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\begin{document}
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\twocolumn[\hsize\textwidth\columnwidth\hsize\csname
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\title{D-braneworld cosmology}

\author{Tetsuya Shiromizu$^{(1,2)}$, Takashi Torii$^{(2)}$ and Tomoko Uesugi$^{(3)}$}


\address{$^{(1)}$ Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan}

\address{$^{(2)}$Advanced Research Institute for Science and Engineering,
Waseda University, Tokyo 169-8555, Japan}

\address{$^{(3)}$Institute of Humanities and Sciences and Department of Physics, 
Ochanomizu University, Tokyo 112-8610, Japan}



\date{\today}

\maketitle

%======================================%
%<<<<<<<<<<<<< ABSTRACT >>>>>>>>>>>>>>>%
%======================================%
\begin{abstract}
We discuss D-braneworld cosmology, that is, the brane is described by the 
Born-Infeld action. Compared with the usual Randall-Sundrum braneworld cosmology 
where the brane action is the Nambu-Goto one, we can see some 
drastic changes at the very early universe: (i)universe may experience the 
rapid accelerating phase 
(ii)the closed universe may avoid the initial 
singularity. We also briefly address the dynamics of the cosmology in the open 
string metric, which might be  favorer than the induced metric from the 
view point of the D-brane. 


\end{abstract}
\vskip2pc]

%\pacs{ }

\vskip1cm

%======================================%
%<<<<<<<<<<<< SECTION I  >>>>>>>>>>>>>>%
%======================================%
%\baselineskip25pt

\section{Introduction}

One of the motivation for the braneworld is originated by the genius of D-brane, that is, 
open strings describing the Standard Model particles stick to the brane. So if one is serious 
about this, we must employ the effective action for D-brane, Born-Infeld action, not 
the Nambu-Goto membrane action\cite{BI}. In the presence of the matters on the brane, 
the difference between the above two actions will appear. 

The simplest braneworld model was proposed by Randall and Sundrum\cite{RSI,RSII} and 
subsequently extended 
to the cosmology by many peoples\cite{Tess,Roy,cosmos}. 
However, the action for the brane is often assumed to be 
the Nambu-Goto action and the action for the matters on the brane is simply added by 
direct sum. In this paper we explore the cosmology on the D-brane (Related to the 
tachyon condensation, the tachyon matter on the brane has been 
also considered in the braneworld\cite{Mukohyama}). 
As seen soon, our starting point is the 5-dimensional Einstein-Hilbert action plus the  
Born-Infeld action (we call this D-braneworld). Then we consider the radiation 
dominated universe on the D-brane. In this situation the brane action is described by Born-Infeld 
one where the matter term is automatically included. 

Here reminded that 
there is non-trivial aspect for the interpretation on the D-brane. According to Seiberg and 
Witten\cite{SW}, the metric for the gauge field on the brane is given by $\stac{s}{g}_{\mu\nu}
=g_{\mu\nu}-(2\pi \alpha')^2(F^2)_{\mu\nu}$, not just the induced metric $g_{\mu\nu}$. $F$ 
represents the field strength of the gauge fields on the brane. 
For this we will discuss the cosmology in the both 
metric. See Ref.~\cite{Causal} for the causality issue.  

Another motivation to think of the D-braneworld is the magnetogenesis in the very early universe
(See Ref.~\cite{PMF} for the comprehensive review and references). 
The coherent magnetic field over the horizon scale currently has the limit 
$B < 10^{-9}{\rm Gauss}$ from the cosmic microwave background anisotropy\cite{Silk}. 
If such magnetic field with long coherent length actually exists, we must seek for 
its primordial origin, especially in the inflating stage of the universe.  It is well-known 
that the magnetic field cannot be generated due to the conformal invariance of 
the Maxwell theory in four dimension\cite{Parker}. But, the conformal invariance is broken due to 
the non-linearity in the Born-Infeld 
theory. Thus we might be able to have a magnetogenesis scenario during the inflation on the
D-braneworld. 

The rest of this paper is organized as follows. In Sec.~\ref{Setup}, we describe the setup for 
D-braneworld. We also comment on a holographic aspects in the D-braneworld. 
In Sec.~\ref{Cosmological}, we will focus on the radiation dominated cosmology in the 
induced metric.  First, we will average the energy-momentum tensor over the volume to 
obtain the equation of state. And then we will see that the universe may be accelerating at the very 
early stage and the closed universe could avoid the initial singularities. 
In Sec.~\ref{Cosmology}, we briefly 
reconsider the cosmology in the open string metric. In Sec.~\ref{Summary}, we give the summary and 
discussion.  See Refs.~\cite{BIcosmos,BIcosmos2,BIBH} 
for other issues on the Born-Infeld cosmology/black holes and Ref.~\cite{Gas} for 
D-brane effect in the different context. 



%======================================%
%<<<<<<<<<<<< SECTION II  >>>>>>>>>>>>>%
%======================================%
\section{Setup}
\label{Setup}

The total action is composed of the bulk and the D-brane action:
%===========<Equation>============%
%
\begin{eqnarray}
S=S_{\rm bulk} +S_{\rm BI},
\end{eqnarray}
%
%=================================%
where $S_{\rm bulk}$ is the 5-dimensional Einstein-Hilbert action with the negative 
cosmological constant and $S_{\rm BI}$ is the Born-Infeld action for D-brane:
%===========<Equation>============%
%
\begin{eqnarray}
S_{\rm BI}=-\sigma \int d^4 x {\sqrt {-{\rm det}(g_{\mu\nu}+ (2\pi \alpha')F_{\mu\nu}) }},
\end{eqnarray}
%
%=================================%
where $F_{\mu\nu}$ is the field strength of the Maxwell theory on the brane. So $F_{\mu\nu}$ will 
correspond to 
the cosmic microwave background radiation if one thinks of the homogeneous and isotropic 
universe as discussed later. In the Born-Infeld action the matter part is automatically 
included. In the usual braneworld, on the other hand, 
we assume that the action on the brane is given by the 
Nambu-Goto action, $S_{\rm NG} \sim -\sigma \int d^4 {\sqrt{-g}}$, plus the matter 
action $S_{\rm matter}$, in particular, $S_{\rm matter} \propto \int d^4 x {\sqrt {-g}}F^2$ 
for the Maxwell field on the brane.  

In four dimensions $S_{\rm BI}$ becomes 
%===========<Equation>============%
%
\begin{eqnarray}
S_{\rm BI} & = & -\sigma \int d^4 x {\sqrt {-g}} \biggl[
1-\frac{1}{2}(2\pi\alpha')^2{\rm Tr}(F^2) \nonumber \\
 & & ~ +\frac{1}{8}(2\pi\alpha')^4 \bigl({\rm Tr}(F^2)\bigr)^2-\frac{1}{4}(2\pi\alpha')^4 {\rm
Tr}(F^4)\biggr]^{1/2} 
\nonumber \\
& = & -\sigma \int d^4x {\sqrt {-g}}\biggl[1-\frac{1}{4}(2\pi\alpha')^2{\rm Tr}(F^2) \nonumber \\
& & ~+\frac{1}{32}(2\pi\alpha')^4\bigl({\rm Tr}(F^2)\bigr)^2-\frac{1}{8}(2\pi\alpha')^4{\rm
Tr}(F^4) 
\nonumber \\
& & ~+O(\alpha'^6) \biggr]. \label{BIaction}
\end{eqnarray}
%
%=================================%
{}From the first to the second line we expanded the square root and wrote down the 
expression up to the order of $O(\alpha'^4)$. For the practice and simplicity, hereafter, 
we employ this approximated action to discuss the {\it whole} history of cosmology. 
The energy-momentum tensor on the brane is given by 
%===========<Equation>============%
%
\begin{eqnarray}
\stac{({\rm BI})}{T}_{\mu\nu}& = & -\sigma g_{\mu\nu}+4\pi \sigma 
(2\pi \alpha')^2 \stac{({\rm em})}{T}_{\mu\nu} \nonumber \\
& & ~+\frac{1}{4}\sigma(2\pi \alpha')^4 {\rm Tr}(F^2) \biggl[ (F^2)_{\mu\nu}
-\frac{1}{8}g_{\mu\nu}{\rm Tr}(F^2) \biggr] \nonumber \\
& & ~-\sigma (2\pi\alpha')^4 \biggl[ (F^4)_{\mu\nu}-\frac{1}{8}g_{\mu\nu}{\rm Tr}(F^4)\biggr]
\nonumber \\
& =: & -\sigma g_{\mu\nu}+T_{\mu\nu},
\end{eqnarray}
%
%=================================%
where 
%===========<Equation>============%
%
\begin{eqnarray}
\stac{({\rm em})}{T}_{\mu\nu}=\frac{1}{4\pi} \biggl( F_\mu^{~\alpha} F_{\nu\alpha}
-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\biggr).
\end{eqnarray}
%
%=================================%
Hereafter we call $T_{\mu\nu}$ the energy-momentum tensor of the Born-Infeld matter. 
To regard the above $ \stac{({\rm em})}{T}_{\mu\nu}$ as the energy-momentum tensor of 
the usual Maxwell field on the brane, we set 
%===========<Equation>============%
%
\begin{eqnarray}
\sigma (2\pi\alpha')^2=1.
\end{eqnarray}
%
%=================================%
At the low energy limit $T_{\mu\nu}$ becomes 
$T_{\mu\nu} \simeq -\sigma g_{\mu\nu}+\stac{({\rm em})}{T}_{\mu\nu}$ which is often 
used in the usual braneworld. 


Since we are interested in what happens on the brane, it is useful to consult with 
the gravitational equation on the brane\cite{Tess} (See also \cite{Roy}):
%===========<Equation>============%
%
\begin{eqnarray}
{}^{(4)}G_{\mu\nu}=8\pi G T_{\mu\nu}+\kappa^4 \pi_{\mu\nu}-E_{\mu\nu},
\end{eqnarray}
%
%=================================%
where 
%===========<Equation>============%
%
\begin{eqnarray}
8\pi G = \frac{\kappa^2}{\ell},\label{scale}
\end{eqnarray}
%
%=================================%
%===========<Equation>============%
%
\begin{equation}
\pi_{\mu\nu} = -\frac{1}{4}T_{\mu\alpha}T^{\;\alpha}_{\nu} +\frac{1}{12}TT_{\mu\nu} 
+\frac{1}{8}g_{\mu\nu}T^{\alpha}_{\;\beta} T^{\;\beta}_{\alpha}-\frac{1}{24}g_{\mu\nu}T^2
\end{equation}
%
%=================================%
and
%===========<Equation>============%
%
\begin{eqnarray}
E_{\mu\nu}=C_{\mu\alpha\nu\beta}n^\alpha n^\beta.
\end{eqnarray}
%
%=================================%
In the above we supposed the Randall-Sundrum fine-tuning, that is, 
%===========<Equation>============%
%
\begin{eqnarray}
\frac{1}{\ell}=\frac{\kappa^2}{6}\sigma,
\end{eqnarray}
%
%=================================%
where $\ell$ is the curvature length of the five dimensional anti-deSitter spacetime. 
Under this tuning, the net-cosmological constant on the brane vanishes. 
We stress, however, that the tuning is not necessary for the discussion in this paper. 

In general the above system is not closed on the brane except for the homogeneous-isotropic 
universe due to the presence of 
a part of the five dimensional Weyl tensor $E_{\mu\nu}$. 
In the weak field limit, we can check
that the four dimensional Einstein gravity can be recovered\cite{Tama}.

Finally, it is worth noting that the trace-part of $\pi_{\mu\nu}$ is related to the trace part 
of $T_{\mu\nu}$ as 
%===========<Equation>============%
%
\begin{eqnarray}
-\frac{\kappa^4}{3}\pi^\mu_\mu = \frac{\kappa^2}{\ell}T^\mu_\mu = 
-\frac{\kappa^4}{12}\biggl[{\rm Tr}(F^4)-\frac{1}{4}({\rm Tr}(F^2))^2 \biggr].
\end{eqnarray}
%
%=================================%
This is a realisation of the holography in the braneworld, that is, 
$\pi^\mu_\mu$ represents a part of the quantum correction to the electromagnetic field theory. 
This is  because the Born-Infeld theory is a sort of phenomenological one for the quantum 
electrodynamics\cite{Hei}. In some cases we can show that $\pi^\mu_\mu$ is identical 
to the trace-anomaly of the quantum field theory on the brane\cite{holo1,holo2}. 

%======================================%
%<<<<<<<<<<<< SECTION III  >>>>>>>>>>>>%
%======================================%
\section{Cosmological models}
\label{Cosmological}

Let us focus on the homogeneous and isotropic universe. For simplicity, we consider the 
single brane model. Then the metric on the brane is 
%===========<Equation>============%
%
\begin{eqnarray}
ds^2=-dt^2+a^2(t) \gamma_{ij}dx^i dx^j,
\end{eqnarray}
%
%=================================%
where $\gamma_{ij}$ is the metric of three dimensional unit sphere or unit hyperboloid 
or flat space. 
In this case we know $E^\mu_{\;\nu}=a^{-4}{\rm diag}(3\mu,-\mu,-\mu,-\mu)$ and $\mu$ is 
proportional to the mass of the five-dimensional Schwarzschild-anti-deSitter spacetime 
which is the bulk geometry. Moreover, the gravitational equation is closed on the brane, 
that is, it is completely written in terms of the four dimensional quantities on the 
brane. From now on we set $\mu=0$ which means that the bulk geometry 
is exactly anti-deSitter spacetime\footnote{If the deviation 
from the anti-deSitter or Schwarzschild-anti-deSitter spacetime 
is, the gravitational equation is not 
closed on the brane\cite{Tess}.}. Thus, the modified Friedman equation becomes 
%===========<Equation>============%
%
\begin{eqnarray}
\label{friedman}
\biggl( \frac{\dot a}{a} \biggr)^2= \frac{\kappa^2}{3 \ell}\rho_{\rm BI}+
\frac{\kappa^4}{36}\rho_{\rm BI}^2-\frac{K}{a^2},
\end{eqnarray}
%
%=================================%
and
%===========<Equation>============%
%
\begin{eqnarray}
\label{raychud}
\frac{\ddot a}{a}=-\frac{\kappa^2}{6\ell}(\rho_{\rm BI}+3P_{\rm BI})-\frac{\kappa^4}{36}
\rho_{\rm BI}(2\rho_{\rm BI}+3P_{\rm BI}).
\end{eqnarray}
%
%=================================%

By defining the electric and magnetic fields by 
%===========<Equation>============%
%
\begin{eqnarray}
E^i=F_0^{~i},~~~{\rm and}~~~B^i=\frac{1}{2}\epsilon^{ijk}F_{jk},
\end{eqnarray}
%
%=================================%
$T_{\mu\nu}$ can be rewritten as
%===========<Equation>============%
%
\begin{eqnarray}
T_{00} & = & \frac{1}{2}(E^2+B^2)+\frac{\kappa^2\ell}{12}(E_i B^i)^2 \nonumber \\
& & ~~+\frac{\kappa^2\ell}{12}(E^2-B^2)\biggl[E^2-\frac{1}{4}(E^2-B^2 )\biggr],
\end{eqnarray}
%
%=================================%
%===========<Equation>============%
%
\begin{eqnarray}
T_{0i}=\epsilon_{ijk}E^jB^k \biggl[1+\frac{\kappa^2\ell}{12}(E^2-B^2) \biggr],
\end{eqnarray}
%
%=================================%
and
%===========<Equation>============%
%
\begin{eqnarray}
T_{ij} & = & -\biggl[ E_i E_j +B_i B_j -\frac{1}{2}g_{ij}(E^2+B^2) \biggr] \nonumber \\
& & ~~-\frac{\kappa^2\ell}{12}(E^2-B^2)\biggl[ E_i E_j +B_i B_j -B^2 g_{ij} \nonumber \\
& & ~~-\frac{1}{4}g_{ij}(E^2-B^2) \biggr] -\frac{\kappa^2\ell}{12}g_{ij}(E_k B^k )^2.
\end{eqnarray}
%
%=================================%

Since we identify the Maxwell field as the background radiation, 
the energy density and the pressure of the Born-Infeld matter should be evaluated by averaging 
over volume as  
%===========<Equation>============%
%
\begin{eqnarray}
\rho_{\rm BI}& := & \langle T_{00} \rangle \nonumber \\
& = & \frac{1}{2}(E^2+B^2)+\frac{\kappa^2\ell}{36}B^2 E^2 
\nonumber \\
& & ~~~+\frac{\kappa^2\ell}{12}(E^2-B^2)\biggl[ E^2-\frac{1}{4}(E^2-B^2)\biggr],
\end{eqnarray}
%
%=================================%
and
%===========<Equation>============%
%
\begin{eqnarray}
P_{\rm BI}:=\frac{1}{3}\langle T^i_i \rangle & = & \frac{1}{6}(E^2+B^2)-\frac{\kappa^2\ell}{36}B^2 E^2
\nonumber \\
& & ~~-\frac{\kappa^2\ell}{144}(E^2-B^2)( E^2-5B^2 ).
\end{eqnarray}
%
%=================================%
In the above we assumed $\langle E_i E_j \rangle = (1/3)g_{ij}E^2$, 
$\langle B_i B_j \rangle = (1/3)g_{ij}B^2$, $\langle E_i \rangle = \langle B_i \rangle =0$ 
and $\langle E_i B_j \rangle =0$ which are natural in the homogeneous and isotropic universe. 
In addition, it is natural to assume ``equipartition"
\footnote{Propery speaking, we must confirm this following the process to the equilibrium 
state based on the Boltzmann-like equation.  

If $F_{\mu\nu}$ corresponds to the primordial magnetic field, 
it is natural to assume $E^2=0$ and $B^2 =2\epsilon \neq 0$. In this situation, 
the energy density and pressure are given by 
$\rho_{\rm BI}=\epsilon-\frac{\kappa^2\ell}{12}\epsilon^2$ and $P_{\rm BI}=
\frac{1}{3}\epsilon -\frac{5\kappa^2\ell}{36}\epsilon^2$. In Ref.~\cite{BIcosmos} 
similar Born-Infeld fluid has been considered, but the authors  
did not considered the D-braneworld. Just cosmology with the non-linear Maxwell field.}:
%===========<Equation>============%
%
\begin{eqnarray}
E^2(t)=B^2(t)=:\epsilon.
\end{eqnarray}
%
%=================================%
Thus the energy density and the pressure are simply given by 
%===========<Equation>============%
%
\begin{eqnarray}
\label{eq-rho}
\rho_{\rm BI}=\epsilon + \frac{\kappa^2\ell}{36}\epsilon^2,
\end{eqnarray}
%
%=================================%
and
%===========<Equation>============%
%
\begin{eqnarray}
\label{eq-p}
P_{\rm BI}=\frac{1}{3}\epsilon - \frac{\kappa^2\ell}{36}\epsilon^2.
\end{eqnarray}
%
%=================================%
By combining Eqs.~(\ref{eq-rho}) and (\ref{eq-p}), we obtain the equation of state
%===========<Equation>============%
%
\begin{eqnarray}
\label{eos}
P_{\rm BI}=(\gamma_{BI}-1)\rho_{BI}
\end{eqnarray}
%
%=================================%
with an effective adiabatic index
%===========<Equation>============%
%
\begin{eqnarray}
\gamma_{BI}=\frac{4}{3}\frac{1}{1+\frac{\kappa^2\ell}{36}\epsilon}.
\end{eqnarray}
%
%=================================%
At the low energy scale such $\epsilon \ll 36/\kappa^2\ell$, 
the Born-Infeld matter behaves as just radiation fluid with $\gamma_{BI}\sim 4/3$ and
$\rho_{\rm BI} \sim (1/3)P_{\rm BI}$. 
Note that the Born-Infeld matter looks like the 
time-dependent cosmological constant if the second terms are dominated, i.e., $\gamma_{BI}\sim 0$.
 
Now we have the equations of motion (\ref{friedman}) and (\ref{raychud}) and the equation of 
state (\ref{eos}) for the Born-Infeld matter. It should be noted that these equations can be 
scaled by defining new variables $\bar{t}:=t/\ell$, 
$\bar{K}:=K\ell^2$, $\bar{\epsilon}:=\kappa^2\ell\epsilon$,
$\bar{\rho}_{BI}:=\kappa^2\ell\rho_{BI}$ and $\bar{P}_{BI}:=\kappa^2\ell P_{BI}$. 
We can see the scale factor dependence of the energy density using the  energy-conservation law,
$\dot \rho_{\rm BI}+3H(\rho_{\rm BI}+P_{\rm BI})=0$, on the  brane: 
%===========<Equation>============%
%
\begin{eqnarray}
\biggl( 1+\frac{\kappa^2\ell}{18} \epsilon \biggr) \dot \epsilon = -4H \epsilon.
\end{eqnarray}
%
%=================================%
It is easy to integrate the above equation and then
%===========<Equation>============%
%
\begin{eqnarray}
\epsilon e^{\frac{\kappa^2\ell}{18}\epsilon}
=\frac{\epsilon_0}{a^4}.
\end{eqnarray}
%
%=================================%
See Fig.~\ref{fig1} for $\epsilon$. Therein we also draw the ordinary radiation case 
of $\epsilon \propto a^{-4}$. In the early universe, the Born-Infeld matter is 
significantly suppressed  compared to the ordinary radiation fluid. 

%=================================%
\begin{figure}[t]
\vspace*{+2mm}
\begin{center}
\epsfxsize=3.0in
\epsffile{dbrane1.eps}
\end{center}
\vspace*{-1mm}
\caption{The log-plot of $\epsilon$. The horizontal axis is the scale factor. The dotted
line is the case of the ordinary radiation  fluid. The solid line is the case of the Born-Infeld
matter.}
\label{fig1}
\end{figure}
%=================================%




Here note that the matter does not always satisfy the ``strong energy 
condition" $\rho+3P \geq 0$. For the D-brane matter, 
%===========<Equation>============%
%
\begin{eqnarray}
\rho_{\rm BI}+3P_{\rm BI}=2\epsilon-\frac{\kappa^2\ell}{18}\epsilon^2,
\end{eqnarray}
%
%=================================%
and
%===========<Equation>============%
%
\begin{eqnarray}
2\rho_{\rm BI}+3P_{\rm BI}=3\epsilon-\frac{\kappa^2\ell}{36}\epsilon^2.
\end{eqnarray}
%
%=================================%
When $\gamma_{BI} < 2/3$, i.e., $\epsilon  > 36/\kappa^2\ell  \sim (10^3 {\rm GeV})^4 
(M_5/10^8{\rm GeV})^6$, the ``strong energy condition" is presumably 
broken\footnote{$M_5$ is the fundamental scale of five dimensional gravity and then 
$M_5 \sim \kappa^{-1/3} \sim (M_{\rm pl}^2\ell^{-1})^{1/3}\sim 10^8(1{\rm mm}/\ell)^{1/3}{\rm GeV}$ 
in the single brane models. The string length becomes 
$\ell_s = {\sqrt {\alpha'}} \sim \sigma^{-1/4} \sim 
(\ell_{\rm pl} \ell)^{1/2} \sim 10^{-17} \times (\ell/1{\rm mm})^{1/2} {\rm cm}$}. 
Thus the universe is accelerating 
during such period, $\ddot a/a >0$. Furthermore, we have the opportunity that 
the initial singularity can be avoided.  



To see the quantitative feature of the dynamics of cosmology 
it is useful to write down the generalized Friedman equation 
as usual:
%===========<Equation>============%
%
\begin{eqnarray}
\dot a^2+V(a)=-K,
\end{eqnarray}
%
%=================================%
where 
%===========<Equation>============%
%
\begin{eqnarray}
V(a):=-\frac{\kappa^2}{3\ell}a^2 \rho_{\rm BI} 
\biggl(1+\frac{\kappa^2\ell}{12}\rho_{\rm BI} \biggr).
\end{eqnarray}
%
%=================================%


See Fig.~\ref{fig2} for the potential profile. Surprisingly, there is the minimum. 
In addition, as shown analytically, the potential is zero at $a=0$. 
Let us look at around $a=0$. $\epsilon$ can be approximately solved as  
%===========<Equation>============%
%
\begin{eqnarray}
\frac{\kappa^2 \ell}{18}\epsilon \sim -4 {\rm log}a.
\end{eqnarray}
%
%=================================%
We can see that the potential, indeed, is zero  at $a=0$ as seen in Fig.~\ref{fig2}. In the 
current approximation, we see 
%===========<Equation>============%
%
\begin{eqnarray}
V(a) \sim -\frac{576}{\ell^2}a^2({\rm log}a)^4 \to  0, ~~~ (a\to 0).
\end{eqnarray}
%
%=================================%
%=================================%
\begin{figure}[t]
\vspace*{2mm}
\begin{center}
\epsfxsize=3.0in
\epsffile{dbrane2.eps}
\end{center}
\vspace*{-1mm}
\caption{The potential profile of $\bar{V}(a):=V\ell^2$. The potential has the minimum and
vanishes at $a=0$.}
\label{fig2}
\end{figure}
%=================================%
As a result the closed universes is bounced around $a=0$! The flat or open universe 
has the initial singularity. Near $a=0$, the behavior of the scale factor in the flat universe 
becomes
%===========<Equation>============%
%
\begin{eqnarray}
a(t) \sim e^{-\frac{\ell}{24t}},
\end{eqnarray}
%
%=================================%
and then $\ddot a /a \sim \frac{\ell^2}{576}\frac{1}{t^2} >0$, that is, the 
universe is accelerating. 

For $\epsilon \ll 36/\kappa^2\ell$, as should be so, the universe is described by 
the ordinary radiation dominated model. See Fig.~\ref{fig3} for the behavior of the scale factor. 

%=================================%
\begin{figure}[t]
\vspace*{2mm}
\begin{center}
\epsfxsize=3.0in
\epsffile{dbrane3.eps}
\end{center}
\vspace*{-1mm}
\caption{The behavior of the scale factor of the closed (solid line), flat (dotted line) 
and open (dot-dashed line) cases in the D-braneworld.}
\label{fig3}
\end{figure}
%=================================%



%======================================%
%<<<<<<<<<<<< SECTION IV  >>>>>>>>>>>>>%
%======================================%
\section{Cosmology in stringy view}
\label{Cosmology}

So far we investigated the D-braneworld in terms of the induced metric 
$g_{\mu\nu}$. For the gauge field on the brane like photons, however, 
the propagation of the field is described by the metric 
%===========<Equation>============%
%
\begin{eqnarray}
\stac{s}{g}_{\mu\nu}=g_{\mu\nu}-(2\pi \alpha')^2(F^2)_{\mu\nu}.
\end{eqnarray}
%
%=================================%
Hence it is fair to consider the gravitational 
equation and cosmology in terms of $\stac{s}{g}_{\mu\nu}$. In the case of the 
radiation dominated universe, the corresponding metric is given by  
%===========<Equation>============%
%
\begin{eqnarray}
d\tilde s^2 & = & \stac{s}{g}_{\mu\nu}dx^\mu dx^\nu= -d\tilde t^2+\tilde a(\tilde t)^2
\gamma_{ij}dx^i dx^j,
\nonumber \\
& = & -\biggl[1+(2\pi \alpha')^2 \langle (F^2)_{00} \rangle \biggr]dt^2 \nonumber \\
& & ~~~~+ 
\biggl[g_{ij}-(2\pi\alpha')^2 \langle (F^2)_{ij} \rangle \biggr]dx^i dx^j, \nonumber \\
& = & -\biggl( 1-\frac{\kappa^2\ell}{6}\epsilon \biggr)dt^2 +a^2 
\biggl(1+\frac{\kappa^2\ell}{18}\epsilon \biggr) \gamma_{ij}dx^i dx^j,
\end{eqnarray}
%
%=================================%
%Thus we have the transformation between the cosmic times and scale factors as follows:  
%%===========<Equation>============%
%%
%\begin{eqnarray}
%d\tilde t = {\sqrt {1-(2\pi\alpha')^2E^2}}dt 
%\end{eqnarray}
%%
%%=================================%
%and
%%===========<Equation>============%
%%
%\begin{eqnarray}
%\tilde a (\tilde t) =a(t) {\sqrt {1+\frac{1}{3}(2\pi\alpha')^2E^2}}.
%\end{eqnarray}
%%
%%=================================%
where $\tilde t$ and $\tilde a (\tilde t)$ are the cosmic time and the scale factor 
of the open string metric, respectively. 
As discussed in Ref.~\cite{Causal}, the light-cone with respect to the open string 
metric is smaller than that in the induced metric.  
Let $n^\mu$ to be null vector for the induced metric. For the concreteness, 
$n=\partial_t+(1/a)\partial_x$. Then 
%===========<Equation>============%
%
\begin{eqnarray}
\stac{s}{g}_{\mu\nu}n^\mu n^\nu & = & -(2\pi\alpha')^2(F^2)_{\mu\nu}n^\mu n^\nu,  \nonumber \\
& = & \frac{2\kappa^2\ell}{9}\epsilon >0, 
\end{eqnarray}
%
%=================================%
and this means that $n$ is the spacelike in the open string metric. 

We point out that the singularity appears at the {\it finite} value of 
$\epsilon=\epsilon_c:= \frac{12}{\kappa^2\ell}$. The physical energy measured 
also diverges because it is proportional to $1/{\sqrt {1-\kappa^2 \ell \epsilon /12}}$. 
Anyway the universe evolves keeping  
$\epsilon < \epsilon_c$. This means that there is no 
drastic changes in the open string metric, but 
slight modification from the ordinary radiation dominated universe. 

At first glance, we cannot examine the 
interesting region where the feature of Born-Infeld becomes essential. However, 
it might be  
better to say that this is because of the limitation of the stringy metric.



%======================================%
%<<<<<<<<<<<< SECTION V  >>>>>>>>>>>>>%
%======================================%
\section{Summary and discussion}
\label{Summary}

 In this paper we have considered the Randall-Sundrum D-braneworld cosmology. Therein 
the matter on the brane is described by the Born-Infeld action. As a first step, we 
considered only the $U(1)$ gauge field and treated it as a sort of radiation fluids. 
Then we examined the radiation dominated universe on the D-brane. We found that 
the strong energy condition is broken in the very early stage and the universe is 
more accelerated than the ordinary inflation. Furthermore the initial singularity is avoided 
in closed universes. Thus we can conclude that we have the different history about the 
early stage from the ordinary braneworld scenario if we are living on the D-brane, not on 
the Nambu-Goto membrane.  In the acceleration phase, the ratio of the two scale factors at 
different times is given by 
$ a(t_f)/a(t_i) \sim e^{\frac{\kappa^2\ell}{72}(\epsilon_i -\epsilon_f)}$. 
For the horizon problem $a(t_f)/a(t_i)> 10^{13} \times (100{\rm km}/s/{\rm Mpc}/H_0)
 ( T/10^3{\rm GeV})$ is required. 
Then if $\frac{\kappa^2\ell}{72}\Delta \epsilon > 30 $ the horizon problem is solved. 
However, the origin of the density fluctuation is not provided just in this scenario. 

We should remark that we employed the approximated action in the second line of  
Eq.~(\ref{BIaction}). We discussed the high energy regime where the 
expansion is broken and the approximated 
action may not be appropriate. Without  
such approximation, however, we must treat the infinite series expansion due to the 
volume averaging. Although our treatment contains this kind of problem, we could obtain 
the important tendency. The future improvement for the treatment of the higher derivative 
terms is desired. 
     
We also addressed the D-braneworld in the open string metric, that is, {\it stringy view}. 
As a result, the singularity appears just at the time when the non-trivial effects of the 
D-brane are dominated. In this sense we might not be able to expect the significant 
contribution from the D-braneworld specialties in the stringy view. 

Since we obtained the nature that the Born-Infeld matter 
contains vacuum/dark energy part, the current accelerating universe can be explained 
in the D-braneworld without introducing the additional exotic fields like quitessence.  




\section*{Acknowledgments}

TS would like to thank Kouji Hashimoto and Norisuke Sakai for fruitful discussions. 
To complete this work, the discussion during and after the YITP workshops YITP-W-01-15 and 
YITP-W-02-19 were useful. TS's work is supported by Grant-in-Aid for Scientific
Research from Ministry of Education, Science, Sports and Culture of
Japan(No. 13135208, No.14740155 and No.14102004).



%\appendix
%\section{Derivation of the covariant curvature formalism}


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\end{document}

