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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)
[1] J. S. Lewis, Physics and chemistry of the Solar system (Elsevier Acad. Press, 2−nd Edition, 2004).
[2] E. Epelbaum, H.-W. Hammer, Ulf-G. Meiβner,
Modern theory of nuclear forces,
Rev. Mod. Phys. 81, 1773 (2009);
arXiv:0811.1338.
[3] D. R. Tilley, C. M. Cheves, J. L. Godwina, G. M. Hale, H. M. Hofmann, J. H. Kelley, C. G. Sheu, and H. R. Weller,
Energy levels of light nuclei A = 5, 6, 7, Nucl. Phys. A 708, 3–163 (2002).
[4] G. S. Bisnovatyi-Kogan, Relativistic astrophysics and physical cosmology (Moskva, Krasand, 2011), 376 pp. [in Russian].
[5] G. S. Bisnovatyi-Kogan, Physical problems of the theory of stellar evolution (Moscow, Nauka, 1989), 488 pp. [in Russian].
[6] R. Kippenhahn, A. Weigert, and A. Weiss, Stellar structure and Evolution (Springer Berlin Heidelberg, 2nd Ed., 2012),
604 pp.
[7] K. Wildermuth and Y. C. Tang, A Unifierd Theory of the Nucleus, (Vieweg, 1977).
[8] Y. C. Tang, M. LeMere, and D. R. Thompson,
Resonating-group method for nuclear many-body problems,
Physics
Reports 47, 167–223 (1978).
[9] Y. C. Tang, in Topics in Nuclear Physics II, Lecture Notes in Physics, Vol. 145 (Springer, Berlin, 1981), pp. 571–692.
[10] H. Horiuchi, Prog. Theor. Phys. Suppl. 62, 90 (1977).
15
[11] Y. K. Gambhir, P. Ring, and A. Thimet, Relativistic mean field theory for finite nuclei, Ann. Phys. 198, 132–179 (1990).
[12] P. Ring, Relativistic Mean Field Theory in Finite Nuclei, Prog. Part. Nucl. Phys. 37, 193–263 (1996).
[13] P. Ring, A. V. Afanasjev, Relativistic Description of High Spin States, Prog. Part. Nucl. Phys. 38, 137–146 (1997).
[14] P. Ring, Nuclear Structure Far from the Valley of Stability, Prog. Part. Nucl. Phys. 46, 165–174 (2001).
[15] M. Serra, A. Rummel, and P. Ring, Nuclear Structure Far from the Valley of Stability, Prog. Part. Nucl. Phys. 46, 175–176
(2001).
[16] P. Ring, E. Litvinova, D. Vretenar, Covariant density functional theory and nuclear dynamics far from stability, Prog.
Part. Nucl. Phys. 59, 393–400 (2007).
[17] D. Pena Arteaga, P. Ring, Relativistic quasiparticle random phase approximation in deformed nuclei, Prog. Part. Nucl.
Phys. 59, 314–316 (2007).
[18] T. Niksic, D. Vretenar, P. Ring, Relativistic nuclear energy density functionals: Mean-field and beyond, Prog. Part. Nucl.
Phys. 66, 519–548 (2011).
[19] D. Vretenar, A. V. Afanasjev, G. A. Lalazissis, and P. Ring, Relativistic Hartree-Bogoliubov theory: static and dynamic
aspects of exotic nuclear structure, Phys. Rep. 409, 101–259 (2005).
[20] J. Boguta and A. R. Bodmer,
Relativistic calculation of nuclear matter and the nuclear surface,
Nucl. Phys. A 292,
413–428 (1977).
[21] P. Ring and J. Speth, Nuclear structure calculations with a density-dependent force in 208Pb, Nucl. Phys. A 235, 315–351
(1974).
[22] G. A. Lalazissis, J. K¨onig, and P. Ring, New parametrization for the Lagrangian density of relativistic mean field theory,
Phys. Rev. C 55, 540 (1997).
[23] G. A. Lalazissis, S. Raman, and P. Ring, Ground-state properties of even-even nuclei in the relativistic mean-field theory,
Atom.Data Nucl.Data Tabl. 71, 1–40 (1999).
[24] B. R. Barrett, P. Navratil, and J. P. Vary, Ab initio no core shell model, Prog. Part. Nucl. Phys. 69, 131–181 (2013).
[25] M. Oertel, M. Hempel, T. Kl¨ahn, and S. Typel, Equations of state for supernov? and compact stars, Rev. Mod. Phys. 89,
015007 (2017). arXiv:1610.03361.
[26] P. A. M. Guichon, J. R. Stone, and A. W. Thomas, QuarkMeson-Coupling (QMC) model for finite nuclei, nuclear matter
and beyond, Prog. Part. Nucl. Phys. 100, 262–297 (2018).
[27] P. Danielewicz, R. Lacey, and W. G. Lynch, Determination of the equation of state of dense matter, Science 298, 1592–1596
(2002). nucl-th/0208016.
[28] S. A. Chin, A relativistic many body theory of high density matter, Ann. Phys. 108, 301–367 (1977). nucl-th/0208016.
[29] H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Relativistic equation of state of nuclear matter for supernova and
neutron star, Nucl. Phys. A 637, 435–450 (1998). nucl-th/9805035.
[30] S. K. Singh, S. K. Biswal, M. Bhuyan, and S. K. Patra, Effect of isospin asymmetry in a nuclear system, J. Phys. G: Nucl.
Part. Phys. 41, 055201 (2014).
[31] A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, Phys. Rev. C 58, 1804 (1998).
[32] V. R. Pandharipande, Hyperonic matter, Nucl. Phys. A178, 123–144 (1971).
[33] V. R. Pandharipande, Dense neutron matter with realistic interactions, Nucl. Phys. A174, 641–656 (1971).
[34] V. R. Pandharipande, Nuclear variational calculations with improved mixed-density approximation, Nucl. Phys. A166,
317–329 (1971).
[35] P. J. Siemens and V. R. Pandharipande,
Neutron matter computations in Bruckner and variational theories,
Nucl.
Phys. A173, 561–570 (1971).
[36] S. P. Maydanyuk, P.-M. Zhang, and S. V. Belchikov, Quantum design using a multiple internal reflections method in a
study of fusion processes in the capture of alpha-particles by nuclei, Nucl. Phys. A 940, 89–118 (2015); arXiv:1504.00567.
[37] S. P. Maydanyuk, P.-M. Zhang, and L.-P. Zou, New quasibound states of the compound nucleus in α-particle capture by
the nucleus, Phys. Rev. C96, 014602 (2017); arXiv:1711.07012.
[38] S. P. Maydanyuk, P.-M. Zhang, and L.-P. Zou, New approach for obtaining information on the many-nucleon structure in
α decay from accompanying bremsstrahlung emission, Phys. Rev. C93, 014617 (2016); arXiv:1505.01029.
[39] A. Yu. Potekhin, Physics of neutron stars, Physics – Uspehi 53 (12), 1279 (2010).
[40] A. I. Steshenko, G. F. Filippov, Size and shape of neutron-excess nuclei, Yad. Fiz. 14, 715 (1971) [Sov. J. Nucl. Phys. 14,
715 (1971)].
[41] G. F. Filippov, A. I. Steshenko, The form of light even-even nuclei, Ukr. Phys. J. 15, 625 (1970).
[42] A. I. Steshenko, Properties of ground states of atomic nuclei (Kiev, 1971).
[43] A. I. Steshenko, On a model of deformed oscillator shells for calculation of bound states of atomic nuclei, Preprint (BITP,
Kiev, 1976).
[44] G. F. Filippov, and I. P. Okhrimenko, Yad. Fiz. 32, 932 (1980) [Sov. J. Nucl. Phys. 32, 480 (1980)].
[45] G. F. Filippov, Yad. Fiz. 33, 928 (1981) [Sov. J. Nucl. Phys. 33, 488 (1981)].
[46] V. S. Vasilevsky, and I. Yu. Rybkin, Yad. Fiz. 50, 662 (1989) [Sov. J. Nucl. Phys. 50, 411 (1989)].
[47] V. S. Vasilevsky, I. Yu. Rybkin, and G. F. Filippov, Yad. Fiz. 51, 112 (1990) [Sov. J. Nucl. Phys. 51, 71 (1990)].
[48] G. F. Filippov, V. S. Vasilevsky, and L. L. Chopovsky, Fiz. Elem. Chastits At. Yadra 16, 349 (1985) [Sov. J. Part. Nucl.
16, 153 (1985)].
[49] G. F. Filippov, S. P. Kruchinin, V. S. Vasilevsky, and L. L. Chopovsky, Yad. Fiz. 43, 843 (1986) [Sov. J. Nucl. Phys. 43,
536 (1986)].
[50] G. F. Filippov, V. S. Vasilevsky, and A. V. Nesterov, Nucl. Phys. A426, 327 (1984).
[51] G. F. Filippov, V. S. Vasilevsky, and A. V. Nesterov, Yad. Fiz. 40, 1418 (1984) [Sov. J. Nucl. Phys. 40, 901 (1984)].
16
[52] F. Arickx, P. Van Leuven, V. S. Vasilevsky, and A. V. Nesterov, in New Perspectives in Nuclear Structure, Proceedings
of the 5th International Spring Seminar in Nuclear Physics, Ravello, Italy, 1995, edited by A. Covello (World Scientific,
Singapore, 1996), p. 111.
[53] V. S. Vasilevsky, A. V. Nesterov, F. Arickx, and P. Van Leuven, Yad. Fiz. 60, 413 (1997) [Phys. At. Nucl. 60, 343 (1997)].
[54] G. F. Filippov, A. V. Nesterov, I. Yu. Rybkin, and S. V. Korennov, Fiz. El. Chastits At. Yadra 25, 1347 (1994) [Phys.
Part. Nucl. 25, 569 (1994)].
[55] V. Vasilevsky, A. V. Nesterov, F. Arickx, and J. Broeckhove, Phys. Rev. C63, 034606 (2001).
[56] V. Vasilevsky, F. Arickx, W. Vanroose, and J. Broeckhove, Phys. Rev. C85, 034318 (2012).
[57] V. Vasilevsky, F. Arickx, W. Vanroose, and J. Broeckhove, Phys. Rev. C85, 034318 (2012).
[58] Yu. A. Lashko, V. S. Vasilevsky, G. F. Filippov, Properties of a potential energy matrix in oscillator basis, Annals Phys.
409, 167930 (2019), arXiv:1902.08759.
[59] V. S. Vasilevsky, K. Kat¯o, N. Takibayev, Formation and decay of resonance state in 9Be and 9B nuclei. Microscopic
three-cluster model investigations, Phys. Rev. C 96 (3), 034322 (2017), arXiv:1706.05514.
[60] V. S. Vasilevsky, Yu. A. Lashko, G. F. Filippov, Two- and three-cluster decays of light nuclei within a hyperspherical
harmonics approach, Phys. Rev. C 97 (6), 064605 (2018), arXiv:1706.04127.
[61] L. D. Landau and E. M. Lifshitz,
Kvantovaya Mehanika, kurs Teoreticheskoi Fiziki (Quantum mechanics, course of
Theoretical Physics), Vol. 3 (Nauka, Mockva, 1989) p. 768 — [in Russian; eng. variant: Oxford, Uk, Pergamon, 1982].
[62] S. P. Maydanyuk,
Model for bremsstrahlung emission accompanying interactions between protons and nuclei from low
energies up to intermediate energies: Role of magnetic emission, Phys. Rev. C86, 014618 (2012), arXiv:1203.1498.
[63] S. P. Maydanyuk and P.-M. Zhang,
New approach to determine proton-nucleus interactions from experimental
bremsstrahlung data, Phys. Rev. C 91, 024605 (2015); arXiv:1309.2784.
[64] S. P. Maydanyuk, P.-M. Zhang, and L.-P. Zou, Manifestation of the important role of nuclear forces in the emission of
photons in pion scattering offnuclei, Phys. Rev. C98, 054613 (2018); arXiv:1809.10403.
[65] S. P. Maydanyuk, P.-M. Zhang, and L.-P. Zou,
Nucleon microscopy in proton-nucleus scattering via analysis of
bremsstrahlung emission, Phys. Rev. C 99, 064602 (2019); arXiv:1812.07180.
[66] X. Liu, S. P. Maydanyuk, P.-M. Zhang, and L. Liu, First investigation of hypernuclei in reactions via analysis of emitted
bremsstrahlung photons, Phys. Rev. C 99, 064614 (2019); arXiv:1810.11942.
[67] S. P. Maydanyuk, P.-M. Zhang, and L. Liu, Search of cluster structure in nuclei via analysis of bremsstrahlung emission,
arXiv:1907.07954.
[68] S. P. Maydanyuk, P.-M. Zhang, and L.-P. Zou, New approach for obtaining information on the many-nucleon structure in
α decay from accompanying bremsstrahlung emission, Phys. Rev. C93, 014617 (2016); arXiv:1505.01029.
[69] S. P. Maydanyuk, A multipolar approach for the description of bremsstrahlung during α-decay, Jour. Phys. Study. 13 (3),
3201 (2009).
[70] S. P. Maydanyuk,
Multipolar approach for description of bremsstrahlung during α-decay and unified formula of the
bremsstrahlung probability, Open Nucl. Part. Phys. J. 2, 17–33 (2009); arXiv:0904.2247.
[71] S. P. Maydanyuk, V. S. Olkhovsky, G. Giardina, G. Fazio, G. Mandaglio, and M. Manganaro, Bremsstrahlung emission
accompanying α-decay of deformed nuclei, Nucl. Phys. A 823, Iss. 1–4, 38–46 (2009).
[72] S. P. Maydanyuk, V. S. Olkhovsky, G. Mandaglio, M. Manganaro, G. Fazio, and G. Giardina, Bremsstrahlung emission
of high energy accompanying spontaneous fission of 252Cf, Phys. Rev. C82, 014602 (2010).
[73] S. P. Maydanyuk, Multipolar model of bremsstrahlung accompanying proton decay of nuclei, J. Phys. G 38 (8), 085106
(2011); arXiv:1102.2067.
[74] S. P. Maydanyuk, V. S. Olkhovsky, G. Mandaglio, M. Manganaro, G. Fazio, G. Giardina, and C. Sacc´a, Bremsstrahlung
emission of photons accompanying ternary fission of 252Cf, Journ. Phys.: Conf. Ser. 282, 012016 (2011).
[75] S. P. Maydanyuk and V. S. Olkhovsky, Angular analysis of bremsstrahlung in α-decay,
Eur. Phys. Journ. A 28 (3),
283–294 (2006); arXiv:nucl-th/0408022.
[76] G. Giardina, G. Fazio, G. Mandaglio, M. Manganaro, C. Sacc´a, N. V. Eremin, A. A. Paskhalov, D. A. Smirnov, S. P. May-
danyuk, and V. S. Olkhovsky, Bremsstrahlung emission accompanying the α-decay of 214Po, Europ. Phys. Journ. A36 (1),
31–36 (2008).
[77] G. Giardina, G. Fazio, G. Mandaglio, M. Manganaro, S. P. Maydanyuk, V. S. Olkhovsky, N. V. Eremin, A. A. Paskhalov,
D. A. Smirnov, and C. Sacc´a, Bremsstrahlung emission during α-decay of 226Ra, Mod. Phys. Lett. A 23 (31), 2651–2663
(2008); arXiv:0804.2640.
[78] M. J. van Goethem, L. Aphecetche, J. C. S. Bacelar, H. Delagrange, J. Diaz, D. d’Enterria, M. Hoefman, R. Holzmann,
H. Huisman, N. Kalantar-Nayestanaki, A. Kugler, H. L¨ohner, G. Martinez, J. G. Messchendorp, R. W. Ostendorf, S. Schad-
mand, R. H. Siemssen, R. S. Simon, Y. Schutz, R. Turrisi, M. Volkerts, V. Wagner, and H. W. Wilschut, Suppresion of
soft nuclear bremsstrahlung in proton-nucleus collisions, Phys. Rev. Lett. 88 (12), 122302 (2002).
[79] J. Clayton, W. Benenson, M. Cronqvist, R. Fox, D. Krofcheck, R. Pfaff, M. F. Mohar, C. Bloch, D. E. Fields, High energy
gamma ray production in proton-induced reactions at 104, 145, and 195 MeV, Phys. Rev. C45, 1815 (1992).
[80] J. E. Clayton, High energy gamma ray production in proton induced reactions at energies of 104, 145, and 195 MeV, PhD
thesis (Michigan State University, 1991).
[81] F. D. Becchetti, Jr., and G. W. Greenlees, Phys. Rev. 182 (4), 1190–1209 (1969). |
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
[1] A. H. Guth, Phys. Rev. D 23, 347 (1981) ; A. D. Linde, Phys.
Lett. B 108, 389 (1982) ; A. Albrecht and P. J. Steinhardt, Phys.
Rev. Lett. 48, 1220 (1982).
[2] M. Tegmark et al. [SDSS Collaboration], Phys. Rev. D 74,
123507 (2006) [arXiv:astro-ph/0608632] ;
[3] E. Komatsu et al. [WMAP Collaboration], arXiv:0803.0547
[astro-ph].
[4] R.
Kallosh,
Lect.
Notes
Phys.
738
(2008)
119
[arXiv:hep-th/0702059] ; J. M. Cline, arXiv:hep-th/0612129 ;
C. P. Burgess, PoS P2GC (2006) 008 [Class. Quant. Grav. 24
(2007) S795] [arXiv:0708.2865 [hep-th]]. L. McAllister and
E. Silverstein, Gen. Rel. Grav. 40 (2008) 565 [arXiv:0710.2951
[hep-th]].
[5] [Planck Collaboration], arXiv:astro-ph/0604069.
[6] http://cmbpol.uchicago.edu/
;
http://www.astro.caltech.edu/˜lgg/spider_front.htm
[7] http://nasascience.nasa.gov/missions/jdem/
[8] http://sci.esa.int/science-e/www/object/index.cfm
?fobjectid=42266
[9] G. R. Dvali and S. H. H. Tye, Phys. Lett. B 450, 72 (1999)
[arXiv:hep-ph/9812483].
[10] S.
Kachru,
R.
Kallosh,
A.
Linde,
J.
M.
Maldacena,
L. P. McAllister and S. P. Trivedi, JCAP 0310, 013 (2003)
[arXiv:hep-th/0308055].
[11] S. Gukov, C. Vafa and E. Witten, Nucl. Phys. B 584 (2000) 69
[Erratum-ibid. B 608 (2001) 477] [arXiv:hep-th/9906070].
[12] S. B. Giddings, S. Kachru and J. Polchinski, Phys. Rev. D 66
(2002) 106006 [arXiv:hep-th/0105097] ; S. Kachru, R. Kallosh,
A. Linde and S. P. Trivedi, Phys. Rev. D 68, 046005 (2003)
[arXiv:hep-th/0301240].
[13] P. Candelas and X. C. de la Ossa, Nucl. Phys. B 342, 246
(1990).
[14] A. Ceresole, G. Dall’Agata, R. D’Auria and S. Ferrara, Phys.
Rev. D 61, 066001 (2000) [arXiv:hep-th/9905226].
[15] D. Baumann, A. Dymarsky, S. Kachru, I. R. Klebanov and
L. McAllister, JHEP 0903, 093 (2009) [arXiv:0808.2811 [hep-
th]].
[16] R. Bean, S. E. Shandera, S. H. Henry Tye and J. Xu, JCAP
0705 (2007) 004 [arXiv:hep-th/0702107] ; S. E. Shandera and
S. H. Tye, JCAP 0605 (2006) 007 [arXiv:hep-th/0601099] ;
L. Lorenz, J. Martin and C. Ringeval, JCAP 0804, 001 (2008)
[arXiv:0709.3758 [hep-th]].
[17] L. Hoi and J. M. Cline, Phys. Rev. D 79, 083537 (2009)
[arXiv:0810.1303 [hep-th]].
[18] D. Baumann, A. Dymarsky, I. R. Klebanov and L. McAllister,
JCAP 0801 (2008) 024 [arXiv:0706.0360 [hep-th]].
[19] C. P. Burgess, J. M. Cline, K. Dasgupta and H. Firouzjahi, JHEP
0703, 027 (2007) [arXiv:hep-th/0610320].
[20] A.
Krause
and
E.
Pajer,
JCAP
0807
(2008)
023
[arXiv:0705.4682 [hep-th]].
12
[21] G. German, G. G. Ross and S. Sarkar, Nucl. Phys. B 608, 423
(2001) [arXiv:hep-ph/0103243].
[22] D. H. Lyth and D. Wands, Phys. Lett. B 524, 5 (2002)
[arXiv:hep-ph/0110002] ; T. Moroi and T. Takahashi, Phys.
Lett. B 522, 215 (2001) [Erratum-ibid. B 539, 303 (2002)]
[arXiv:hep-ph/0110096].
[23] K. Y. Choi, J. O. Gong and D. Jeong, JCAP 0902, 032 (2009)
[arXiv:0810.2299 [hep-ph]].
[24] H. Y. Chen, J. O. Gong and G. Shiu, JHEP 0809, 011 (2008)
[arXiv:0807.1927 [hep-th]].
[25] O. DeWolfe and S. B. Giddings, Phys. Rev. D 67, 066008
(2003) [arXiv:hep-th/0208123].
[26] G. Shiu, G. Torroba, B. Underwood and M. R. Douglas, JHEP
0806, 024 (2008) [arXiv:0803.3068 [hep-th]] ; M. R. Dou-
glas and G. Torroba, JHEP 0905, 013 (2009) [arXiv:0805.3700
[hep-th]] ;
A. R. Frey, G. Torroba, B. Underwood and
M. R. Douglas, JHEP 0901, 036 (2009) [arXiv:0810.5768 [hep-
th]] ; F. Marchesano, P. McGuirk and G. Shiu, JHEP 0904, 095
(2009) [arXiv:0812.2247 [hep-th]].
[27] S. B. Giddings and A. Maharana, Phys. Rev. D 73, 126003
(2006) [arXiv:hep-th/0507158].
[28] D. Baumann, A. Dymarsky, I. R. Klebanov, J. M. Maldacena,
L. P. McAllister and A. Murugan, JHEP 0611, 031 (2006)
[arXiv:hep-th/0607050] ; O. J. Ganor, Nucl. Phys. B 499 (1997)
55 [arXiv:hep-th/9612077] ; M. Berg, M. Haack and B. Kors,
Phys. Rev. D 71, 026005 (2005) [arXiv:hep-th/0404087].
[29] I. R. Klebanov and M. J. Strassler, JHEP 0008 (2000) 052
[arXiv:hep-th/0007191].
[30] S. Kuperstein, JHEP 0503, 014 (2005) [arXiv:hep-th/0411097].
[31] C.
P.
Herzog,
I.
R.
Klebanov
and
P.
Ouyang,
arXiv:hep-th/0108101 ; C. P. Herzog, I. R. Klebanov and
P. Ouyang, arXiv:hep-th/0205100.
[32] C. P. Burgess, R. Kallosh and F. Quevedo, JHEP 0310, 056
(2003) [arXiv:hep-th/0309187].
[33] J. O. Gong and E. D. Stewart, Phys. Lett. B 510, 1 (2001)
[arXiv:astro-ph/0101225].
[34] A. Ali, R. Chingangbam, S. Panda and M. Sami, Phys. Lett. B
674, 131 (2009) [arXiv:0809.4941 [hep-th]].
[35] D.
H.
Lyth,
Phys.
Rev.
Lett.
78
(1997)
1861
[arXiv:hep-ph/9606387] ; D. Baumann and L. McAllister,
Phys. Rev. D 75 (2007) 123508 [arXiv:hep-th/0610285].
[36] A. R. Liddle and S. M. Leach, Phys. Rev. D 68, 103503 (2003)
[arXiv:astro-ph/0305263] ; L. Alabidi and D. H. Lyth, JCAP
0605, 016 (2006) [arXiv:astro-ph/0510441].
[37] R.
Kallosh
and
A.
Linde,
JHEP
0412,
004
(2004)
[arXiv:hep-th/0411011].
[38] H. Y. Chen, L. Y. Hung and G. Shiu, JHEP 0903, 083 (2009)
[arXiv:0901.0267 [hep-th]]. |
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.
REFERENCES
1.
D. C. Morton and L. Spitzer, Jr., ApJ 144, 1-12 (1966).
2.
A. L. Kinney, R. C. Bohlin, D. Calzetti, N. Panagia, and R. F. G. Wyse, ApJS 86, 5-93 (1993).
3.
M. Giavalisco, ARAA 40, 579-641 (2002).
4.
C. M. Schwartz, C. L. Martin, R. Chandar, C. Leitherer, T. M. Heckman, and M. S. Oey, ApJ 646,
858-871 (2006).
5.
E. Gawiser et al., ApJ 671, 278-284 (2007).
6.
A. Gil de Paz et al., ApJS 173, 185-255 (2007).
7.
J. D. Goldader, G. Meurer, T. M. Heckman, M. Seibert, D. B. Sanders, D. Calzetti and C. C.
Steidel, ApJ 568, 651-679 (2002).
8.
C. Leitherer, J. R. S. Leão, T. M. Heckman, D. J. Lennon, M. Pettini and C. Robert, ApJ 550, 724-
736 (2001).
9.
R. M. González Delgado, C. Leitherer, G. StasiĔska and T. M. Heckman, ApJ 580, 824-843 (2002).
10. R. Chandar, C. Leitherer, C. A. Tremonti, D. Calzetti, A. Aloisi, G. R. Meurer and D. de Mello,
ApJ 628, 210-230 (2005).
11. R. de Grijs and G. Parmentier, CJAA 7, 155-186 (2007).
12. J. P. Grimes, T. M. Heckman, A. Aloisi, D. Calzetti, C. Leitherer, C. L. Martin, G. Meurer, K.
Sembach and D. K. Strickland, ApJS (in press; arXiv:0811.3618v1).
13. T. M. Heckman, RevMexAA (Serie de Conferencias) 17, 47-55 (2003).
14. M. Damen, I. Labbé, M. Franx, P. G. van Dokkum, E. N. Taylor and E. J. Gawiser, ApJ 690, 937-
943 (2009).
15. S. Charlot and S. M. Fall, ApJ 414, 580-588 (1993).
16. D. Kunth, J. M. Mas-Hesse, E. Terlevich, R. Terlevich, J. Lequeux and S. M. Fall, A&A 334, 11-20
(1998).
17. A. E. Shapley, C. C. Steidel, M. Pettini and K. L. Adelberger, ApJ 588, 65-89 (2003).
18. M. Hayes, G. Östlin, H. Atek, D. Kunth, J. M. Mas-Hesse, C. Leitherer, E. Jiménez-Bailón and A.
Adamo, MNRAS 382, 1465-1480 (2007).
19. C. Leitherer, D. Schaerer, J. D. Goldader, R. M. González, C. Robert, D. Foo Kune, D. F. de Mello,
D. Devost and T. M. Heckman, ApJS 123, 3-40 (1999).
20. C. Leitherer, in Starbursts: From 30 Doradus to Lyman Break Galaxies, edited by R. de Grijs and
R. M. González Delgado, Dordrecht: Springer, 2005, pp. 89-96.
21. C. C. Steidel, M. Pettini and K. L. Adelberger, ApJ 546, 665-671 (2001).
22. A. E. Shapley, C. C. Steidel, M. Pettini, K. L. Adelberger and D. K. Erb, ApJ 651, 688-703 (2006). |
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.
REFERENCES
Aller, M. F. et al. 1997, in AIP Conference Series, Vol. 410,
Proceedings of the Fourth Compton Symposium, ed. C. D.
Dermer, M. S. Strickman, & J. D. Kurfess, 1423–+
Angione, R. J. 1971, AJ, 76, 412
Barthelmy, S. D. et al. 2005, Space Science Reviews, 120, 143
Beckwith, S. et al. 1976, ApJ, 208, 390
Bessell, M. S., Castelli, F., & Plez, B. 1998, A&A, 333, 231
B¨ottcher, M. 2007, Ap&SS, 309, 95
Burrows, D. N. et al. 2005, Space Science Reviews, 120, 165
Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345,
245
Celotti, A. & Ghisellini, G. 2008, MNRAS, 385, 283
Craine, E. R. 1977, A handbook of quasistellar and BL Lacertae
objects (Astronomy and Astrophysics Series, Tucson: Pachart,
1977)
Dermer, C. D. & Schlickeiser, R. 1993, ApJ, 416, 458
Djorgovski, S. G. et al. 2008, The Astronomer’s Telegram, 1684, 1
Edelson, R. A. & Krolik, J. H. 1988, ApJ, 333, 646
Fiorucci, M., Tosti, G., & Rizzi, N. 1998, PASP, 110, 105
Fossati, G. et al. 1998, MNRAS, 299, 433
Gasparrini, D. et al. 2008, The Astronomer’s Telegram, 1592, 1
Gehrels, N. et al. 2004, ApJ, 611, 1005
Ghisellini, G. et al. 2007, MNRAS, 382, L82
Ghisellini, G. & Madau, P. 1996, MNRAS, 280, 67
Hartman, R. C.et al. 1999, ApJS, 123, 79
Hartman, R. C. et al. 1993, ApJ, 407, L41
Jones, T. W., O’dell, S. L., & Stein, W. A. 1974, ApJ, 188, 353
Konigl, A. 1981, ApJ, 243, 700
Michelson, P. F. 2007, in AIP Conference Series, Vol. 921, The
First GLAST Symposium, ed. S. Ritz, P. Michelson, & C. A.
Meegan, 8–12
M¨ucke, A. & Protheroe, R. J. 2001, Astroparticle Physics, 15, 121
M¨ucke, A. et al. 2003, Astroparticle Physics, 18, 593
Pian, E. et al. 2006, A&A, 449, L21
Poole, T. S. et al. 2008, MNRAS, 383, 627
Raiteri, C. M. et al. 2007, A&A, 473, 819
Raiteri, C. M. et al. 2008, A&A, 485, L17
Roming, P. W. A. et al. 2005, Space Science Reviews, 120, 95
Sambruna, R. M., Maraschi, L., & Urry, C. M. 1996, ApJ, 463,
444
Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500,
525
Sikora, M., Begelman, M. C., & Rees, M. J. 1994, ApJ, 421, 153
Tavani, M. et al. 2008, Nuclear Instruments and Methods in
Physics Research A, 588, 52
Tosti, G. et al. 2008, The Astronomer’s Telegram, 1628, 1
Urry, C. M. & Mushotzky, R. F. 1982, ApJ, 253, 38
Urry, C. M. & Padovani, P. 1995, PASP, 107, 803
Vaughan, S. et al. 2003, MNRAS, 345, 1271
Vercellone, S. et al. 2008, ApJ, 676, L13
Vercellone, S. et al. 2009, ApJ, 690, 1018
Villata, M. et al. 2007, A&A, 464, L5
Villata, M. et al. 2006, A&A, 453, 817
Villata, M. et al. 2008, The Astronomer’s Telegram, 1625, 1
Vittorini, V. et al. 2008, The Astronomer’s Telegram, 1581, 1
White, R. J. & Peterson, B. M. 1994, PASP, 106, 879
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. |
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