#  KEK-TH-2421 KYUSHU-HET-240 Spacetime evolution during moduli stabilization
in radiation dominated era beyond 4D effective theory

Hajime Otsukaa*** E-mail address<EMAIL_ADDRESS>and
Yutaka Sakamurab,c††† E-mail address<EMAIL_ADDRESS>
aDepartment of Physics, Kyushu University,
744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
bKEK Theory Center, Institute of Particle and Nuclear Studies, KEK,
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
cGraduate University for Advanced Studies (Sokendai),
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan.

Abstract

We investigate the time evolution of the background spacetime during the
moduli stabilization process, which is assumed to occur in the radiation
dominated era. The setup is basically the Salam-Sezgin model, but we add a
potential term for the dilaton in order to stabilize the moduli completely. We
numerically solve the higher-dimensional background field equations, including
a case that the stabilization process cannot be described within the 4D
effective theory. In contrast to the conventional 4D effective theory
analysis, we find that when the mass scale of the stabilization is larger than
the compactification scale, the radiation contribution to the total energy
density remains to be non-negligible for a much longer time than the
stabilization time scale. As a result, the non-compact 3D space expands slower
than the matter dominated universe. We also find the equation of state for the
radiation $w_{\rm rad}$ remains to be smaller than 1/3 for a long time, which
indicates that the radiation still feels the extra dimensions for a while even
after the moduli are stabilized.

###### Contents

  1. 1 Introduction
  2. 2 6D supergravity compactified on a sphere
    1. 2.1 Setup
    2. 2.2 Salam-Sezgin model
    3. 2.3 Nishino-Maeda model
  3. 3 Thermodynamic quantities and moduli potential
    1. 3.1 Thermodynamic quantities
    2. 3.2 Conservation law and entropy
    3. 3.3 Approximation of $Q_{i}$ ($i=1,2,3$)
    4. 3.4 Effective potential
  4. 4 Numerical analyses
    1. 4.1 Evolution equations
    2. 4.2 Parameter choice and initial values
    3. 4.3 Numerical results
      1. 4.3.1 Moduli stabilization
      2. 4.3.2 Energy densities of radiation and moduli oscillation
      3. 4.3.3 Equation of state for radiation
    4. 4.4 4D Planck unit
  5. 5 Conclusions

##  1 Introduction

Current and future gravitational wave observations will give impacts on
several stages of the cosmological history of our universe. There is a
possibility to detect signatures of a fundamental theory of particle physics
and cosmology, e.g., a higher-dimensional gravity arising from string theory,
from the spectrum of the gravitational waves [1, 2] since their spectrum
depends on the spacetime in which the gravitational waves have propagated.
This direction of the search for the extra dimensions is complementary to the
collider experiments that have searched for the Kaluza-Klein (KK) modes. In
the early universe, the three-dimensional (3D) space was much smaller, and we
expect that various effects due to the existence of the extra dimensions were
larger than those in the current universe. Hence, in order to extract some
signatures of the extra dimensions from the gravitational wave spectrum, we
need to know the detailed time evolution of the background spacetime at early
times.

The stabilization of the moduli to some finite values is an important process
to realize the standard four-dimensional (4D) cosmology at late times. In this
paper, we investigate the time evolution of the space during the moduli
stabilization, which is assumed to occur in the radiation dominated era. When
the moduli stabilization can be described within the 4D effective theory, it
was well known that the oscillation of the moduli around the potential minimum
rapidly dominates the energy density and the 3D space expands as the matter
dominated universe during the oscillation [3, 4, 5, 6, 7]. However, in such a
case, we must restrict the parameter space so that the typical mass scale of
the stabilization $m$ is lower than the compactification scale $m_{\rm KK}$,
in order for the 4D effective theory analysis to be valid. The purpose of this
paper is to extend the analysis to a more general choice of the parameters,
including cases in which $m\gtrsim m_{\rm KK}$. We numerically pursue the
spacetime evolution by solving the higher-dimensional field equations
including the radiation that fills in the bulk, instead of analyzing the 4D
effective theory.

As a setup, we consider a gauged six-dimensional (6D) supergravity [10, 11]
compactified on a sphere $S^{2}$. Since the 6D spacetime allows the background
tensor fields such as fluxes on the internal space, the 6D gauged supergravity
has a rich structure, such as the existence of chiral fermions[12, 13], the
Green-Schwarz anomaly cancellation[14], and a self-tuning of the four-
dimensional vacuum energy[15, 16, 17, 18]. It is well known that a static
solution is realized by balancing between the $S^{2}$ curvature and the $U(1)$
magnetic flux in a specific model, which is called the Salam-Sezgin model
[19]. It was generalized to incorporate codimension-two branes [15, 16, 17,
18, 20, 21, 22]. The time-dependent extension of the Salam-Sezgin background
solution was discussed in Ref. [23] in the vacuum and the radiation dominated
situations. They obtained a solution that asymptotically approaches the 4D
radiation dominated universe. In their analysis, however, the equation of
state is set to a constant by hand. Besides, there remains a modulus that is
not stabilized in their setup. In this work, we treat the pressure and the
energy density independently, and pursue their time evolutions in the presence
of a moduli-stabilization potential that completely fixes the moduli.

The paper is organized as follows. In Sec. 2, we briefly review the Salam-
Sezgin model [19] and its time-dependent extension: the Nishino-Maeda model
[23]. In Sec. 3, we derive the expressions for various thermodynamic
quantities in the five-dimensional (5D) space and the moduli potential by
introducing an additional dilaton potential. In Sec. 4, we numerically solve
the background field equations including the radiation contribution, and see
the time evolutions of the background space and the radiation contribution to
the total energy density. Finally, we conclude this paper in Sec. 5.

##  2 6D supergravity compactified on a sphere

In this section, we briefly review the known background solutions of 6D
supergravity compactified on a sphere.

###  2.1 Setup

We focus on a bosonic part of the gauged 6D ${\cal N}=(1,0)$ supergravity [10,
11], following the convention of Ref. [25]. The bosonic fields of our interest
consist of a gravity multiplet including the metric ($g_{MN}$), the self-dual
antisymmetric Kalb-Ramond field ($B_{MN}$), the dilaton ($\sigma$), and a
$U(1)$ gauge field ($A_{M}$). The indices $M,N=0,1,2,\cdots,5$ are the 6D
coordinate indices. The 6D gravitational anomalies are canceled by the
introduction of hypermultiplets or additional vector multiplets, but we assume
that they do not affect the dynamics of background geometry at the moment.
Then, the 6D effective action is given by

$\displaystyle\begin{split}S&=\int
d^{6}x\sqrt{-g^{(6)}}\biggl{[}-\frac{1}{2}R-\frac{1}{2}g^{MN}\partial_{M}\sigma\partial_{N}\sigma-\frac{g^{2}e^{\sigma}}{4}F^{MN}F_{MN}-\frac{g^{4}e^{2\sigma}}{12}G_{MNP}G^{MNP}-V(\sigma)\biggl{]},\end{split}$
(2.1)

where $g^{(6)}\equiv\det(g_{MN})$ is the determinant of the 6D metric,
$F_{MN}\equiv\partial_{M}A_{N}-\partial_{N}A_{M}$ denotes the $U(1)$ gauge
field strength with the gauge coupling $g$, and the three-form field strength
for $B_{MN}$ is given by $G_{MNP}$ containing the Chern-Simons term:
$G_{MNP}\equiv\partial_{[M}B_{NP]}+F_{[MN}A_{P]}$. Note that the three-form
field strength $G_{MNP}$ including the Chern-Simons contribution is
constructed in a $U(1)$ invariant way:

$\displaystyle\delta A_{M}=\partial_{M}\lambda,\quad\delta B_{MN}=-\lambda
F_{MN}.$ (2.2)

Here, we adopt the metric signature $(-,+,...,+)$ and the unit of the 6D
gravitational constant $\kappa_{6}^{2}=8\pi G_{6}=1$ throughout this paper. We
follow the Weinberg’s curvature conventions for the Riemann curvature tensor
as well as the covariant derivative of the gauge potential. The scalar
potential is given by

$\displaystyle V(\sigma)=2e^{-\sigma},$ (2.3)

which is induced as a consequence of the gauging of the $U(1)_{R}$ symmetry.

The equations of motions for $\\{\sigma,B_{MN},F_{MN},g_{MN}\\}$ are
summarized as follows:

$\displaystyle\begin{split}&\frac{1}{\sqrt{-g^{(6)}}}\partial_{M}(\sqrt{-g^{(6)}}\partial^{M}\sigma)-\frac{g^{2}e^{\sigma}}{4}F^{MN}F_{MN}-\frac{g^{4}e^{2\sigma}}{6}G_{MNP}G^{MNP}-\partial_{\sigma}V(\sigma)=0,\\\
&\partial_{M}\left(\sqrt{-g^{(6)}}e^{2\sigma}G^{MNP}\right)=0,\\\
&\partial_{M}\left(\sqrt{-g^{(6)}}e^{\sigma}F^{MN}\right)+g^{2}e^{2\sigma}G^{MNP}F_{MP}=0,\\\
&R_{MN}-\frac{1}{2}g_{MN}R-T_{MN}=0.\end{split}$ (2.4)

The last equation is the Einstein equation. The explicit form of the energy-
momentum tensor $T_{MN}$ is provided by

$\displaystyle\begin{split}T_{MN}=&\frac{2}{\sqrt{-g}}\frac{\delta S_{\rm
matter}}{\delta g^{MN}}\\\
=&-\partial_{M}\sigma\partial_{N}\sigma+\frac{1}{2}g_{MN}\partial_{L}\sigma\partial^{L}\sigma-g^{2}e^{\sigma}F_{MC}F_{N}^{\,\,C}+\frac{g^{2}e^{\sigma}}{4}g_{MN}F^{AB}F_{AB}\\\
&-\frac{g^{4}e^{2\sigma}}{2}G_{MBC}G_{N}^{\,\,BC}+\frac{g^{4}e^{2\sigma}}{12}g_{MN}G_{ABC}G^{ABC}+g_{MN}V(\sigma).\end{split}$
(2.5)

###  2.2 Salam-Sezgin model

We first review the famous Salam-Sezgin model [19] as a solution of 6D SUGRA
on the sphere. The background fields are chosen as

$\displaystyle\begin{split}g_{MN}&=\begin{pmatrix}\eta_{\mu\nu}&0\\\
0&g_{mn}(y)\end{pmatrix},\\\ \sigma&=\mbox{(constant)},\\\ B_{MN}&=0,\\\
F_{MN}&=\begin{pmatrix}0&0\\\ 0&F_{mn}(y)\end{pmatrix},\end{split}$ (2.6)

where $\eta_{\mu\nu}$ ($\mu,\nu=0,1,2,3$) is the 4D Minkowski metric, and
$y=(x^{4},x^{5})$ denotes the coordinates of the compact space. The background
metric of $S^{2}$ is

$\displaystyle\begin{split}g_{mn}(y)&=b^{2}r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),\end{split}$ (2.7)

where $m,n=4,5$, and $r$ is some fixed length scale. In this coordinate
$\\{x^{4},x^{5}\\}=\\{\theta,\phi\\}$, the radius of $S^{2}$ is given by $br$,
and the lightest fluctuation mode of $b$ corresponds to the radion from the 4D
point of view. In the following, we choose $r$ as the 6D Planck length, i.e.,
$r=1$. Here, we assumed that $B_{MN}$ does not have a nontrivial background,
but the $U(1)$ gauge field $A_{M}$ has the monopole background allowed by
$S^{2}$ geometry.

$\displaystyle F_{mn}(y)$
$\displaystyle=f\epsilon_{mn}\equiv\frac{n}{2gb^{2}}\epsilon_{mn},\;\;\;\;\;\left(n\in{\mathbb{Z}}\right)$
(2.8)

where

$\displaystyle\epsilon_{45}$
$\displaystyle=-\epsilon_{54}=\sqrt{\det(g_{mn})}=b^{2}\sin\theta,$ (2.9)
$\displaystyle\epsilon_{44}$ $\displaystyle=\epsilon_{55}=0.$ (2.10)

This satisfies the equation of motion and the Dirac quantization condition,

$\displaystyle g\int_{S^{2}}d^{2}y\;F_{45}$
$\displaystyle=g\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\;\frac{n}{2g}\sin\theta=2\pi
n.$ (2.11)

The monopole charge $n$ is fixed to satisfy the equations of motion for the
dilaton $\sigma$ and the internal metric $g_{mn}$:

$\displaystyle\begin{split}\frac{g^{2}e^{\sigma}}{4}F^{mn}F_{mn}&=-\partial_{\sigma}V(\sigma)=V(\sigma),\\\
R_{mn}&=-g^{2}e^{\sigma}F_{mp}F_{n}^{\;\,p}=-g^{2}e^{\sigma}f^{2}g_{mn},\end{split}$
(2.12)

which are rewritten as

$\displaystyle\begin{split}e^{-\sigma}&=\frac{n^{2}}{4}b^{-2},\\\
e^{-2\sigma}&=\frac{n^{2}}{16}b^{-4},\end{split}$ (2.13)

respectively. We have used the relation: $R_{mn}=b^{-2}g_{mn}$. Hence, these
equations are consistent with each other iff the monopole charge is chosen as

$\displaystyle n^{2}=1,$ (2.14)

as stated in Ref. [19]. Note that the scalar potential (2.3) only stabilizes a
linear combination of the dilaton $\sigma$ and the size of $S^{2}$ (radion)
$b$. Namely, it only relates these fields as

$\displaystyle e^{\sigma}=4b^{2},$ (2.15)

and does not fix their individual values.

###  2.3 Nishino-Maeda model

We next discuss time-dependent solutions in the Salam-Sezgin setup [23, 24].
We allow the time-dependence of the background, and modify (2.6) as

$\displaystyle\begin{split}g_{MN}&=\begin{pmatrix}g_{\mu\nu}(t)&0\\\
0&g_{mn}(t,y)\end{pmatrix},\\\ \sigma&=\sigma(t),\\\ B_{MN}&=0,\\\
F_{MN}&=\begin{pmatrix}0&0\\\ 0&F_{mn}(t,y)\end{pmatrix},\end{split}$ (2.16)

where the background metric is assumed as

$\displaystyle ds^{2}$ $\displaystyle=g_{MN}dx^{M}dx^{N}$
$\displaystyle=-dt^{2}+a(t)^{2}\left\\{(dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2}\right\\}+b(t)^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (2.17)

with the scale factors $\\{a(t),b(t)\\}$, and the $U(1)$ magnetic field takes
the value:

$\displaystyle F_{mn}(t,y)$
$\displaystyle=f\epsilon_{mn}=\frac{n}{2gb^{2}(t)}\epsilon_{mn},\;\;\;\;\;(n=\pm
1)$ (2.18)

satisfying the equation of motion for $F_{mn}$ and the Dirac quantization
condition (2.11). In the following analysis, we set the monopole charge $n=1$.
There are two independent classes of solutions depending on the initial
conditions.

  1. 1.

Static solution:

$\displaystyle\begin{split}a&=a_{0},\\\ b&=b_{0},\\\
\sigma&=\sigma_{0},\end{split}$ (2.19)

where $\\{a_{0},b_{0},\sigma_{0}\\}$ are constants and this corresponds to the
Salam-Sezgin solution.

  2. 2.

Time-dependent solution:

$\displaystyle\begin{split}a&=a_{0}(t-t_{0})^{\frac{9\pm 4\sqrt{3}}{33}},\\\
b&=b_{0}(t-t_{0})^{\frac{1\mp 2\sqrt{3}}{11}},\\\
\sigma&=\sigma_{0}+\frac{2(1\mp 2\sqrt{3})}{11}\ln(t-t_{0}),\end{split}$
(2.20)

where $\\{a_{0},b_{0},\sigma_{0},t_{0}\\}$ are constants. Noting that
$(9+4\sqrt{3})/33\simeq 0.48$, one of these solutions has a similar expansion
property to the 4D radiation dominated one. But the compact extra space
continues to shrink in that case.

In addition to the above solutions, the authors of Refs. [23, 24] also
introduce the radiation to the original action,

$\displaystyle S\rightarrow S+S_{\rm rad}.$ (2.21)

The radiation consists of the gravitino and/or the matter fields that are
required by the cancellation of the 6D gravitational anomalies. Such fields
are supposed to be in the thermal equilibrium. They assume that the energy-
momentum tensor for the radiation has the form of

$\displaystyle\left(T_{\rm rad}\right)_{M}^{\;\;\;N}$
$\displaystyle=\begin{pmatrix}\rho_{\rm rad}&&\\\ &-p_{{\rm
rad},3}\mbox{\boldmath$1_{3}$}&\\\ &&-p_{{\rm
rad},2}\mbox{\boldmath$1_{2}$}\end{pmatrix}.$ (2.22)

They also assume that the equations of state for the radiation are constants.
For example, when

$\displaystyle p_{\rm rad,3}$ $\displaystyle=\frac{\rho_{\rm
rad}}{3},\;\;\;\;\;p_{\rm rad,2}=0,$ (2.23)

they found that the background spacetime asymptotically approaches the 4D
radiation dominated universe at late times.

$\displaystyle a(t)$ $\displaystyle\to a_{0}t^{1/2},\;\;\;\;\;b(t)\to
b_{0},\;\;\;\;\;\sigma(t)\to\sigma_{0},$ $\displaystyle F_{\mu\nu}$
$\displaystyle=F_{\mu
m}=0,\;\;\;\;\;F_{mn}(t)=\frac{1}{2gb^{2}(t)}\epsilon_{mn},\;\;\;\;\;B_{MN}=0,$
(2.24)

where $a_{0}$, $b_{0}$ and $\sigma_{0}$ are constants. However, the equations
of state $w_{\rm rad,3}\equiv p_{\rm rad,3}/\rho_{\rm rad}$ and $w_{\rm
rad,2}\equiv p_{\rm rad,2}/\rho_{\rm rad}$ should be determined by the
dynamics, and thus change with time. Thus, in the following sections, we will
numerically solve the evolution equations for the background, and see how the
background spacetime and $w_{\rm rad,3}$ evolve with time.

##  3 Thermodynamic quantities and moduli potential

In this section, we derive the expressions for the thermodynamic quantities
for the 6D radiation and the scalar potential for the moduli.

###  3.1 Thermodynamic quantities

The physical volume of the compact space $S^{2}$ is given by

$\displaystyle{\cal V}_{2}$ $\displaystyle=4\pi b^{2}.$ (3.1)

Since the comoving volume for the 3D space is

$\displaystyle{\cal V}_{3}$ $\displaystyle=a^{3},$ (3.2)

the 5D comoving volume is given by

$\displaystyle\begin{split}{\cal V}_{5}={\cal V}_{3}{\cal V}_{2}=4\pi
a^{3}b^{2}.\end{split}$ (3.3)

The dispersion relation of a 6D massless particle on this background:

$\displaystyle\begin{split}P_{M}P^{M}=-p_{0}^{2}+\frac{1}{a^{2}}\vec{p}^{\;2}+\frac{1}{b^{2}}p_{\theta}^{2}+\frac{1}{b^{2}\sin^{2}\theta}p_{\phi}^{2}=0\end{split}$
(3.4)

implies that the energy of the particle with the 3D momentum
$\vec{p}=(p_{1},p_{2},p_{3})$ and the angular momentum $l$ on $S^{2}$ is

$\displaystyle\begin{split}{\cal E}_{p,l}\equiv
p_{0}=\sqrt{\frac{p^{2}}{a^{2}}+\frac{l(l+1)}{b^{2}}},\end{split}$ (3.5)

where $p\equiv\sqrt{\vec{p}^{\;2}}$. Since each one-particle state is
specified by $\vec{p}$, $l$ and the ‘magnetic quantum number’ $m=-l,\cdots,l$,
we have $(2l+1)$ degenerate energy eigenstates for each $\vec{p}$ and $l$.

The grand potential is then expressed as

$\displaystyle J(\beta,\mu,{\cal V}_{3},{\cal V}_{2})$
$\displaystyle=\pm\sum_{l=0}^{\infty}\frac{g_{\rm
dof}(2l+1)}{2\pi^{2}\beta}\int_{0}^{\infty}dp\;p^{2}\ln\left(1\mp
e^{-\beta({\cal E}_{p,l}-\mu)}\right),$ (3.6)

where $g_{\rm dof}$ denotes the degrees of freedom for the 6D relativistic
particles, $\beta$ is the inverse temperature, and $\mu$ is the chemical
potential. The upper sign in the logarithm corresponds to the case of bosons,
while the lower is that of fermions.

From (3.6), various thermodynamic quantities are calculated as follows.

  * •

Energy density:

$\displaystyle\rho_{\rm rad}$ $\displaystyle=\frac{1}{{\cal
V}_{5}}\left(\partial_{\beta}-\frac{\mu}{\beta}\partial_{\mu}\right)(\beta
J)=\sum_{l=0}^{\infty}\frac{g_{\rm dof}(2l+1)}{2\pi^{2}{\cal
V}_{5}}\int_{0}^{\infty}dp\;\frac{p^{2}{\cal E}_{p,l}}{e^{\beta({\cal
E}_{p,l}-\mu)}\mp 1}$ $\displaystyle=\sum_{l=0}^{\infty}\frac{g_{\rm
dof}(2l+1)}{2\pi^{2}\beta^{4}{\cal
V}_{2}}\int_{0}^{\infty}dk\;\frac{k^{2}\sqrt{k^{2}+c_{l}^{2}}}{e^{\sqrt{k^{2}+c_{l}^{2}}-\beta\mu}\mp
1}.$ (3.7)

In the second line, we have rescaled the integration variable and the KK
masses as

$\displaystyle k$
$\displaystyle\equiv\frac{\beta}{a}p,\;\;\;\;\;c_{l}\equiv\beta\sqrt{\frac{4\pi
l(l+1)}{{\cal V}_{2}}}.$ (3.8)

  * •

Pressures:
The pressure in the 3D non-compact space is given by

$\displaystyle P_{3}$ $\displaystyle=-\frac{\partial J}{\partial{\cal
V}_{3}}=\sum_{l=0}^{\infty}\frac{g_{\rm
dof}(2l+1)}{6\pi^{2}a^{5}}\int_{0}^{\infty}dp\;\frac{p^{4}}{{\cal
E}_{p,l}(e^{\beta({\cal E}_{p,l}-\mu)}\mp 1)}$
$\displaystyle=\sum_{l=0}^{\infty}\frac{g_{\rm
dof}(2l+1)}{6\pi^{2}\beta^{4}}\int_{0}^{\infty}dk\;\frac{k^{4}}{\sqrt{k^{2}+c_{l}^{2}}(e^{\sqrt{k^{2}+c_{l}^{2}}-\beta\mu}\mp
1)},$ (3.9)

and that in the $S^{2}$ compact space is given by

$\displaystyle P_{2}$ $\displaystyle=-\frac{\partial J}{\partial{\cal
V}_{2}}=\sum_{l=0}^{\infty}\frac{g_{\rm dof}l(l+1)(2l+1)}{\pi{\cal
V}_{2}^{2}}\int_{0}^{\infty}dp\;\frac{p^{2}}{{\cal E}_{p,l}(e^{\beta({\cal
E}_{p,l}-\mu)}\mp 1)}$ $\displaystyle=\sum_{l=1}^{\infty}\frac{g_{\rm
dof}l(l+1)(2l+1){\cal V}_{3}}{\pi{\cal
V}_{2}^{2}\beta^{2}}\int_{0}^{\infty}dk\;\frac{k^{2}}{\sqrt{k^{2}+c_{l}^{2}}(e^{\sqrt{k^{2}+c_{l}^{2}}-\beta\mu}\mp
1)}.$ (3.10)

  * •

Entropy:

$\displaystyle{\cal S}_{\rm rad}$ $\displaystyle=\beta^{2}\frac{\partial
J}{\partial\beta}=\sum_{l=0}^{\infty}\frac{g_{\rm
dof}(2l+1)}{2\pi^{2}}\int_{0}^{\infty}dp\;p^{2}\left\\{\mp\ln\left(1\mp
e^{-\beta({\cal E}_{p,l}-\mu)}\right)+\frac{\beta({\cal
E}_{p,l}-\mu)}{e^{\beta({\cal E}_{p,l}-\mu)}\mp 1}\right\\}$
$\displaystyle=\sum_{l=0}^{\infty}\frac{g_{\rm dof}(2l+1){\cal
V}_{3}}{2\pi^{2}\beta^{3}}\int_{0}^{\infty}dk\;k^{2}\left\\{\mp\ln\left(1\mp
e^{-\sqrt{k^{2}+c_{l}^{2}}+\beta\mu}\right)+\frac{\sqrt{k^{2}+c_{l}^{2}}-\beta\mu}{e^{\sqrt{k^{2}+c_{l}^{2}}-\beta\mu}\mp
1}\right\\}.$ (3.11)

The equation (3.6) is rewritten as

$\displaystyle J(\beta,\mu,{\cal V}_{3},{\cal V}_{2})$
$\displaystyle=\mp\frac{g_{\rm dof}{\cal V}_{3}}{\pi^{2}\beta^{4}}{\rm
Li}_{4}(\pm e^{\beta\mu})\pm\sum_{l=1}^{\infty}\frac{g_{\rm dof}(2l+1){\cal
V}_{3}}{2\pi^{2}\beta^{4}}\int_{0}^{\infty}dk\;k^{2}\ln\left(1\mp
e^{-\sqrt{k^{2}+c_{l}^{2}}+\beta\mu}\right),$ (3.12)

where ${\rm Li}_{s}(z)$ is the polylogarithmic function. The first term is the
contribution of the $l=0$ state. In the following, we consider a situation in
which $e^{-c_{l}+\beta\mu}\ll 1$ ($l\geq 1$). Then, the grand potential can be
approximated as

$\displaystyle J(\beta,\mu,{\cal V}_{3},{\cal V}_{2})$
$\displaystyle\simeq\mp\frac{g_{\rm dof}{\cal V}_{3}}{\pi^{2}\beta^{4}}{\rm
Li}_{4}(\pm e^{\beta\mu})-\sum_{l=1}^{\infty}\frac{g_{\rm dof}(2l+1){\cal
V}_{3}}{2\pi^{2}\beta^{4}}\int_{0}^{\infty}dk\;k^{2}e^{-\sqrt{k^{2}+c_{l}^{2}}+\beta\mu}$
$\displaystyle=-\frac{g_{\rm dof}{\cal V}_{3}}{2\pi^{2}\beta^{4}}\left\\{\pm
2{\rm Li}_{4}(\pm
e^{\beta\mu})+e^{\beta\mu}Q_{1}\left(\beta\sqrt{\frac{4\pi}{{\cal
V}_{2}}}\right)\right\\},$ (3.13)

where

$\displaystyle Q_{1}(x)$
$\displaystyle\equiv\sum_{l=1}^{\infty}x^{2}l(l+1)(2l+1)K_{2}\left(x\sqrt{l(l+1)}\right).$
(3.14)

Here $K_{2}(z)$ is the modified Bessel function of the second kind. Then the
above quantities are approximated as

$\displaystyle\rho_{\rm rad}$ $\displaystyle\simeq\rho_{\rm rad}^{\rm
ap}\equiv\frac{g_{\rm dof}}{2\pi^{2}\beta^{4}{\cal V}_{2}}\left\\{\pm 6{\rm
Li}_{4}(\pm e^{\beta\mu})+e^{\beta\mu}(3Q_{1}+Q_{2})\right\\},$ $\displaystyle
P_{3}$ $\displaystyle\simeq P_{3}^{\rm ap}\equiv\frac{g_{\rm
dof}}{2\pi^{2}\beta^{4}}\left(\pm 2{\rm Li}_{4}(\pm
e^{\beta\mu})+e^{\beta\mu}Q_{1}\right),$ $\displaystyle P_{2}$
$\displaystyle\simeq P_{2}^{\rm ap}\equiv\frac{g_{\rm dof}e^{\beta\mu}{\cal
V}_{3}}{4\pi^{2}\beta^{4}{\cal V}_{2}}Q_{2},$ $\displaystyle{\cal S}_{\rm
rad}$ $\displaystyle\simeq{\cal S}^{\rm ap}_{\rm rad}\equiv\frac{g_{\rm
dof}{\cal V}_{3}}{2\pi^{2}\beta^{3}}\left\\{\pm 8{\rm Li}_{4}(\pm
e^{\beta\mu})+e^{\beta\mu}(4Q_{1}+Q_{2})-\beta\mu\left(\pm 2{\rm Li}_{3}(\pm
e^{\beta\mu})+e^{\beta\mu}Q_{1}\right)\right\\},$ (3.15)

where the arguments of $Q_{i}$ ($i=1,2,3$) are $\beta\sqrt{4\pi/{\cal
V}_{2}}$, and

$\displaystyle Q_{2}(x)$
$\displaystyle\equiv\sum_{l=1}^{\infty}x^{3}l^{3/2}(l+1)^{3/2}(2l+1)K_{1}\left(x\sqrt{l(l+1)}\right).$
(3.16)

Here we have used that

$\displaystyle Q_{1}^{\prime}(x)$ $\displaystyle=-\frac{Q_{2}(x)}{x},$ (3.17)

which follows from the formula for the modified Bessel functions:

$\displaystyle K^{\prime}_{\nu}(z)=-K_{\nu-1}(z)-\frac{\nu}{z}K_{\nu}(z).$
(3.18)

Note that $p_{\rm rad,3}$ and $p_{\rm rad,2}$ in (2.22) should be identified
as

$\displaystyle p_{\rm rad,3}$ $\displaystyle=\frac{P_{3}}{{\cal V}_{2}}\simeq
p_{\rm rad,3}^{\rm ap}\equiv\frac{P_{3}^{\rm ap}}{{\cal V}_{2}},$
$\displaystyle p_{\rm rad,2}$ $\displaystyle=\frac{P_{2}}{{\cal V}_{3}}\simeq
p_{\rm rad,2}^{\rm ap}\equiv\frac{P_{2}^{\rm ap}}{{\cal V}_{3}}.$ (3.19)

From (3.15), we find that 111 From (3.7), (3.9) and (3.10), the first relation
also holds for the exact expressions:  $\displaystyle\rho_{\rm rad}$
$\displaystyle=3p_{\rm rad,3}+2p_{\rm rad,2}.$

$\displaystyle\rho_{\rm rad}^{\rm ap}$ $\displaystyle=3p_{\rm rad,3}^{\rm
ap}+2p_{\rm rad,2}^{\rm ap},$ $\displaystyle s^{\rm ap}_{\rm rad}$
$\displaystyle\equiv\frac{{\cal S}^{\rm ap}_{\rm rad}}{{\cal
V}_{5}}=\beta\left\\{(4-\beta\mu)p_{\rm rad,3}^{\rm ap}+2p_{\rm rad,2}^{\rm
ap}\right\\}.$ (3.20)

The (radiation) energy density for the 4D effective theory is

$\displaystyle\rho_{\rm rad}^{\rm(4D)}$ $\displaystyle=\rho_{\rm rad}{\cal
V}_{2}.$ (3.21)

In the case that the energy density is dominated by $\rho_{\rm rad}$ and
$p_{\rm rad,3}^{\rm ap}\gg p_{\rm rad,2}^{\rm ap}$, we have the usual equation
of state in the 3D space:

$\displaystyle w_{\rm rad}\equiv\frac{P_{3}}{\rho_{\rm
rad}^{\rm(4D)}}=\frac{p_{\rm rad,3}}{\rho_{\rm rad}}\simeq\frac{p_{\rm
rad,3}^{\rm ap}}{\rho_{\rm rad}^{\rm ap}}\simeq\frac{1}{3}.$ (3.22)

If the pressure is isotropic in the whole 5D space, i.e., $p_{\rm rad,3}^{\rm
ap}\simeq p_{\rm rad,2}^{\rm ap}$, this ratio becomes

$\displaystyle w_{\rm rad}$ $\displaystyle\simeq\frac{1}{5}.$ (3.23)

Namely, $w_{\rm rad}^{-1}$ measures the dimension of the space that the
radiation feels.

Using (3.15) and (3.20), this ratio is expressed as

$\displaystyle w_{\rm rad}^{-1}$ $\displaystyle\simeq 3+\frac{2p_{\rm
rad,2}^{\rm ap}}{p_{\rm rad,3}}=3+\frac{Q_{2}(x)}{\pm 2e^{-\beta\mu}{\rm
Li}_{4}(\pm e^{\beta\mu})+Q_{1}(x)},$ (3.24)

where $x=\beta\sqrt{4\pi/{\cal V}_{2}}=\beta/b$. In the case that $\beta\mu\ll
1$ and negligible, $w_{\rm rad}^{-1}$ is a function of only $\beta/b$, which
is plotted in Fig. 1.222 We can numerically check that the approximate
expression (3.24) well agrees with the exact expression used by (3.9) and
(3.10) even for a small $\beta/b$.  From this plot, we can see that the
radiation feels almost all space dimensions when the extra space is large or
the temperature is high ($\beta/b\ll 1$), but eventually feels only the non-
compact 3D space at low temperature after the compact space is stabilized to a
finite value ($\beta/b\gg 1$). However, if the compact space $S^{2}$ continues
to expand and the ratio $\beta/b$ remains to be ${\cal O}(1)$, the value
$w_{\rm rad}^{-1}$ takes some intermediate value between 3 and 5.

Figure 1: The ratio $w_{\rm rad}^{-1}$ in (3.24). The red solid line
corresponds to the bosonic case, while the blue dashed line corresponds to the
fermionic case.

In order to check the accuracy of the approximation in (3.15), we define the
ratios:

$\displaystyle{\cal R}_{P_{3}}$ $\displaystyle\equiv\frac{P_{3}}{P_{3}^{\rm
ap}},\;\;\;\;\;{\cal R}_{P_{2}}\equiv\frac{P_{2}}{P_{2}^{\rm ap}}.$ (3.25)

In the case that $\beta\mu\ll 1$ and negligible, we can see from (3.9), (3.10)
and (3.15) that they are functions of only $\beta\sqrt{4\pi/{\cal
V}_{2}}=\beta/b$. Namely, we have

$\displaystyle{\cal R}_{P_{3}}(x)\equiv$ $\displaystyle\frac{1}{\pm 2{\rm
Li}_{4}(\pm
1)+Q_{1}(x)}\sum_{l=0}^{\infty}\frac{2l+1}{3}\int_{0}^{\infty}dk\frac{k^{4}}{\sqrt{k^{2}+x^{2}l(l+1)}\left(e^{\sqrt{k^{2}+x^{2}l(l+1)}}\mp
1\right)},$ $\displaystyle{\cal R}_{P_{2}}(x)\equiv$
$\displaystyle\frac{x^{2}}{Q_{2}(x)}\sum_{l=1}^{\infty}l(l+1)(2l+1)\int_{0}^{\infty}dk\;\frac{k^{2}}{\sqrt{k^{2}+x^{2}l(l+1)}\left(e^{\sqrt{k^{2}+x^{2}l(l+1)}}\mp
1\right)},$ (3.26)

where $x\equiv\beta/b$. Note that

$\displaystyle{\rm Li}_{4}(1)=\zeta(4),\;\;\;\;\;{\rm
Li}_{4}(-1)=-\frac{7}{8}\zeta(4).$ (3.27)

Fig. 2 shows the ratios in (3.25) as functions of $\beta/b$. From these plots,
we can see that the approximated quantities in (3.15) can be used for
$\beta/b\gtrsim{\cal O}(1)$ with good accuracy.

Figure 2: The ratios ${\cal R}_{P_{3}}$ (left plot) and ${\cal R}_{P_{2}}$
(right plot) in (3.26). The upper (red) solid lines correspond to the bosonic
case, while the lower (blue) dashed line to the fermionic case.

###  3.2 Conservation law and entropy

The energy-momentum conservation law is

$\displaystyle\nabla_{M}T^{M}_{\;\;N}\equiv\partial_{M}T^{M}_{\;\;N}+\Gamma^{M}_{\;\;ML}T^{L}_{\;\;N}-\Gamma^{L}_{\;\;MN}T^{M}_{\;\;L}=0,$
(3.28)

where

$\displaystyle T^{t}_{\;\;t}$
$\displaystyle=\frac{1}{2}\dot{\sigma}^{2}+\frac{e^{\sigma}}{8b^{4}}+2e^{-\sigma}+V_{\rm
stab}(\sigma)+\rho_{\rm rad}\equiv\rho_{\rm tot},$ $\displaystyle
T^{i}_{\;\;j}$
$\displaystyle=\delta^{i}_{\;\;j}\left\\{-\frac{1}{2}\dot{\sigma}^{2}+\frac{e^{\sigma}}{8b^{4}}+2e^{-\sigma}+V_{\rm
stab}(\sigma)-p_{\rm rad,3}\right\\}\equiv-\delta^{i}_{\;\;j}p_{\rm tot,3},$
$\displaystyle T^{4}_{\;\;4}$
$\displaystyle=T^{5}_{\;\;5}=-\frac{1}{2}\dot{\sigma}^{2}-\frac{e^{\sigma}}{8b^{4}}+2e^{-\sigma}+V_{\rm
stab}(\sigma)-p_{\rm rad,2}\equiv-p_{\rm tot,2}.$ (3.29)

From the case of $N=t$ in (3.28), we have 333 This is also obtained from the
Einstein equations in (4.1).

$\displaystyle\dot{\rho}_{\rm tot}+\frac{3\dot{a}}{a}\left(\rho_{\rm
tot}+p_{\rm tot,3}\right)+\frac{2\dot{b}}{b}\left(\rho_{\rm tot}+p_{\rm
tot,2}\right)=0,$ (3.30)

where the dot denotes the time derivative. The other components hold
trivially, and do not provide any information. The conservation law (3.30) is
rewritten as

$\displaystyle\frac{d}{dt}\left(\rho_{\rm tot}{\cal V}_{5}\right)$
$\displaystyle=-p_{\rm tot,3}{\cal V}_{2}\frac{d{\cal V}_{3}}{dt}-p_{\rm
tot,2}{\cal V}_{3}\frac{d{\cal V}_{2}}{dt}.$ (3.31)

Since $U_{\rm tot}\equiv\rho_{\rm tot}{\cal V}_{5}$ is the total energy in the
5D comoving volume and

$\displaystyle dU_{\rm tot}$ $\displaystyle=\frac{1}{\beta}d{\cal S}_{\rm
tot}-P_{3}d{\cal V}_{3}-P_{2}d{\cal V}_{2}+\mu dN,$ (3.32)

where $N$ is the particle number, the total entropy in the 5D comoving volume
${\cal S}_{\rm tot}$ is conserved when we neglect the chemical potential
$\mu$.

$\displaystyle\dot{\cal S}_{\rm tot}$ $\displaystyle=0.$ (3.33)

By using the dilaton equation of motion, the equation (3.30) is reduced to

$\displaystyle\dot{\rho}_{\rm
rad}+\left(\frac{3\dot{a}}{a}+\frac{2\dot{b}}{b}\right)\rho_{\rm
rad}+\frac{3\dot{a}}{a}p_{\rm rad,3}+\frac{2\dot{b}}{b}p_{\rm rad,2}$
$\displaystyle=0.$ (3.34)

Thus, the conservation law holds only for the radiation. Namely, the radiation
part of the entropy ${\cal S}_{\rm rad}$ is individually conserved when $\mu$
is negligible.

$\displaystyle\dot{\cal S}_{\rm rad}$ $\displaystyle=0.$ (3.35)

Plugging the expressions in (3.15) into (3.34), we obtain

$\displaystyle\frac{\dot{\beta}}{\beta}\left\\{\pm 24{\rm Li}_{4}(\pm
e^{\beta\mu})+e^{\beta\mu}\left(12Q_{1}+5Q_{2}+Q_{3}\right)\right.$
$\displaystyle\hskip 14.22636pt\left.-\beta\mu\left(\pm 6{\rm Li}_{3}(\pm
e^{\beta\mu})+e^{\beta\mu}(3Q_{1}+Q_{2})\right)\right\\}$ $\displaystyle=$
$\displaystyle\frac{3\dot{a}}{a}\left\\{\pm 8{\rm Li}_{4}(\pm
e^{\beta\mu})+e^{\beta\mu}(4Q_{1}+Q_{2})\right\\}+\frac{\dot{b}}{b}e^{\beta\mu}(2Q_{2}+Q_{3}),$
(3.36)

where

$\displaystyle Q_{3}(x)$
$\displaystyle\equiv\sum_{l=1}^{\infty}x^{4}l^{2}(l+1)^{2}(2l+1)K_{0}\left(x\sqrt{l(l+1)}\right).$
(3.37)

We have used that

$\displaystyle Q_{2}^{\prime}(x)$
$\displaystyle=\frac{1}{x}\left\\{2Q_{2}(x)-Q_{3}(x)\right\\},$ (3.38)

which is followed from the formula (3.18). Eq.(3.36) is the evolution equation
for the inverse temperature $\beta$.

###  3.3 Approximation of $Q_{i}$ ($i=1,2,3$)

The definitions of the functions in (3.14), (3.16) and (3.37) involve infinite
sums for $l$. In fact, we need to sum the summands up to large numbers of $l$
in order to reproduce the correct values of $Q_{i}(x)$ $(i=1,2,3)$ for $x\ll
1$ with high accuracy. However, the contributions from large $l$ become less
important as $x$ increases. Thus, we approximate them by the following
functions:

$\displaystyle Q_{1}^{\rm ap}(x)$
$\displaystyle\equiv\begin{cases}q_{1}(x)&(0<x<1.0)\\\
\tilde{Q}_{1}(x;11)&(1.0\leq x<2.0)\\\ \tilde{Q}_{1}(x;5)&(2.0\leq x<5.0)\\\
\tilde{Q}_{1}(x;2)&(x\geq 5.0)\end{cases},$ $\displaystyle Q_{2}^{\rm ap}(x)$
$\displaystyle\equiv\begin{cases}q_{2}(x)&(0<x<2.0)\\\
\tilde{Q}_{2}(x;6)&(2.0\leq x<5.0)\\\ \tilde{Q}_{2}(x;2)&(x\geq
5.0)\end{cases},$ $\displaystyle Q_{3}^{\rm ap}(x)$
$\displaystyle\equiv\begin{cases}q_{3}(x)&(0<x<3.0)\\\
\tilde{Q}_{3}(x;5)&(3.0\leq x<6.0)\\\ \tilde{Q}_{3}(x;2)&(x\geq
6.0)\end{cases},$ (3.39)

where

$\displaystyle q_{1}(x)$ $\displaystyle\equiv
16.0x^{-2}+0.00869x^{-1}-1.37+0.0842x-0.0477x^{2}+0.0340x^{3},$ $\displaystyle
q_{2}(x)$ $\displaystyle\equiv
32.0x^{-2}+0.0302x^{-1}-0.120+0.208x-0.229x^{2}+0.0322x^{3},$ $\displaystyle
q_{3}(x)$ $\displaystyle\equiv
128x^{-2}+0.395x^{-1}-0.946+0.836x-0.221x^{2}-0.00938x^{3},$ (3.40)

and

$\displaystyle\tilde{Q}_{1}(x;l_{\rm max})$ $\displaystyle=\sum_{l=1}^{l_{\rm
max}}x^{2}l(l+1)(2l+1)K_{2}\left(x\sqrt{l(l+1)}\right),$
$\displaystyle\tilde{Q}_{2}(x;l_{\rm max})$ $\displaystyle=\sum_{l=1}^{l_{\rm
max}}x^{3}l^{3/2}(l+1)^{3/2}(2l+1)K_{1}\left(x\sqrt{l(l+1)}\right),$
$\displaystyle\tilde{Q}_{3}(x;l_{\rm max})$ $\displaystyle=\sum_{l=1}^{l_{\rm
max}}x^{4}l^{2}(l+1)^{2}(2l+1)K_{0}\left(x\sqrt{l(l+1)}\right).$ (3.41)

Fig. 3 shows the profiles of $x^{2}Q_{i}(x)$ ($i=1,2,3$) and the ratios
$Q_{i}^{\rm ap}(x)/Q_{i}(x)$. The plots shows the above functions well
approximate $Q_{i}(x)$ with the errors less than $10^{-3}$. We will use these
approximate functions for the numerical calculations.

Figure 3: The profiles of $x^{2}Q_{i}(x)$ ($i=1,2,3$) (left plots), and the
ratios to their approximate functions defined in (3.39) (right plots). We have
used $\tilde{Q}_{i}(x;300)$ for $Q_{i}(x)$ ($i=1,2,3$).

###  3.4 Effective potential

Before going to analyze the time evolution of the 6D thermal background, let
us discuss the stabilization of the compact space $S^{2}$. As we mentioned at
the end of Sec. 2.2, the dilaton potential (2.3) does not stabilize the radius
of $S^{2}$. To fix it, we add the stabilization potential $V_{\rm
stab}(\sigma)$ to (2.3). Once the dilaton $\sigma$ is stabilized to a finite
value, the radion $b$ is also fixed through the relation (2.15).

We expect that the stabilization potential $V_{\rm stab}(\sigma)$ would be
generated by dilaton-dependent quantum corrections allowed by the gauged
$U(1)_{R}$ symmetry. It is quite important to derive $V_{\rm stab}(\sigma)$
from the string theory, but still a challenging issue. We leave the derivation
of 6D gauged supergravity with the stabilization potential for a future
work.444For earlier studies of the realization of lower-dimensional gauged
supergravities from the string theory, see e.g., Ref. [26]. The stabilization
of moduli fields was also discussed in the context of 4D low-energy effective
field theory[25].

For the purpose of discussing the time evolution of the 5D space around and
after the moduli stabilization, we do not need to know the complete functional
form of $V_{\rm stab}(\sigma)$. The only relevant properties of $V_{\rm
stab}(\sigma)$ are the position of the minimum and the curvature around it.
Thus, we simply introduce the following form of $V_{\rm stab}(\sigma)$ as an
approximation around the potential minimum.

$\displaystyle V_{\rm
stab}(\sigma)=\frac{m^{2}}{2}\left(\sigma-\sigma_{\ast}\right)^{2},$ (3.42)

where $m$ and $\sigma_{\ast}$ are real parameters.

The KK mass scale (or the compactification scale) is characterized by

$\displaystyle m_{\rm KK}$ $\displaystyle\equiv\frac{1}{b}.$ (3.43)

From (3.5), the first excited KK mass is $\sqrt{2}m_{\rm KK}$. Thus, there are
two possible cases:

  * •

$m\gtrsim m_{\rm KK}$ (The moduli are already fixed in the 4D effective
theory.)

  * •

$m\ll m_{\rm KK}$ (The moduli stabilization can be treated in the 4D effective
theory.)

In the latter case, the dynamics of the moduli stabilization can be described
within the 4D effective theory. From the 6D Lagrangian (2.1) and the potential
(3.42), the potential for the moduli is described by

$\displaystyle V_{\rm mod}(b,\sigma)$
$\displaystyle=-\frac{1}{2}g^{mn}R_{mn}+\frac{g^{2}e^{\sigma}}{4}F^{mn}F_{mn}+2e^{-\sigma}+V_{\rm
stab}(\sigma)$
$\displaystyle=-b^{-2}+\frac{e^{\sigma}}{8b^{4}}+2e^{-\sigma}+\frac{m^{2}}{2}\left(\sigma-\sigma_{\ast}\right)^{2}$
$\displaystyle=2e^{-\sigma}\left(1-\frac{e^{\sigma}}{4b^{2}}\right)^{2}+\frac{m^{2}}{2}(\sigma-\sigma_{*})^{2}.$
(3.44)

Minimizing this potential, we find that both $b$ and $\sigma$ are stabilized
at

$\displaystyle b_{\rm vac}$
$\displaystyle=\frac{e^{\sigma_{\ast}/2}}{2},\;\;\;\;\;\;\sigma_{\rm
vac}=\sigma_{\ast}.$ (3.45)

Note that the potential energy at the minimum (3.45) is zero. Thus the
geometry approaches the Minkowski spacetime at late times.

Recall that one direction in the moduli space $\\{b,\sigma\\}$, which is
specified by (2.15), is already stabilized in the original Salam-Sezgin model.
Now the other direction is also fixed by the stabilization potential $V_{\rm
stab}(\sigma)$ whose typical scale is characterized by $m$. We assume that
$V_{\rm stab}(\sigma)$ is generated at some scale well below the 6D Planck
mass scale by some mechanism, i.e., $m\ll 1$. Since we are interested in the
behavior around the vacuum values (3.45), the size modulus $b$ takes values of
${\cal O}(b_{\rm vac})$. Hence, the relation $m\ll m_{\rm KK}$ indicates that
$m^{2}b_{\rm vac}^{2}=m^{2}e^{\sigma_{\ast}}/4\ll 1$. When $m\gtrsim m_{\rm
KK}$, the moduli stabilization process should be analyzed in the context of
the full 6D theory.

Fig. 4 shows the moduli potential $V_{\rm mod}(b,\sigma)$ for different values
of $m^{2}b_{\rm vac}$. In the left plot, the stabilized values of the moduli
are $(b_{\rm vac},\sigma_{\rm vac})=(10.0,6.0)$, and $m^{2}b_{\rm
vac}^{2}=0.01\ll 1$. In this case, the two directions corresponding to the
mass eigenvalues have hierarchically different curvatures. In the right plot,
the minimum is $(b_{\rm vac},\sigma_{\rm vac})=(100,10.6)$, and $m^{2}b_{\rm
vac}^{2}=1.0$. So the 4D effective theory analysis cannot be used in this
case.

Figure 4: The moduli potential $V_{\rm mod}(b,\sigma)$ with
$(m,\sigma_{\ast})=(10^{-2},6.0)$ (left plot) and
$(m,\sigma_{\ast})=(10^{-2},10.6)$ (right plot). For the left plot, the
minimum is $(B,\sigma)=(2.3,6.0)$, and $m^{2}b_{\rm vac}^{2}=0.01$. For the
right plot, they are $(4.6,10.6)$ and 1.0, respectively.

Around the minimum, the moduli potential $V_{\rm mod}$ has the form of

$\displaystyle V_{\rm mod}$ $\displaystyle=-\frac{1}{2}(\delta
b,\delta\sigma)\begin{pmatrix}64e^{-2\sigma_{\ast}}&-16e^{-\frac{3}{2}\sigma_{\ast}}\\\
-16e^{-\frac{3}{2}\sigma_{\ast}}&m^{2}+4e^{-\sigma_{\ast}}\end{pmatrix}\begin{pmatrix}\delta
b\\\ \delta\sigma\end{pmatrix}+\cdots,$ (3.46)

where $\delta b\equiv b-b_{\rm vac}$ and $\delta\sigma\equiv\sigma-\sigma_{\rm
vac}$, and the ellipsis denotes higher order terms in $\delta b$ or
$\delta\sigma$. When $b_{\rm vac}$ is large enough, the mixing between $\delta
b$ and $\delta\sigma$ is negligible as shown in Fig. 5. In this case, the
curvature at the potential minimum along the $\delta\sigma$ direction is
determined by the parameter $m$, while that along the $\delta b$ direction is
suppressed by $b_{\rm vac}^{-2}$.

Figure 5: The moduli potential $V_{\rm mod}(b,\sigma)$ with
$(m,\sigma_{\ast})=(10^{-2},15.2)$. The minimum is $(B,\sigma)=(6.8,15.2)$,
and $b_{\rm vac}=1000$.

##  4 Numerical analyses

###  4.1 Evolution equations

We are now ready for the numerical analysis of the background time evolution.
Including the radiation contribution to the energy-momentum tensor $T_{MN}$,
which is expressed in terms of the thermodynamic quantities in Sec. 3.1, the
equations of motion (2.4) with the background ansatz (2.16)-(2.18) are
summarized as follows:

$\displaystyle\frac{3\dot{a}^{2}}{a^{2}}+\frac{\dot{b}^{2}}{b^{2}}+\frac{6\dot{a}\dot{b}}{ab}+\frac{1}{b^{2}}-\frac{1}{2}\dot{\sigma}^{2}-\frac{e^{\sigma}}{8b^{4}}-2e^{-\sigma}-V_{\rm
stab}(\sigma)-\rho_{\rm rad}=0,$
$\displaystyle\frac{2\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}+\frac{2\ddot{b}}{b}+\frac{\dot{b}^{2}}{b^{2}}+\frac{4\dot{a}\dot{b}}{ab}+\frac{1}{b^{2}}+\frac{1}{2}\dot{\sigma}^{2}-\frac{e^{\sigma}}{8b^{4}}-2e^{-\sigma}-V_{\rm
stab}(\sigma)+p_{\rm rad,3}=0,$
$\displaystyle\frac{3\ddot{a}}{a}+\frac{3\dot{a}^{2}}{a^{2}}+\frac{\ddot{b}}{b}+\frac{3\dot{a}\dot{b}}{ab}+\frac{1}{2}\dot{\sigma}^{2}+\frac{e^{\sigma}}{8b^{4}}-2e^{-\sigma}-V_{\rm
stab}(\sigma)+p_{\rm rad,2}=0,$
$\displaystyle\ddot{\sigma}+\left(\frac{3\dot{a}}{a}+\frac{2\dot{b}}{b}\right)\dot{\sigma}+\frac{e^{\sigma}}{8b^{4}}-2e^{-\sigma}+\partial_{\sigma}V_{\rm
stab}(\sigma)=0,$ (4.1)

where $\rho_{\rm rad}$, $p_{\rm rad,3}$ and $p_{\rm rad,2}$ are given by
(3.7), (3.9) and (3.10) with (3.19), respectively. The first equation is the
$(t,t)$-component of the Einstein equation. Since this does not contain the
second order $t$-derivatives, it is regarded as a constraint on the initial
conditions of the time-evolution equations. The second and the third equations
come from the diagonal components for the 3D non-compact space and the $S^{2}$
space, respectively. The other components of the Einstein equation vanish. The
last equation is the dilaton equation of motion.

For the radiation energy density $\rho_{\rm rad}$ and the pressures $p_{\rm
rad,3}$ and $p_{\rm rad,2}$, we use the approximated forms in (3.15), and
assume that the chemical potential $\mu$ is negligible. Then, (4.1) and (3.36)
are rewritten as 555 The last equation in (4.2) can also be obtained from the
entropy-conservation condition (3.35) by neglecting the dependence of the
chemical potential $\mu$.

$\displaystyle\begin{split}\ddot{A}=&-\frac{1}{4}\left(9\dot{A}^{2}-\dot{B}^{2}+2\dot{A}\dot{B}-e^{-2B}+\frac{1}{2}\dot{\sigma}^{2}+\frac{3}{8}e^{\sigma-4B}-2e^{-\sigma}-V_{\rm
stab}(\sigma)-p_{\rm rad,3}^{\rm ap}+2p_{\rm rad,2}^{\rm
ap}\right),\end{split}$ $\displaystyle\ddot{B}=$
$\displaystyle-\frac{1}{4}\left(-3\dot{A}^{2}+7\dot{B}^{2}+6\dot{A}\dot{B}+3e^{-2B}+\frac{1}{2}\dot{\sigma}^{2}-\frac{5}{8}e^{\sigma-4B}-2e^{-\sigma}-V_{\rm
stab}(\sigma)+3p_{\rm rad,3}^{\rm ap}-2p_{\rm rad,2}^{\rm ap}\right),$
$\displaystyle\ddot{\sigma}=$
$\displaystyle-\left(3\dot{A}+2\dot{B}\right)\dot{\sigma}-\frac{1}{8}e^{\sigma-4B}+2e^{-\sigma}-\partial_{\sigma}V_{\rm
stab}(\sigma),$ $\displaystyle\frac{\dot{\beta}}{\beta}=$
$\displaystyle\frac{3\dot{A}\left\\{\pm 8{\rm Li}_{4}(\pm
1)+4Q_{1}+Q_{2}\right\\}+\dot{B}\left(2Q_{2}+Q_{3}\right)}{\pm 24{\rm
Li}_{4}(\pm 1)+12Q_{1}+5Q_{2}+Q_{3}},$ (4.2)

where the arguments of $Q_{i}$ ($i=1,2,3$) are $\beta e^{-B}$, and

$\displaystyle A$ $\displaystyle\equiv\ln a,\;\;\;\;\;B\equiv\ln b,$ (4.3)

and

$\displaystyle\rho_{\rm rad}^{\rm ap}$ $\displaystyle=\frac{g_{\rm
dof}e^{-2B}}{8\pi^{3}\beta^{4}}\left\\{\pm 6{\rm Li}_{4}(\pm
1)+3Q_{1}+Q_{2}\right\\},$ $\displaystyle p_{\rm rad,3}^{\rm ap}$
$\displaystyle=\frac{g_{\rm dof}e^{-2B}}{8\pi^{3}\beta^{4}}\left\\{\pm 2{\rm
Li}_{4}(\pm 1)+Q_{1}\right\\},\;\;\;\;\;p_{\rm rad,2}^{\rm ap}=\frac{g_{\rm
dof}e^{-2B}}{16\pi^{3}\beta^{4}}Q_{2},$ (4.4)

with the constraint:

$\displaystyle
3\dot{A}^{2}+\dot{B}^{2}+6\dot{A}\dot{B}+e^{-2B}-\frac{1}{2}\dot{\sigma}^{2}-\frac{1}{8}e^{\sigma-4B}-2e^{-\sigma}-V_{\rm
stab}(\sigma)-\rho_{\rm rad}^{\rm ap}=0.$ (4.5)

These are the evolution equations we will numerically solve in the following.

Notice that the above evolution equations do not depend on the non-derivative
value of $A$. Namely, if $A_{\rm sol}(t)$ is a solution, $A_{\rm
sol}(t)+A_{0}$ ($A_{0}$: constant) is also a solution. This is because only
the relative ratio of the scale factors $a(t)$ at different times is
physically meaningful.666 The value of $a$ at a specific time can be freely
changed by the 3D coordinate $x^{i}$ ($i=1,2,3$). In contrast, the value of
$b(t)$ has the physical meaning, i.e., the radius of the extra space
$S^{2}$.777 Recall that we have chosen the coordinate $y$ so that $r=1$.  In
fact, the evolution equations explicitly depend on the non-derivative $B(t)$.

###  4.2 Parameter choice and initial values

In the following numerical calculations, we choose the stabilization scale $m$
and the degrees of freedom for the radiation particles $g_{\rm dof}$ as

$\displaystyle m$ $\displaystyle=0.01\,\,\mbox{(except for
Fig.\,\ref{bsgm:m001})},\;\;\;\;\;g_{\rm dof}=100.$ (4.6)

The parameter $\sigma_{\ast}$ is used to control the stabilized moduli values
in (3.45).

The initial values at $t=0$ are chosen as

$\displaystyle\beta(0)$ $\displaystyle=10\,\,\mbox{(except for
Fig.\,\ref{plots:winv1-bt100})},$ $\displaystyle a(0)$
$\displaystyle=1,\;\;\;\;\;b(0)=b_{\rm
vac}-\delta,\;\;\;\;\;\sigma(0)=\sigma_{\rm vac}+\delta,\;\;\;\;\;$
$\displaystyle\dot{a}(0)$ $\displaystyle=\frac{1}{\sqrt{3}}\sqrt{V_{\rm
mod}(b,\sigma)+\rho_{\rm rad}^{\rm
ap}},\;\;\;\;\;\dot{b}(0)=\dot{\sigma}(0)=0,$ (4.7)

where $\\{b_{\rm vac},\sigma_{\rm vac}\\}$ are the stabilized moduli values
defined in (3.45), $V_{\rm mod}(b,\sigma)$ is defined in (3.44), and $\delta$
is chosen as 0.1. The value of $\dot{a}(0)$ is determined by the constraint
(4.5).

###  4.3 Numerical results

####  4.3.1 Moduli stabilization

Figs. 6 and 7 show the time evolution of the scale factors $\\{a,b\\}$, the
inverse temperature $\beta$, and the background dilaton value $\sigma$ for
various values of $m^{2}b_{\rm vac}^{2}$. From these plots, we can see that
the moduli $b$ and $\sigma$ settle down at the stabilized values until $t\sim
1000$ in all cases. Namely, the time scale required for the moduli
stabilization does not depend on the value of $\sigma_{\ast}$ (or $b_{\rm
vac}$). We can also see that the moduli oscillate around the stabilized values
in (3.45). This consists of two oscillations with different periodicities,
corresponding to the two eigenvalues of
$\partial_{\varphi}\partial_{\chi}V_{\rm mod}$ ($\varphi,\chi=b,\sigma$) at
the minimum of the potential. As shown in Fig. 4, these periodicities are
hierarchical when $m^{2}b_{\rm vac}^{2}\ll 1$ while they are comparable when
$m^{2}b_{\rm vac}^{2}={\cal O}(1)$.

Figure 6: Time evolution of the scale factors $\\{a,b\\}$, the inverse
temperature $\beta$, and the dilaton $\sigma$. The value of $\sigma_{\ast}$ is
chosen from the bottom to the top as $\sigma_{\ast}=3.7$, 4.8, 6.0, which
correspond to $m^{2}b_{\rm vac}^{2}=0.001$, 0.003, 0.01, respectively. The
other parameters are chosen as (4.7). Figure 7: Time evolution of the scale
factors $\\{a,b\\}$, the inverse temperature $\beta$, and the dilaton
$\sigma$. The value of $\sigma_{\ast}$ is chosen from the bottom to the top as
$\sigma_{\ast}=7.1$, 8.3, 9.4, which correspond to $m^{2}b_{\rm
vac}^{2}=0.03$, 0.1, 0.3, respectively. The other parameters are chosen as
(4.7).

Fig. 8 shows the time evolution of the moduli in the case of $m=0.001$. We can
see that the time scale of the moduli stabilization is 10 times larger than
the case of $m=0.01$ in Figs. 6 and 7. This indicates that this time scale
$t_{\rm stab}$ is estimated as about $10m^{-1}$.

Figure 8: Time evolution of the scale factor $b$ and the dilaton $\sigma$ in
the case of $m=0.001$. The value of $\sigma_{\ast}$ is chosen from the bottom
to the top as $\sigma_{\ast}=7.1$, 8.3, 9.4, which correspond to $m^{2}b_{\rm
vac}^{2}=3\times 10^{-4}$, $1\times 10^{-3}$, $3\times 10^{-3}$, respectively.
The other parameters are chosen as (4.7).

We should note that the above properties are not sensitive to the choice of
the initial values. In fact, similar results are obtained even if we choose
$\delta=0$ in (4.7) since $\dot{a}(0)$ is non-zero due to the existence of the
radiation.

####  4.3.2 Energy densities of radiation and moduli oscillation

At late times when $b$ is stabilized, its time derivative can be neglected.
Then, the Einstein equations:

$\displaystyle R_{tt}-\frac{1}{2}g_{tt}R$ $\displaystyle=T_{tt},$
$\displaystyle R_{ij}-\frac{1}{2}g_{ij}R$ $\displaystyle=T_{ij},$ (4.8)

which correspond to the first and the second equations in (4.1), become

$\displaystyle\frac{3\dot{a}^{2}}{a^{2}}+\frac{1}{b^{2}}$
$\displaystyle=\frac{1}{2}\dot{\sigma}^{2}+\frac{e^{\sigma}}{8b^{4}}+2e^{-\sigma}+V_{\rm
stab}(\sigma)+\rho_{\rm rad}=\rho_{\rm tot},$
$\displaystyle\frac{2\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}+\frac{1}{b^{2}}$
$\displaystyle=-\frac{1}{2}\dot{\sigma}^{2}+\frac{e^{\sigma}}{8b^{4}}+2e^{-\sigma}+V_{\rm
stab}(\sigma)-p_{\rm rad,3}=-p_{\rm tot,3}.$ (4.9)

The ‘constant’ terms $1/b^{2}$ in the LHSs come from the scalar curvature (in
the second terms in LHSs of (4.8)). For the purpose of seeing the
correspondence to the 4D Einstein equations, it is convenient to redefine the
energy density and the 3D pressure as

$\displaystyle\tilde{\rho}_{\rm tot}$ $\displaystyle\equiv\rho_{\rm
tot}-\frac{1}{b^{2}}=\frac{1}{2}\dot{\sigma}^{2}+V_{\rm
mod}(b,\sigma)+\rho_{\rm rad},$ $\displaystyle\tilde{p}_{\rm tot,3}$
$\displaystyle\equiv p_{\rm
tot,3}+\frac{1}{b^{2}}=\frac{1}{2}\dot{\sigma}^{2}-V_{\rm
mod}(b,\sigma)+p_{\rm rad,3},$ (4.10)

where $V_{\rm mod}(b,\sigma)$ is defined in (3.44). Then (4.9) has the same
form as the 4D Friedmann equations. In the following, we use these quantities
even when $b$ has not been stabilized yet, i.e., it is time-dependent.

Fig. 9 shows the time evolution of $\rho_{\rm tot}$ defined in (3.29). We can
see that it converges to a constant value at the same time scale as the moduli
stabilization (i.e., $\sim 10m^{-1}$). We have checked that the radiation
contribution $\rho_{\rm rad}$ is rapidly damped and becomes negligible at the
moduli-stabilized time $t_{\rm stab}$. The constant value which $\rho_{\rm
tot}$ approach is read off as $1/b_{\rm vac}^{2}$. Hence $\tilde{\rho}_{\rm
tot}$ is damped to zero at late times.

Figure 9: Time evolution of $\rho_{\rm tot}$ defined in (3.29). The parameter
is chosen from the bottom to the top as $\sigma_{\ast}=3.7$, 4.8, 6.0, which
correspond to $m^{2}b_{\rm vac}^{2}=0.001$, 0.003, 0.01 in the left plot, and
$\sigma_{\ast}=7.1$, 8.3, 9.4, which correspond to $m^{2}b_{\rm
vac}^{2}=0.03$, 0.1, 0.3 in the right plot, respectively. The other parameters
are chosen as (4.7).

Fig. 10 shows the time evolution of the ratio of the radiation energy density
$\rho_{\rm rad}$ to the total energy density $\tilde{\rho}_{\rm tot}$. In the
case that $m^{2}b_{\rm vac}^{2}\ll 1$, the moduli stabilization can be
described in the 4D effective theory. In such a case, it is known that the
moduli oscillation around the potential minimum behaves as a non-relativistic
matter, and thus its contribution to the energy density dominates over that of
the radiation because $\rho_{\rm mod}\propto a^{-3}$ and $\rho_{\rm
rad}\propto a^{-4}$, where $\rho_{\rm mod}\equiv\tilde{\rho}_{\rm
tot}-\rho_{\rm rad}$ is the contribution of the moduli to the energy density.
In fact, we can see that $\rho_{\rm rad}/\tilde{\rho}_{\rm tot}$ is rapidly
damped when $\sigma_{\ast}=6.0$, i.e., $m^{2}b_{\rm vac}^{2}=0.01$. However,
in the case that $m^{2}b_{\rm vac}^{2}\gtrsim 1$, the situation changes. We
cannot describe the moduli stabilization process from the viewpoint of the 4D
effective theory any more. From the plots, we can see that the radiation
contriubtion to the total energy density becomes non-negligible as the
stabilized size of $S^{2}$ is increased (the left plot in Fig. 10). For
$b_{\rm vac}=1000$, for example, the radiation energy density oscillates but
remains to be larger than 40% of the total one until about $100t_{\rm stab}$.

Figure 10: Time evolution of the ratio $\rho_{\rm rad}/\tilde{\rho}_{\rm tot}$
for various values of $\sigma_{*}$. The parameter is chosen from the bottom to
the top as $\sigma_{\ast}=6.0$ (red), 8.3 (orange), 10.6 (green), which
correspond to $b_{\rm vac}=10$, 31.7, 100 in the left plot, and
$\sigma_{\ast}=12.9$ (blue), 15.2 (purple thin), 16.0 (brown fine dashed),
16.8 (black coarse dashed), which correspond to $b_{\rm
vac}=316,1000,1490,2223$ in the right plot, respectively. The other parameters
are chosen as (4.7).

For $b_{\rm vac}>1000$, the lower values of the oscillation for this ratio
start to decrease, and $\rho_{\rm rad}$ becomes negligible at late times when
$b_{\rm vac}\geq 2220$ (the right plot in Fig. 10). This is because the
modulus $b$ is no longer stabilized and continues to increase for $b_{\rm
vac}\geq 2220$ (see Fig. 11). Namely, $\tilde{\rho}_{\rm tot}$ is not damped
to zero, but approaches the value $1/b_{\rm vac}^{2}$. Note that the potential
curvature along the $b$-direction at the minimum $\partial_{b}^{2}V_{\rm
mod}(b_{\rm vac},\sigma_{\rm vac})$ becomes smaller as $b_{\rm vac}$
increases, as mentioned at the end of Sec. 3. Thus if $b_{\rm vac}$ is large
enough, the modulus $b$ cannot fall into the bottom of the potential $V_{\rm
mod}$ due to the Hubble friction.888 The dilaton $\sigma$ falls into the
potential minimum because $\partial_{\sigma}^{2}V_{\rm mod}(b_{\rm
vac},\sigma_{\rm vac})$ is bounded from below by $m^{2}$.  In such a case, the
non-vanishing vacuum energy density forces to expand the universe. As a
result, the size modulus $b$ increases further. We should note that this
behavior depends on the structure of the moduli potential, and is not a
general property of the moduli stabilization. If we assume a stabilization
mechanism that also works for larger stabilized value $b_{\rm vac}$, it is
expected that the situation such that $\rho_{\rm rad}/\tilde{\rho}_{\rm
tot}\simeq 1$ is realized.

Figure 11: Time evolutions of $b$ (left plot) and $\sigma$ (right plot). The
parameter $\sigma_{*}$ is chosen from the bottom to the top as
$\sigma_{\ast}=15.2$ (purple), 16.0 (brown fine dashed), 16.7 (gray fine
dashed), 16.8 (black coarse dashed), which correspond to $b_{\rm
vac}=1000,1490,2115,2223$, respectively. The other parameters are chosen as
(4.7).

In the standard 4D cosmology, the 3D space expands as
$a(t)\propto(t-t_{0})^{2/3}$ for the matter-dominated universe while
$a(t)\propto(t-t_{0})^{1/2}$ for the radiation dominated one. Due to the non-
negligible contribution of the radiation to the energy density, the 3D space
expands slower than the matter-dominated case. In order to see a rough
behavior of the scale factor $a(t)$, we fit its numerical data with the
function:

$\displaystyle a(t)$ $\displaystyle=N_{a}(t-t_{0})^{u},$ (4.11)

where $N_{a}$, $u$ and $t_{0}$ are real constants. The fitted values for
various $b_{\rm vac}$ are listed in Table 1. We can see that the power $u$ is
close to $2/3$ when the size of the compact space is small. However, its value
decreases until $b_{\rm vac}=316$, and then turns to increase due to the
destabilization of the size modulus $b$ mentioned above.

$b_{\rm vac}$ | $N_{a}$ | $u$ | $t_{0}$
---|---|---|---
10 | 0.09825 | 0.6633 | 210.8
31.6 | 0.04073 | 0.6590 | -146.4
100 | 0.03466 | 0.6243 | -325.1
316 | 0.03817 | 0.6061 | -264.1
500 | 0.03738 | 0.6122 | -181.8
1000 | 0.03477 | 0.6405 | 270.7
Table 1: The fitted values of $N_{a}$, $u$ and $t_{0}$ in (4.11) for various
values of $b_{\rm vac}$.

####  4.3.3 Equation of state for radiation

Finally, we will see the equation of state for the radiation $w_{\rm rad}$. As
mentioned in Sec. 3.1, $w_{\rm rad}^{-1}$ measures the dimension of the space
that the radiation feels. Fig. 12 shows the time evolution of $w_{\rm
rad}^{-1}$ for various values of $\sigma_{\ast}$ (or $b_{\rm vac}$). From
these plots, we can see that it takes a longer time until $w_{\rm rad}^{-1}$
settles down at the 3D space value 3 as $b_{\rm vac}$ increases. Namely, when
the size of $S^{2}$ is set to be large ($mb_{\rm vac}\gg 1$), the radiation
feels the extra dimensions for a long time even after the moduli are
stabilized. Also from this aspect, the 4D effective theory should not be used
to analyze the stabilization procedure in such a case.

Figure 12: Time evolution of $w_{\rm rad}^{-1}$ for various values of
$\sigma_{*}$. The parameter is chosen from the bottom to the top as
$\sigma_{\ast}=4.6$ (red), 6.0 (orange), 6.8 (green), 8.3 (blue), which
correspond to $b_{\rm vac}=5.0$, 10.0, 15.0, 31.7 in the left plot, and
$\sigma_{\ast}=6.8$ (green), 8.3 (blue), 10.6 (purple), which correspond to
$b_{\rm vac}=15.0$, 31.7, 100 in the right plot. The other parameters are
chosen as (4.7).

Since $w_{\rm rad}^{-1}$ is a monotonically decreasing function of $\beta/b$
(see Fig. 1), it approaches 3 more rapidly if we choose a larger initial value
of $\beta$. Fig. 13 shows the time evolution of $w_{\rm rad}^{-1}$ in the case
of $\beta(t=0)=100$.

Figure 13: Time evolution of $w_{\rm rad}^{-1}$ for various values of
$\sigma_{*}$ in the case of $\beta(t=0)=100$. The parameter is chosen from the
bottom to the top as $\sigma_{\ast}=8.3$ (blue), 10.6 (purple), 12.9 (gray)
which correspond to $b_{\rm vac}=31.7$, 100, 316, respectively. The other
parameters are chosen as (4.7).

###  4.4 4D Planck unit

So far we have worked in the 6D Planck unit $M_{6}=1$, where $M_{6}$ is the 6D
Planck mass. Since the 4D Planck mass $M_{4}$ is defined from the current
gravitational coupling after the size of the compact space has been
stabilized, the relation between $M_{4}$ and $M_{6}$ is given by

$\displaystyle M_{4}^{2}$ $\displaystyle=4\pi(b_{\rm vac}r)^{2}M_{6}^{4}=4\pi
b_{\rm vac}^{2}M_{6}^{2},$ (4.12)

where $b_{\rm vac}$ defined in (3.45) is the stabilized value of $b$. Note
that we have chosen the coordinate radius $r$ as the 6D Planck length, i.e.,
$r=1/M_{6}$. Namely, the values of all dimensionful quantities in the previous
subsections are measured in the unit of $M_{6}$. For example, since the
inverse temperature $\beta$, the stabilization mass scale $m$ and the dilaton
$\sigma$ have mass-dimensions $-1$, 1 and 1, respectively, they should be
understood as

$\displaystyle\beta$
$\displaystyle\to\frac{\beta}{M_{6}}=\beta\sqrt{\frac{4\pi b_{\rm
vac}^{2}}{M_{4}^{2}}}=\frac{2\sqrt{\pi}b_{\rm
vac}\beta}{M_{4}}=\frac{\beta^{(4)}}{M_{4}},$ $\displaystyle m$
$\displaystyle\to mM_{6}=\frac{mM_{4}}{2\sqrt{\pi}b_{\rm vac}}=m^{(4)}M_{4},$
$\displaystyle\sigma$ $\displaystyle\to\sigma M_{6}=\frac{\sigma
M_{4}}{2\sqrt{\pi}b_{\rm vac}}=\sigma^{(4)}M_{4},$ (4.13)

where $\beta^{(4)}$, $m^{(4)}$ and $\sigma^{(4)}$ are the values measured in
the unit of $M_{4}$. Namely, if we change the unit of the mass scale from
$M_{6}$ to $M_{4}$, their values are changed as 999 The parameter $\sigma_{*}$
denotes the value measured by $M_{6}$.

$\displaystyle\beta^{(4)}$ $\displaystyle=2\sqrt{\pi}b_{\rm
vac}\beta=\sqrt{\pi}e^{\sigma_{*}/2}\beta,$ $\displaystyle m^{(4)}$
$\displaystyle=\frac{m}{2\sqrt{\pi}b_{\rm
vac}}=\frac{e^{-\sigma_{*}/2}}{\sqrt{\pi}}m,$ $\displaystyle\sigma^{(4)}$
$\displaystyle=\frac{\sigma}{2\sqrt{\pi}b_{\rm
vac}}=\frac{e^{-\sigma_{*}/2}}{\sqrt{\pi}}\sigma.$ (4.14)

As for the dimensionless quantities such as the scale factors $a$ and $b$,
their values are unchanged. The physical radius of the compact space $S^{2}$
is

$\displaystyle br=\frac{b}{M_{6}}=\frac{2\sqrt{\pi}b_{\rm vac}}{M_{4}}b.$
(4.15)

At late times after the moduli are stabilized, this approaches

$\displaystyle\frac{R_{\rm phys}^{(4)}}{M_{4}}\equiv\frac{2\sqrt{\pi}b_{\rm
vac}^{2}}{M_{4}},$ (4.16)

where $R_{\rm phys}$ is the value of the stabilized radius measured by
$M_{4}$.

For the plots in the previous subsection, we can use the 3D scale factor
$a(t)$ instead of the time $t$.101010 Since we have chosen the initial value
as $a(0)=1$, this equals $e^{N(t)}$, where $N(t)$ is the e-folding number for
the expanding 3D space.  Fig. 14 shows the inverse temperature measured by
$M_{4}$, i.e., $\beta_{(4)}$, the scale factor for $S^{2}$, i.e., $b$, and the
ratio $\rho_{\rm rad}/\tilde{\rho}_{\rm tot}$ as functions of the 3D scale
factor $a$. The parameters are chosen as (4.6) and $\sigma_{\ast}=12.9$, which
correspond to

$\displaystyle m^{(4)}$ $\displaystyle=8.92\times 10^{-6},$ $\displaystyle
m_{\rm KK}^{(4)}$ $\displaystyle\equiv\frac{1}{R_{\rm
phys}^{(4)}}=\frac{1}{2\sqrt{\pi}b_{\rm
vac}^{2}}=\frac{2e^{-\sigma_{\ast}}}{\sqrt{\pi}}=2.82\times 10^{-6},$ (4.17)

and the initial values are chosen as (4.7). In particular, $\beta(0)=10$
corresponds to

$\displaystyle\beta^{(4)}(0)=1.12\times 10^{4}.$ (4.18)

Figure 14: The 3D scale factor $a$ as a function of $t$, and the 2D scale
factor $b$, the inverse temperature measured by $M_{4}$, i.e., $\beta^{(4)}$
and the ratio $\rho_{\rm rad}/\tilde{\rho}_{\rm tot}$ as functions of the 3D
scale factor $a$. The parameters are chosen as (4.17) and (4.18). Note that
$a=41$ corresponds to $t=10^{5}$.

From the plots, we can see that the modulus $b$ is stabilized in about 3
e-foldings for the 3D space expansion. The stabilization time scale for the
other modulus $\sigma$ is shorter than it when $m\gtrsim m_{\rm KK}$.

##  5 Conclusions

We investigated the time evolutions of a higher-dimensional space during the
process of the moduli stabilization, which is assumed to occur in the
radiation dominated era. In many of the previous related works, the analyses
have been performed within 4D effective field theories, e.g., [3, 4, 5, 6, 7,
8, 9]. This means that only restricted parameter spaces, in which the moduli
stabilization mass scale $m$ is smaller than the compactification scale
$m_{\rm KK}$, have been investigated. In this paper, we have studied the
spacetime evolution by solving the higher dimensional (background) field
equations in order to extend the analysis beyond the 4D effective theory. As a
previous research along this direction, the authors of Ref. [23] discussed the
spacetime evolution of the Salam-Sezgin model [19] in the presence of the
radiation. However, they have set the equation of state for the radiation to a
constant by hand. Besides, one combination of the moduli remains to be unfixed
in their model. Here, we have treated the energy density and the pressures as
independent thermodynamic quantities, and introduced the dilaton potential
$V_{\rm stab}(\sigma)$ in order to stabilize all the moduli.

We have checked that when the mass scale of the stabilization potential is
much smaller than the compactification scale ($m\ll m_{\rm KK}$), the moduli
oscillation around the potential minimum rapidly dominates the energy density
over the radiation, and the non-compact 3D space expands as the matter-
dominated universe $a(t)\sim t^{2/3}$. This is a well known result obtained
from the analysis of the 4D effective theory. In contrast, when $m\gtrsim
m_{\rm KK}$, the situation changes. In this case, the 4D effective theory
approach is no longer valid, and we have found that the energy density of the
radiation $\rho_{\rm rad}$ remains to be non-negligible for a long time even
after the moduli have been stabilized. In fact, the time scale for the moduli
stabilization is mainly determined by the mass scale of the stabilization
potential $V_{\rm stab}(\sigma)$, i.e., $t_{\rm stab}\sim 10m^{-1}$, while the
time necessary for the moduli oscillation to dominate the energy density is
determined by the stabilized size of the compact space $b_{\rm vac}$. For the
parameters that realize $m>m_{\rm KK}$, the latter can be much larger than the
former. As a result, the 3D space expands slower than the usual matter-
dominated universe. In such a situation, we have also checked that the
equation of state for the radiation $w_{\rm rad}$ remains smaller than the
3D-space value $1/3$ for much longer time than the moduli-stabilization time
scale $t_{\rm stab}$. Namely, the radiation feels the extra dimensions for a
while even after the moduli are fixed.

In our model, there is an upper bound on the stabilized size of $S^{2}$. This
stems from the specific structure of the stabilization potential we
introduced, and is not a general property of the moduli stabilization. Hence,
with an appropriate stabilization mechanism, we can realize a situation that
$m$ is arbitrarily larger than $m_{\rm KK}$. In such a case, there is enough
time for the deviation of the spacetime evolution from that of the
conventional 4D moduli oscillation to affect the current observable. This can
lead to some signals that suggest the existence of the extra dimensions. We
will discuss this possibility including the generalization of Salam-Sezgin
model [27, 28, 29] in our subsequent works.

In this work, we have focused on the effects of moduli stabilization on the
spacetime evolution. Thus, we just added the potential $V_{\rm stab}(\sigma)$
by hand. Of course, its origin has to be clarified when we construct a
complete model. Furthermore, the moduli fields will eventually decay into
matter fields, and their energy converts into the radiation. In conventional
scenarios, the moduli fields dominate the energy density of the universe.
However, our results suggest the possibility that there is no such a moduli-
dominated era before their decay to the radiation. In order to discuss the
full thermal history of the universe, we need to check whether this is the
case. We also leave these issues for the subsequent papers.

### Acknowledgements

H. O. was supported in part by JSPS KAKENHI Grant Number JP20K14477.

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