency sidebands may reflect longeru2011period dynamical influences within the source system. We further derive a longu2011term decrease in the GW frequency, corresponding to a secular increase in the orbital period, and provide a prediction for the period in 2026. These results demonstrate that phaseu2011sensitive demodulation techniques can extract ultrau2011lowu2011frequency, coherent signals from noisy geophysical data and may offer a complementary pathway for probing continuous GW sources in the microhertz regime. | Herbert Weidner | Astrophysics | https://vixra.org/abs/2602.0111 | RASTI 000, 1–8 (2026)
Preprint February 21, 2026
Compiled using rasti LATEX style file v3.0
Identification of a Continuous Microhertz Signal
Consistent with Gravitational Radiation from Zeta Phoenicis
Herbert Weidner ,1★
1University of applied sciences, Aschaffenburg, Germany
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We investigate whether long-term atmospheric pressure measurements contain a coherent, frequency-stable signal consistent
with the expected gravitational-wave (GW) emission from the eclipsing binary Zeta Phoenicis. The system’s orbital period of
1.6697739 days implies a GW frequency of 13.863 𝜇Hz, a regime inaccessible to conventional interferometric detectors. Using 20
years of hourly pressure data from more than 100 stations, we apply a communications-engineering approach combining coherent
integration, superheterodyne frequency shifting, and iterative phase-demodulation to isolate weak, structured oscillations. After
compensating for frequency drift and multiple phase modulations, we recover a narrow, persistent spectral feature at the
predicted frequency. One modulation matches the annual Doppler signature expected from Earth’s orbital motion, while additional
low-frequency sidebands may reflect longer-period dynamical influences within the source system. We further derive a long-term
decrease in the GW frequency, corresponding to a secular increase in the orbital period, and provide a prediction for the period in
2026. These results demonstrate that phase-sensitive demodulation techniques can extract ultra-low-frequency, coherent signals
from noisy geophysical data and may offer a complementary pathway for probing continuous GW sources in the microhertz
regime.
Key words: Data Methods – Zeta Phoenicis – Low frequency – Gravitational wave
1 INTRODUCTION
Gravitational waves (GWs) are a fundamental prediction of general
relativity, and binary stars are expected to emit continuous radiation at
twice their orbital frequency. For systems with periods of order days,
the corresponding GW frequencies fall in the microhertz regime – far
below the sensitivity range of ground-based interferometers, which
are limited to frequencies above several tens of Hertz. Consequently,
the ultra-low-frequency GW domain remains largely unexplored,
despite the abundance of nearby eclipsing binaries that should, in
principle, contribute measurable signals.
Zeta Phoenicis is an Algol-type eclipsing binary with an orbital
period of 1.6697739 days. Its expected GW frequency of 13.863 𝜇Hz
lies in a spectral region where conventional detection techniques are
ineffective, and where long-term geophysical measurements exhibit
several unexplained narrow features. Atmospheric pressure records,
in particular, contain persistent spectral lines whose origin is not
yet understood and which motivate a closer examination of this fre-
quency range.
In this work, we investigate whether long-baseline atmospheric
pressure measurements contain a coherent, frequency-stable compo-
nent consistent with the expected GW signal from Zeta Phoenicis.
To this end, we adopt a communications-engineering approach: we
construct a receiver tuned to the predicted emission frequency, treat
terrestrial sensors as effective antennas, and apply phase-sensitive
demodulation and coherent integration techniques to extract weak,
★herbertweidner@gmx.de
structured signals from noise. This methodology allows us to
probe frequency drifts, annual Doppler modulations, and additional
low-frequency sidebands that may encode dynamical information
about the source system.
Our aim is to assess whether the observed spectral feature at 13.863
𝜇Hz can be attributed to a continuous astrophysical signal and to
characterize its temporal behaviour, including potential modulations
arising from orbital motion within the Zeta Phoenicis system.
2 MYSTERIOUS SIGNAL SOURCES
When a celestial body or a GW deforms the Earth, pressure changes
and accelerations occur on the surface. Sensitive sensors convert
these into electrical signals, which we examine. The longer the mea-
surement period 𝑇𝑚𝑖𝑛, the lower the frequency uncertainty Δ 𝑓. Ac-
cording to Küpfmüller (1993), the following applies:
𝑇𝑚𝑖𝑛· Δ 𝑓≥0.5
(1)
With a minimum period of 𝑇𝑚𝑖𝑛> 20 years, we reach the value
Δ 𝑓≈0.8 nHz. The atmospheric pressure spectrum (Figure 1) shows
several lines of unclear origin. One of these lies at the frequency at
which we expect the GW of the binary star system Zeta Phoenicis.
Neighboring celestial bodies such as the Moon also produce spectral
lines, which are listed in Hartmann (1995). The closest interfering
frequencies are found there at 13.81 𝜇Hz and 13.90 𝜇Hz – far outside
the range shown.
© 2026 The Authors
2
Herbert Weidner
Figure 1. Power spectrum of the atmospheric pressure, one measurement per
hour. The data is based on the average pressure in Germany between 2000
and 2020. The red bar marks 𝑓𝐺𝑊of Zeta Phoenicis.
It is unclear what sources generate the spectral lines in this fre-
quency range. Below 20 𝜇Hz, the amplitude in the spectrum in-
creases approximately proportionally to 𝑓−3 (Weidner 2025). If
this were noise, one would expect a fall-off proportional to 𝑓−1
(Colors of noise 2026). Apparently, there are numerous, as yet un-
known sources causing this inexplicable increase. This phenomenon
is reason to investigate a narrow spectral range and identify possible
sources. Short-term phenomena such as ocean-atmosphere coupling,
Chandler wobble, and annual modes cannot generate waves that are
detectable with extreme frequency stability for over 20 years.
Barometers are not the only devices that react to these special
waves: We found almost identical spectra in the records of super-
conducting gravimeters. However, two factors impair the quality: no
gravimeter has been operated continuously for decades (resulting in
broadened spectral lines), and frequent earthquakes increase the noise
level. Although the two measurement methods differ fundamentally,
they produce strikingly similar spectra. The cause is apparently a
common one – possibly GW. We will not discuss this question in this
paper. Our sole focus is on reception technology and signal analysis.
3 THE CHOICE OF BANDWIDTH
We suspect that radiation from binary stars is the cause of the peculiar
amplitude increase in the spectrum. Binary stars are very common
in our galaxy, and many of them may have planetary systems. If this
assumption applies to Zeta Phoenicis, the planets force the binary
system to move around the center of gravity of the entire system. Then
the generated GW is phase-modulated and accompanying frequen-
cies arise on both sides of 𝑓𝐺𝑊(sidebands). The orbital radii of the
planets determine the modulation frequencies, the masses determine
the modulation indices. Since the orbital period of the binary system
can change over time (drift), the bandwidth (BW) of the receiving
channel should significantly exceed the minimum value Δ 𝑓≈0.8
nHz. It is difficult to determine BW without prior knowledge of the
overall system. Too high a value increases the noise level and im-
pairs reception quality. A value that is too small will delete or distort
possible modulations (see sector 12.3).
In Figure 1 are two weak, symmetrical spectral lines at a distance
of 2 nHz from 𝑓𝐺𝑊. Whether these are sidebands generated by
modulation of 𝑓𝐺𝑊cannot be determined because the spectrum
erases all phase information. The answer requires subtle methods of
communications engineering, which we will discuss below.
Albert Einstein believed that GWs were practically immeasur-
able due to their low amplitude. In fact, no prominent spectral lines
that could be generated by GWs can be found in any spectrum or
frequency range. This changes when long periods of time are con-
sidered and BW is reduced to improve the signal-to-noise ratio. At
a bandwidth of 10 nHz, we see an obviously modulated continuous
Figure 2. Total amplitude of the frequency mixture in the range (13.858 <
𝑓< 13.868) 𝜇Hz as a function of time. The signal-to-noise ratio is sur-
prisingly good considering the questionable ‘antenna’. Cutting off all higher
sideband frequencies with 𝑓𝑚𝑜𝑑𝑢𝑙𝑎𝑡𝑖𝑜𝑛> 5 nHz can distort the envelope,
but improves the signal-to-noise ratio (SNR).
signal instead of the expected noise (Figure 2). Is it the GW from
Zeta Phoenicis? We emphasize that no source in the solar system
generates this signal because Hartmann (1995) contains no corre-
sponding entry in this frequency range. This frequency gap is the
only reason to analyze the GW of Zeta Phoenicis and not the GW of
the closer binary star Algol: The moon causes tides on Earth. One
of these has almost the same frequency as the GW from Algol, and
the two are difficult to separate. The limitation to BW = 10 nHz has
consequences:
Figure 2 gives the impression of an amplitude-modulated oscil-
lation. This is probably an artifact: The arbitrarily chosen BW sup-
presses sideband frequencies outside the BW and therefore distorts
the information transported there. The actual envelope and modula-
tion requires a detailed data analysis, including all relevant sidebands
and adjustment of the BW.
During the measurement period, the Earth orbits the Sun 20 times.
If the signal source is not located near one of the two poles of the
ecliptic, our distance from the source changes periodically. The Earth
moves toward or away from Zeta Phoenicis at a maximum speed of
17.3 km/s. Due to the Doppler effect, the reception frequency 𝑓𝐺𝑊
should fluctuate periodically by a maximum of ±0.8 nHz (Figure 7).
This value is referred to as the frequency deviation of the frequency
or phase modulation. A property of this type of modulation that is
not easy to understand but has been experimentally verified is that it
is not the frequency deviation that appears in the spectrum, but two
sideband frequencies at a distance of ±31.69 nHz from 𝑓𝐺𝑊. The
distance is the modulation frequency (reciprocal of the length of the
year), and the amplitude is a measure of the frequency deviation.
If we suppress these signal components beyond the selected band-
width in order to minimize noise, we delete the information necessary
to detect the periodic annual modulation.
We only examine the immediate vicinity of the unchanged spectral
line at 𝑓𝐺𝑊. The selected bandwidth allows us to detect frequency
drift and slow modulations of the signal at frequency 𝑓𝐺𝑊when
𝑓𝑚𝑜𝑑< 5 nHz. This allows us to detect planets with orbital periods
longer than 6.3 years.
4 COHERENT SIGNAL INTEGRATION
Communications engineering knows several methods for detecting
a signal of defined frequency in a frequency mixture. One is the
coherent addition of successive oscillations, taking into account the
phase relationship. The method can be summarized as follows (sec-
tion 12.1): An oscillator oscillates at the desired frequency with a
RASTI 000, 1–8 (2026)
GW from Zeta Phoenicis
3
Figure 3. Coherent sum of the signal from figure 2 as a function of time.
The resonator responds preferentially to its resonant frequency 13.863 𝜇Hz.
Figure 4. A frequency-modulated signal causes the oscillation period of the
intermediate frequency to oscillate around the target value of 648,000 seconds
(injection pulling). The results of the first two years are omitted because the
small amplitude of the coherent sum does not allow for a precise determination
of the oscillation period.
certain amplitude. If it is disturbed by an injected signal, it reacts in
a characteristic manner:
If the injected signal mixture contains the desired frequency, the
amplitude of the oscillator increases or decreases depending on the
phase difference. This disappears after a sufficiently long time (in-
jection locking).
With an unmodulated signal of the same phase, the coherent sum
increases proportionally to time.
Signals of similar frequency cause injection pulling. The frequency
of the oscillator follows the signal frequency within certain limits.
This can lead to a ripple in the smooth envelope (Figure 3) or to a
periodic frequency change (Figure 4).
Signals with poor SNR usually exhibit high phase noise. Injection
locking provides a means to significantly reduce phase noise.
Interference pulses are always broadband and can be effectively
suppressed by an upstream band filter.
Figure 3 shows the result for a very narrow frequency range around
13.863 𝜇Hz. The almost linear increase in the envelope shows that the
frequency and amplitude of the carrier frequency hardly change dur-
ing the entire 20-year measurement period. Weak signals (sidebands)
of high frequency stability within the selected bandwidth modulate
the envelope.
5 SUPERHET
We suspect the double star Zeta Phoenicis to be the source of the
signal described above. The frequency 𝑓𝐺𝑊could change slowly
(drift), and existing planets cause phase modulation (PM). The aim
of our research is to determine the exact value of 𝑓𝐺𝑊, the frequency
drift, and possible PM. We must eliminate these peculiarities of the
signal by means of an intermediate step, because the coherent signal
integrator expects an unmodulated signal of constant frequency. With
a modified superheterodyne, we can shift the reception channel to a
lower frequency 𝑓𝐼𝐹and simultaneously eliminate the modulations.
We achieve this by modulating a local oscillator of the intermediate
frequency 𝑓𝑙𝑜𝑐in the same way as the signal coming from the an-
tenna. The better the copy, the more constant the difference frequency
𝑓𝐼𝐹is. We check this by determining the oscillation period of 𝑓𝐼𝐹
by counting. If the oscillation period oscillates around a mean value
as in Figure 4, the signal is frequency-modulated. We can estimate
the modulation frequency, modulate the local oscillator with it, and
start a new iteration. Due to a lack of information, we have to guess
the frequencies at which the GW signal could be (phase) modulated.
We start with the usual approach
𝑦= 𝑠𝑖𝑛(2𝜋𝑡· 𝑓𝑠𝑖𝑔𝑛𝑎𝑙+ 𝜙𝑚𝑜𝑑𝑢𝑙𝑎𝑡𝑖𝑜𝑛)
(2)
and adjust the two parameters 𝑓𝑠𝑖𝑔𝑛𝑎𝑙and 𝜙𝑚𝑜𝑑𝑢𝑙𝑎𝑡𝑖𝑜𝑛to the prob-
lem: The frequency 𝑓𝑠𝑖𝑔𝑛𝑎𝑙can change proportionally to time and
𝜙𝑚𝑜𝑑𝑢𝑙𝑎𝑡𝑖𝑜𝑛can be the sum of several sine functions. If the modu-
lation consists of a single frequency 𝑓𝑚𝑜𝑑, the equation is
𝑦= 𝑠𝑖𝑛(2𝜋𝑡( 𝑓𝑠𝑖𝑔𝑛𝑎𝑙+ 𝑡¤𝑓) + 𝜂· 𝑠𝑖𝑛(2𝜋𝑡𝑓𝑚𝑜𝑑+ 𝜑))
(3)
A sinusoidal PM causes the instantaneous frequency of the signal
to fluctuate periodically between the limits 𝑓𝑠𝑖𝑔𝑛𝑎𝑙+ Δ 𝑓(maximum
blue shift) and 𝑓𝑠𝑖𝑔𝑛𝑎𝑙−Δ 𝑓(maximum red shift). The quantity Δ 𝑓is
referred to as the frequency deviation. The instantaneous frequency
at a given point in time is difficult and inaccurate to measure, as it
is never constant over a longer period of time. It is easier and more
common to determine the modulation index 𝜂using equation (3) and
calculate Δ 𝑓= 𝜂· 𝑓𝑚𝑜𝑑.
Since we are changing the signal frequency using the Superhet
method, we need to adjust the approach (for a single planet):
𝑓𝑙𝑜𝑐= 𝑓𝐺𝑊+ 𝑓𝐼𝐹+ 𝑡¤𝑓𝐺𝑊+ 𝜂𝑚𝑜𝑑· 𝑠𝑖𝑛(2𝜋𝑡𝑓𝑚𝑜𝑑+ 𝜑𝑚𝑜𝑑)
(4)
If multiple planets are suspected, equation (4) is supplemented
with additional summands with adjusted constants 𝜂𝑛, 𝑓𝑛and 𝜑𝑛.
We calculate the difference 𝑓𝐺𝑊−𝑓𝑙𝑜𝑐𝑎𝑙= 𝑓𝐼𝐹using the usual su-
perheterodyne method and verify the frequency-shifted signal with
coherent signal integration. If 𝑓𝐼𝐹changes, we minimize the de-
viations by iterating the characteristic values of 𝑓𝑙𝑜𝑐. After several
steps, the local oscillator 𝑓𝑙𝑜𝑐is modulated in the same way as the
GW signal.
Using the superheterodyne principle, we reduce the signal fre-
quency from 𝑓𝐺𝑊= 13.863 𝜇Hz to a ’smooth’ value, for example
𝑓𝐼𝐹= 1/(360 hours) ≈772 nHz. Then the value of each individual
oscillation can be determined with particular accuracy. With the se-
lected value of 𝑓𝐼𝐹, the time span between adjacent zero crossings
should be exactly 648000 seconds (half the oscillation period of the
intermediate frequency). The intermediate result in Figure 4 shows
that there are apparently further frequency modulations. The reduced
frequency allows for a longer sampling period and faster calculation.
6 RESULTS – 1 (BW = 10 NHZ)
The aim of this work is to identify the GW ofZeta Phoenicis. Since the
frequency is likely to change slowly (drift) and the literature search
(Simbad 1991) provides no clues, we initially select a large bandwidth
of BW = 10 nHz. The measurements show that the choice of BW has
an unexpectedly strong influence on the results (Table 1, column 3).
Periodic frequency changes of 𝑓𝑠𝑖𝑔𝑛𝑎𝑙can be detected because the
corresponding sidebands lie within BW. Therefore, we also measure
RASTI 000, 1–8 (2026)
4
Herbert Weidner
Table 1. The modulations of 𝑓𝑠𝑖𝑔𝑛𝑎𝑙= 13.863 𝜇Hz. 𝑓1 und 𝑓2 are the
modulation frequencies of the signal. 𝜂is the individual modulation index of
the phase modulations. We omit the phases 𝜑for reasons of space.
BW
𝑓𝐺𝑊
drift
𝑓1
𝜂1
𝑓2
𝜂2
(nHz)
(𝜇Hz)
×10−20 𝑠−2
(nHz)
(nHz)
8
13.8636
17.6
2.14
0.614
3.84
0.216
8.2
13.8636
17.74
2.18
0.607
3.84
0.256
8.4
13.8635
23.9
2.16
0.607
3.90
0.269
8.5
13.8626
180.8
1.797
0.923
3.94
0.285
8.6
13.8626
178.6
1.81
0.910
3.94
0.305
8.7
13.8626
175.8
1.811
0.897
3.93
0.320
8.8
13.8626
172.7
1.83
0.884
3.93
0.327
8.9
13.8626
169.4
1.835
0.870
3.931
0.338
9
13.8626
165.7
1.86
0.857
3.94
0.346
9.1
13.8627
162.5
1.91
0.850
3.970
0.342
9.2
13.8627
159.0
1.90
0.834
3.96
0.363
9.3
13.8627
156.6
1.93
0.825
3.96
0.371
9.4
13.8635
339.7
1.67
1.27
4.02
0.39
9.5
13.8616
336.4
1.685
1.25
4.06
0.403
10
13.8616
329.5
1.701
1.20
4.15
0.488
these modulations by adjusting equation (4). The lowest modulation
frequency is approximately 1.9 nHz. Since the oscillation period 𝑇≈
17 years is about as long as the entire measurement period (20 years),
the measurement error calculated with equation (1) is enormous. An
improvement would require extending the measurement period to
many decades.
We define the BW using a windowed sinc filter with an rectangular
passband. Small changes in BW cause sideband frequencies at the
edges of BW to be switched on and off almost digitally. This changes
the results abruptly in some cases (Table 1). Although smoother
transitions can be achieved with other filtering methods, the basic
problem remains: The bandwidth of the analysis has an unduly strong
influence on the results.
Signal processing with a large bandwidth has serious disadvan-
tages: The SNR is poor and most of the sidebands that carry valuable
information are discarded. This includes, in particular, the direction
of the source of the received signal. This is determined by measur-
ing the phase shift of the Doppler shift as a result of the Earth’s
orbit. This measurement requires a minimum bandwidth of 64 nHz,
which drastically worsens the SNR. Depending on the (initially) un-
known modulation index, a significantly larger (Carson) BW may be
necessary, which reduces the SNR even further.
Conclusion: We need a method that allows the signal modulations
to be deciphered even with the smallest possible BW.
7 IMPROVED METHOD
We see only one way to minimize the influence of bandwidth: In a
first step, we eliminate all modulations – including frequency drift.
Then the signal frequency is constant and there are no sidebands
that we need to consider. Another advantage is that we can filter the
signal with the lowest possible BW, thereby improving the SNR. We
decided on BW = 0.4 nHz (see equation (1)).
The original phase-modulated signal consists of a frequency bun-
dle that requires a certain BW (see section 12.3). Each individual
frequency within this bandwidth carries energy, but only part of the
total energy of the signal is accounted for by the carrier frequency
𝑓𝐺𝑊. If we succeed in eliminating all modulations, the total energy
of the signal is concentrated in a very narrow range around 𝑓𝐺𝑊and
the SNR improves.
Every signal coming from a source outside the solar system is
modulated with 𝑓𝑦𝑒𝑎𝑟= 31.69 nHz because the Earth orbits the
Sun. Detecting this modulation requires a minimum signal process-
ing bandwidth of 64 nHz. The spectrum contains evidence of fixed
modulation frequencies (section 12.2), the origin of which is unclear.
In the course of the investigations, it became apparent that we can
only decipher all modulations without distortion with 𝐵𝑊≥2700
nHz (Carson-BW). Such a wide frequency band contains a lot of
noise and signals from other sources that degrade the SNR.
The method outlined here picks out only the frequencies that match
the selected modulations and ignores all other signals within the BW.
No other method has a comparably high selectivity. This method is
described in Weidner (2025) and is only briefly explained here:
The preprocessor reduces the signal frequency from 13.8 𝜇Hz to
1.4 𝜇Hz without changing the modulations. The maximum possible
bandwidth of the signal processing is 2.7 𝜇Hz, which is sufficient to
demodulate the strongly phase-modulated signal with low distortion.
Once the initial results (section 8) are known, these values can be
adjusted. Extending the sampling period from one to 𝑇𝑠= 42 hours
reduces the amount of data and the processing time.
The main loop is repeated many thousands of times:
We phase modulate the frequency of a local auxiliary oscillator
with fixed frequencies 𝑓𝑚1 ... 𝑓𝑚5. We have to guess the individual
start values. In section 12.2, we describe a method for systematically
finding modulation frequencies.
The frequency of the auxiliary oscillator and its drift are adjustable.
The original assumption of a linear drift is not sufficient. The con-
vergence improved after expanding to a quadratic drift.
We mix the signal frequency with the local auxiliary oscillator and
obtain the difference frequency 𝑓𝐼𝐹. We choose ( 𝑓𝐼𝐹= 1/𝑇𝑠/40 ≈
165 nHz) in order to detect even the slightest frequency changes and
reduce the bandwidth of the signal processing to 0.4 nHz.
We vary the drift and modulation frequencies with two objectives:
The amplitude of 𝑓𝐼𝐹must be as high as possible and the value
𝑓𝐼𝐹must be constant. If the errors fall below specified limits, the
modulations of the auxiliary oscillator and the signal are largely
identical.
The key features of this method are: We obtain a high SNR and
we can measure the PM at 31.69 nHz. This allows us to determine
the direction from which the signal is coming. The poor SNR of
𝑓𝑠𝑖𝑔𝑛𝑎𝑙(Figure 1) does not allow us to determine whether the signal
is amplitude-modulated.
8 RESULTS – 2 (BW = 0.4 NHZ)
Figure 5 shows that the amplitude of the intermediate frequency
𝑓𝐼𝐹assumes considerable values because during the demodulation
process, the energy of the sidebands is transferred to the carrier
frequency (see section 12.3). Table 2 shows the modulations. The
accuracy depends heavily on the value of the modulation frequency:
For modulation-1, the measurement period of 20 years comprises
approximately 74 oscillations, while for modulation-5, it is only 1.6
oscillations. This means that an estimated error of ±0.1 oscillations
has a much greater impact.
We place great importance on an accurate measurement of the
frequency drift because no results have been published so far. The
method described in section 6 is too inaccurate, as the chosen band-
width and low-frequency modulations influence the result. The op-
RASTI 000, 1–8 (2026)
GW from Zeta Phoenicis
5
Figure 5. Power spectrum (Welch method) of the vicinity of 𝑓𝐼𝐹≈165 nHz.
The absence of symmetrical structures next to the carrier frequency means
that there are no undiscovered modulations. The increase in the amplitude
of the carrier frequency results from the deletion of the measured sidebands
(Table 2). A comparison of the vicinity of 𝑓𝐼𝐹with the vicinity of the
signal frequency in Figure 1 is meaningless because the demodulation method
(variable frequency of the local auxiliary oscillator) distorts the spectrum in
the vicinity of the carrier frequency.
Figure 6. Orbital period of Zeta Phoenicis as a function of time according to
equation (5) (blue stars). The two black circles at the bottom left mark earlier
results (Simbad 1991). The red bullet at the top right marks our prediction
for the year 2026.
posite is true for the improved method described in section 7: The
intermediate frequency 𝑓𝐼𝐹must be unmodulated and constant. The
slightest changes are noticeable and the frequency drift can be deter-
mined with particular accuracy. The results:
𝑓(𝑡) = 𝑓𝑠𝑖𝑔𝑛𝑎𝑙+ 𝑡¤𝑓+ 𝑡2 ¥𝑓
(5)
𝑓𝑠𝑖𝑔𝑛𝑎𝑙= 13.862353 ± 0.00003 𝜇Hz (Date 2020-01-01).
¤𝑓= −2.975 × 10−18𝑠−2
¥𝑓= −3.5 × 10−28𝑠−3
The good SNR of 𝑓𝐼𝐹(Figure 5) allows for high measurement
accuracy and a prediction: The rotational frequency of the binary
system decreases over time and is expected to be 13.8597 𝜇Hz in
2026. This corresponds to an orbital period of 1.67018 days.
Simbad (1991) and VizieR cite two specific results measured
around 1991 as the orbital period of the binary system: 𝑃1 =
1.6697753 days and 𝑃2 = 1.6697678 days, without specifying any
measurement errors. Figure 6 shows the measured values and the
expected orbital period in 2026.
Table 2. The phase modulations of 𝑓𝑠𝑖𝑔𝑛𝑎𝑙= 13.862353 𝜇Hz (Date: 2020-
01-01). The error of the modulation frequencies Δ 𝑓is estimated.
𝑓𝑚𝑜𝑑
Δ 𝑓
𝑃
𝜑
𝜂
(nHz)
(nHz)
(days)
Modulation-1
118.21
±0.02
97.9
0.41
1.40
Modulation-2
31.27
±0.03
370
3.77
5.24
Modulation-3
28.77
±0.03
402
1.25
2.08
Modulation-4
14.37
±0.05
805
1.25
1.51
Modulation-5
1.96
±0.08
5900
6.10
2.85
Figure 7. Comparison of the measured periodic frequency shift of 𝑓𝑠𝑖𝑔𝑛𝑎𝑙
(blue curve) with the date-dependent orbital speed of the Earth (red stars).
The scale on the right has been adjusted to illustrate the good agreement. The
modulation frequencies 𝑓1, 𝑓3, 𝑓4 and 𝑓5 are hidden.
9 INTERPRETATION OF THE RESULTS
We limited our measurements to the phase modulation of the signal
for several reasons: a) Unlike amplitude modulation, PM exhibits the
well-known phenomenon of demodulation gain with a noticeable im-
provement in SNR. Our measurements confirm this effect (Figure 5).
b) The radiation source Zeta Phoenicis is located outside the solar
system. Therefore, the signal must be phase-modulated with a period
of 365 days. In fact, the period of phase modulation-2 (Table 2) cor-
responds well with the orbital period of the Earth. For the other four
phase modulations, there is insufficient evidence to speculate on the
causes. We suspect that this PM is caused by planets around Zeta
Phoenicis. Below, we will only discuss the results for Modulation-2.
9.1 Direction to the source
Zeta Phoenicis is located at RA = 01 h 08 m 23 s and Dec = −55𝑜
14’ 45". On the 153rd day of the year, the distance to Zeta Phoenicis
decreases at a maximum speed of about 17 km/s; six months later,
it increases by the same amount (Calculate radial velocities 2026).
The Doppler effect causes a periodic phase change in the reception
frequency, which acts like a frequency change (blueshift/redshift).
Comparing the modulation phase-2 (𝜑2) with the phase of the relative
velocity of the Earth in Figure 7 confirms: The signal with a frequency
of 13.86256 𝜇Hz comes from the same RA as the electromagnetic
radiation from Zeta Phoenicis. To check the declination, we need the
propagation speed of the wave.
9.2 Signal speed
The Doppler effect is a fundamental principle of astronomy and
describes the relationship between the frequency shift Δ 𝑓of the
signal and the speed of the transmitter or receiver. If we assume
RASTI 000, 1–8 (2026)
6
Herbert Weidner
that the signal propagates at the speed of light 𝑐, we have to use the
relativistic equation
Δ 𝑓= 𝑓𝑠𝑖𝑔𝑛𝑎𝑙·
√︂
𝑐+ 𝑣
𝑐−𝑣−1
!
(6)
where 𝑣is the relative speed between the source and the Earth. For
the source Zeta Phoenicis, the Earth’s relative speed oscillates in
the range -17 km 𝑠−1 < 𝑣< 17 km 𝑠−1. This limits the Doppler
shift to the range |Δ 𝑓| < 0.8 nHz. We calculate frequency deviations
that were 200 times higher (Figure 7). This discrepancy leaves no
room for ambiguity: The propagation speed of the waves (at this
frequency) is about 200 times lower than the speed of light. Similar
results are found at lower frequencies around 5 𝜇Hz (Weidner 2025).
The attempt to provide a theoretical foundation for 𝑣𝑠𝑖𝑔𝑛𝑎𝑙would go
beyond the context of this paper.
10 DISCUSSION
Some of our findings contradict the properties of gravitational waves
predicted by A. Einstein 110 years ago. This applies in particular to
the unexpectedly low propagation speed and the strong reaction of
atmospheric pressure to these waves. However, we also emphasize
that continuous GW have never been detected before. The speed of
a GW has only been measured once (GW170817) and applies to
frequencies above 100 Hz. We measured at a frequency millions of
times lower and obtained a strongly deviating value. Future research
will have to clarify whether dispersion causes this difference.
We can only partially characterize the waves received:
The ’receiving antenna’ is the pressure at the interface between
the atmosphere and the Earth. Apparently, the wave moves the mas-
sive globe differently than the lighter air envelope, causing periodic
pressure fluctuations.
Since Zeta Phoenicis cannot be seen from Europe, we conclude
that the mass of the Earth has little or no influence on wave propa-
gation.
The period of the radiation from Zeta Phoenicis is about 20 hours,
which corresponds to 5/6 of the length of a day. We have neither
observed nor expected any interference from periods of 24 ± 20
hours. Corresponding measurements would require short sampling
periods and significantly increase the calculation time.
Air pressure is a scalar quantity and the rotation period of the
Earth is smaller than the sampling period 𝑇𝑠= 42 hours. Therefore,
we were unable to experimentally determine whether the measured
fluctuations in air pressure are caused by transverse or longitudinal
waves.
Figure 6 shows that the orbital period of Zeta Phoenicis increases
over the long term. The cause is likely to be tidal friction within the
two closely neighboring suns: both deform each other and thereby
lose rotational energy, which is transferred to the binary system. As
a result, the two suns move away from each other over time and the
orbital frequency decreases ( ¤𝑓< 0). Obviously, this energy transfer
is greater than the energy loss due to the emitted wave that we have
received.
It can be ruled out that the signal is caused by errors in the anal-
ysis program: the signal does not undergo Fourier transformation to
examine its frequency bands. The reasons: Inappropriately selected
parameters of the Fourier method can lead to misinterpretations.
Fourier analysis is well suited to detecting strong and unmodulated
signals. Detecting weak, modulated signals below the noise level
requires more sophisticated communication technologies that allow
the reconstruction and demodulation of existing phase modulations.
11 CONCLUSION
In this study we have explored whether atmospheric pressure records,
when analysed with high-selectivity communication-engineering
techniques, contain evidence of a coherent signal at the expected
GW frequency of Zeta Phoenicis. After compensating for frequency
drift and multiple phase modulations, we identify a persistent,
narrow-band oscillation at 13.863 𝜇Hz whose temporal behaviour
is consistent with an astrophysical origin. The detection of a mod-
ulation matching Earth’s orbital period provides a directional sig-
nature, while additional sidebands suggest further dynamical struc-
ture within the source system. The improved signal-to-noise ratio
achieved through phase-sensitive demodulation enables a precise
measurement of the long-term frequency drift, yielding a prediction
for the binary’s orbital period in 2026.
Several aspects of the recovered signal differ from expectations
based on standard GW theory, particularly the inferred propagation
speed and the strong response of atmospheric pressure. These dis-
crepancies highlight the need for further investigation, both obser-
vational and theoretical, into the behaviour of ultra-low-frequency
waves and their coupling to terrestrial media. Nonetheless, the
methodology presented here – combining coherent integration, su-
perheterodyne demodulation, and long-baseline geophysical data –
demonstrates a promising framework for detecting weak, structured
signals in the microhertz regime and may serve as a foundation for
future studies of continuous GW sources beyond the reach of con-
ventional detectors.
12 METHODS
Data Acquisition: Atmospheric pressure data were obtained from the
German Weather Service (DWD 2021), comprising hourly measure-
ments from over 100 barometric stations across Germany between
2000 and 2020. Only stations with at least ten years of continuous
operation were included. To enhance the signal-to-noise ratio (SNR),
pressure records were coherently summed across all stations.
Sensor Calibration: All pressure sensors used by the DWD are
temperature-compensated and conform to international standards,
with a measurement range of 500-1100 hPa and an accuracy of ±0.1
hPa. Instrumental drift and data gaps remain unchanged because
they may influence the amplitude of the signal, but do not generate
frequency-stable oscillations or phase-modulate them. Data synchro-
nization is accurate to the second and is guaranteed by the DWD.
Sampling: The air pressure is not measured continuously, but at
fixed time intervals of, for example, 𝑇𝑠= 1 hour. The inverse of 𝑇𝑠is
called the sampling frequency 𝑓𝑠. Although the signal does not pass
through an analog low-pass filter with a cutoff frequency of 0.5 𝑓𝑠,
aliasing effects are virtually eliminated when sampling atmospheric
pressure hourly. This is especially true for 𝑓< 15 𝜇Hz, because the
amplitudes in the lowest range are at least a factor of 105 higher
than the spectral lines and the noise in the high frequency range
( 𝑓> 100 𝜇Hz).
Signal Processing: Since we are only looking for phase-modulated
(PM) signals (Haykin 2001), we apply a well-known demodulation
method similar to that used in Betz (2013). The method iteratively
identifies and compensates for constant modulation frequencies and
RASTI 000, 1–8 (2026)
GW from Zeta Phoenicis
7
Figure 8. Integrated amplitude of a phase-modulated signal. The ripple in the
envelope is caused by alternating constructive and destructive interference
of the carrier frequency and sidebands. Without modulation, the amplitude
increase would be linear.
concentrates the signal energy in a narrow spectral line. Unlike con-
ventional FFT-based spectral methods, this approach preserves phase
information and enables the detection of weak, phase-modulated sig-
nals below the noise threshold.
Frequency Resolution: Figure 1 shows that the frequency reso-
lution should be better than 1 nHz. Therefore, we chose 20 years as
the minimum observation time 𝑇min, which corresponds to the uncer-
tainty principle 𝑇min · Δ 𝑓≥0.5 (Küpfmüller 1993). This long-term
integration enables the detection of modulation periods ranging from
months to decades. The frequency drift of the signals is smaller than
the bandwidth of the method and can be measured and compensated
using a standard superheterodyne receiver.
12.1 Coherent Signal Integration
A phase sensitive integrator acts as an extremely narrow band filter
with selectable frequency and a "memory" for the phase of already
processed data. This makes it insensitive to interference. Digital data
is a sequence of samples 𝑧𝑛, 𝑧𝑛+1, 𝑧𝑛+2,... , which are measured at
fixed intervals (sampling time 𝑇𝑠). A sinusoidal signal of a given
frequency 𝑓can be easily detected in this data if, for example, sam-
pling is performed at exactly four times the frequency. This freedom
is rarely available; in most cases, 𝑇𝑠is fixed. Then the phase angle
increases by 𝛼= 2𝜋𝑓𝑇𝑠with each measurement.
One solution for detecting the sinusoidal signal is to construct an
oscillator with exactly this frequency 𝑓and feed in the digital data.
If the phase is correct, the energy supply increases the amplitude of
the oscillator. If the phase is opposite, the amplitude of the oscillator
decreases. If this oscillator is fed with noise, the amplitude fluctuates
irregularly around a mean value.
We program the oscillator with the following two CORDIC equa-
tions, which we calculate alternately:
𝑥𝑛+1 = 𝑐𝑜𝑠(𝛼)𝑥𝑛+ 𝑠𝑖𝑛(𝛼)𝑦𝑛+ 𝑧𝑛
(7)
𝑦𝑛+1 = 𝑐𝑜𝑠(𝛼)𝑦𝑛−𝑠𝑖𝑛(𝛼)𝑥𝑛
(8)
The choice of parameters determines the sequence of calculated
values 𝑥𝑛and 𝑦𝑛:
Without an injected signal (𝑧𝑛= 0) and with initial values 𝑥1 = 0
and 𝑦1 = 1, the two equations calculate a table of values for 𝑥=
𝑠𝑖𝑛(2𝜋𝑡𝑓) and 𝑦= 𝑐𝑜𝑠(2𝜋𝑡𝑓). The amplitudes are constant.
Figure 9. Superposition of the spectrum of the area surrounding the signal
at the central frequency 𝑓𝐼𝐹= 165 nHz with its mirror image (high and low
frequencies are interchanged with respect to 𝑓𝐼𝐹). Significant similarities are
marked. These might be the sidebands caused by a modulation.
Setting 𝑥1 = 𝑦1 = 0 and feeding a monochromatic signal 𝑧𝑛of
frequency 𝑓, the equations calculate an oscillation with frequency
𝑓, whose amplitude increases proportionally in time. The difference
in phase between the signal and the oscillator disappears after a few
oscillations.
If the programmed and injected frequency differ or if the phase
or amplitude of the injected signal 𝑧𝑛changes, the output signal
of the integrator varies and the linear increase of the envelope is
lost. If the signal is in phase opposition, the output signal decreases
proportionally to time.
If we feed a phase- or frequency-modulated signal 𝑧𝑛, the envelope
changes in rhythm with the modulation frequency (Figure 8).
If noise is fed in, the equations calculate an oscillation, whose fre-
quency fluctuates around 𝑓and whose amplitude varies irregularly.
Figure 8 shows a typical result when the integrator processes a
signal that is phase modulated (PM) with a very small modulation
index 𝜂≈10−4. This signal consists of the carrier frequency 𝑓0 and
the two adjacent sidebands 𝑓0 −𝑓𝑚𝑜𝑑and 𝑓0 + 𝑓𝑚𝑜𝑑. It is difficult to
measure this weak PM using other methods because the amplitudes
of the sidebands are much smaller than the amplitude of the carrier.
12.2 How to find modulation frequencies
Modulating a carrier frequency 𝑓0 with 𝑓𝑚𝑜𝑑generates (at least) two
sideband frequencies at 𝑓0 −𝑓𝑚𝑜𝑑and 𝑓0 + 𝑓𝑚𝑜𝑑. These are often
difficult to detect in the spectrum when signals are weak and noisy.
Identification is made easier by superimposing two spectra (image
and mirror image) and searching for symmetries (Figure 9). If the
Weaver method (Waever 1956) is used to shift the frequency of the
signal, a sign change is sufficient to generate the mirror spectrum.
Since random coincidences can also be found in the noise, each
“suspicious” modulation frequency must be confirmed using the
method described in Weidner (2025).
12.3 Phase modulation needs bandwidth
If a signal is phase-modulated with multiple frequencies, the spec-
trum can look like noise (Figure 10, left side). The larger the mod-
ulation indices 𝜂, the more bandwidth the signal requires. A good
demodulator is needed to identify the sideband frequencies in the
noise and to decipher such a broadband signal. If the signal is pro-
cessed with insufficient bandwidth, demodulation will fail.
RASTI 000, 1–8 (2026)
8
Herbert Weidner
Figure 10. Left: Spectrum of the signal after the preprocessor has reduced the
frequency from 13.86 𝜇Hz to 1.4 𝜇Hz. The signal is modulated as specified
in Table 2. No additional noise! Middle: Spectrum with 𝑓𝑚𝑜𝑑= 31.69
nHz alone. Right: Spectrum of the unmodulated signal. The total energy is
identical in each case, and the carrier frequency is 1.4 𝜇Hz in each case.
The middle image in Figure 10 shows how many lines the spectrum
is split into, even though the carrier frequency is phase-modulated
with a single frequency. The modulation index 𝜂determines the
amplitude of each line of the frequency bundle. The spectrum shows
too little information to demodulate the signal because the magnitude
formation inside FFT cancels out the phases of the lines.
The right-hand image in Figure 10 shows the amplitude increase
of the carrier frequency when all PM is eliminated. This difference
can also be seen when comparing Figures 1 and 5.
13 COMPETING INTERESTS
All investigations, including coding, were carried out by H. Weidner,
who is also the sole author. The author declares no conflict of interest.
14 FUNDING
The author declares no funding.
15 AVAILABILITY OF PROGRAMS AND DATA
The German Weather Service (DWD 2021) stores historical measure-
ment results from German weather stations (raw data). The summed,
error-corrected barometer files are also available from the author
upon request.
The homodyne detection code was written in MATLAB R2020a.
The code (ZP22.m) and the datasets (yDWD.mat) are available in
the github repository:
https://github.com/herbertweidner/cGW/tree/main
The programs will be explained by the author upon request.
References
Betz, W., 2021, The Navstar Global Positioning System, Wiley,
https://doi.org/10.1002/9781119458449.ch3
Calculate radial velocities of the GBT,
https://www.gb.nrao.edu/GBT/setups/radvelcalc.html
Colors of noise, https://en.wikipedia.org/wiki/Colors_of_noise
DWD,
https://opendata.dwd.de/climate_environment/CDC/observations_germany
Hartmann T., Wenzel H., 1995, The HW95 tidal potential catalogue,
https://publikationen.bibliothek.kit.edu/160395
Haykin, S., 2001, Communication Systems. Wiley. p. 107, ISBN 0-471-
17869-1.
Küpfmüller K., Kohn G., 1993, Theoretische Elektrotechnik und Elektronik:
Eine Einführung, ISBN 978-3540565000
Simbad zet Phe – Double or Multiple Star,
https://simbad.cds.unistra.fr/simbad/sim-
ref?bibcode=2004AcA....54..207K
Southworth J., 2020, Rediscussion of eclipsing binaries. Paper I. The totally-
eclipsing B-type system Zeta Phoenicis, https://arxiv.org/pdf/2012.05559
Weaver D., A third method of generation and detection of single-sideband
signals, Proceedings of the IRE, 1956, pp. 1703-5.
Weidner H., 2025, Coherent Low-Frequency Atmospheric Pressure Oscilla-
tions of Extraterrestrial Origin? in press (RASTI)
This paper has been typeset from a TEX/LATEX file prepared by the author.
RASTI 000, 1–8 (2026) |
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