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\begin{document}
\begin{titlepage}
\begin{center}
\hfill LBNL-42952 \\
{}~{} \hfill UCB-PTH-99/06 \\
{}~{} \hfill hep-ph/9903319\\
%\vskip .25in
\vskip .3in
{\Large \bf Late Inflation and the
Moduli Problem of Sub-Millimeter Dimensions}
%\vskip 0.3in
\vskip 0.3in
{\bf Csaba Cs\'aki\footnote{Research fellow, Miller Institute for
Basic Research in Science.}, Michael Graesser\footnote{Supported
by the Natural Sciences and Research Council of Canada.}, and John
Terning}
\vskip 0.15in
{\em Theoretical Physics Group\\
Ernest Orlando Lawrence Berkeley National Laboratory\\
University of California, Berkeley, California 94720}
\vskip 0.1in
\vskip 0.1in
{\em Department of Physics\\
University of California, Berkeley, California 94720}
\vskip 0.1in
{\tt ccsaki@lbl.gov, mlgraesser@lbl.gov, terning@alvin.lbl.gov }
\end{center}
\vskip .25in
\begin{abstract}
We consider a recent model with sub-millimeter sized
extra dimensions, where the field that determines the size of the extra
dimensions (the
radion) also acts as an inflaton. The radion is also a stable modulus, and its
coherent oscillations
can potentially overclose the Universe. It has been suggested that a second
round of
late inflation can solve this problem, however we find that this scenario
does not
allow for sufficient reheating of the Universe.
\end{abstract}
\end{titlepage}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\setcounter{footnote}{0}
Recently a remarkable proposal has been made by
Arkani-Hamed et. al. \cite{ADD}, suggesting that the fundamental Planck scale
could be at the TeV scale
provided that there are compact (sub-millimeter sized) extra dimensions that
gravitational fields
can propagate in. The radius of the extra dimensions then acts as a light
(mass of order $10^{-3}$ eV to order MeV) dynamical field that is
referred to as the radion \cite{ADM}.
A very attractive scenario \cite{inflate} was then proposed
in which the radion field itself
can act as an inflaton in the early Universe, i.e. its vacuum energy density
can dominate
the energy density of the Universe and cause the three large spatial
dimensions to
expand exponentially in time. After this inflationary episode the Universe is
reheated to
a temperature of $10-100$ MeV with radiation being the dominant form of
energy.
The remaining energy
density stored in the coherent oscillations of the radion is
severely
constrained so that it does not overclose the Universe. Thus the radion
presents
an example of a cosmological moduli problem, which is referred to as the
radion problem in Refs. \cite{ADM,inflate}.
In Ref. \cite{inflate} it was proposed that a second
round of inflation (a late inflation) with 5 to 6 e-foldings could
sufficiently damp
the radion oscillations. Here we will examine this proposal in some detail.
We find
that this late
inflation generically shifts the minimum of the potential for the small
radius, and that this shift is reliably calculable in these models.
In order to achieve
sufficient damping of the radion, the inflaton mass has to
be extremely
small. Even if such a light inflaton existed, it cannot reheat the
Universe enough to
allow for standard big-bang nucleosynthesis (BBN) to occur at temperatures
around
1 MeV. Thus we argue that late inflation is an unlikely solution
to the radion problem, which remains as one of the most severe
problems for models with sub-millimeter dimensions. Needless to say,
this result does not exclude the appealing framework of Ref.~\cite{inflate},
but rather reiterates the difficulty of the moduli problem.
First we briefly review the effective Lagrangian approach of \cite{ADM}
to the equations of motion for the expanding universe.
These equations are obtained by assuming that the metric of the $4+n$
dimensional spacetime is given by
\begin{equation}
\label{eq:metric}
g_{\mu \nu}= \left( \begin{array}{ccc}
1 \\ & -R(t)^2 g_{IJ} \\
&& -r(t)^2 g_{ij}\\ \end{array} \right),
\end{equation}
where $R(t)$ is the time dependent scale factor of the large
4 dimensional space-time, $r(t)$ is the scale factor of the $n$ extra
dimensions, while the $g_{IJ}$ and $g_{ij}$ are flat metrics in
3 and $n$ dimensions respectively. The 3 large dimensions
can be viewed as a ``brane" or wall in the $4+n$ dimensional space-time.
The effective Lagrangian for the system is obtained by plugging the
background
metric (\ref{eq:metric}) into the action
\begin{equation}
\label{eq:action}
S=-\int d^{4+n}x \sqrt{-g} \left( M_{*}^{n+2} {\cal R} -{\cal L} \right),
\end{equation}
where $g=\det g_{\mu \nu}$, ${\cal R}$ is the curvature scalar,
$M_*$ is the fundamental scale of the theory ($M_* \approx 1$ TeV),
and ${\cal L}$ is the Lagrangian density which includes
matter fields and cosmological constants both in the bulk and on the wall.
By performing the
integrals over the spatial coordinates, and
integrating by parts to eliminate
second time derivatives,
an action dependent on $R$, $r$, and single time
derivatives is obtained.
The resulting Lagrangian for $R(t),r(t)$ is then given by \cite{ADM}
\begin{equation}
\label{eq:effaction}
L_{\rm eff}=-M_{*}^{n+2} R^3 r^n \left( 6
\left(\frac{\dot{R}}{R}\right)^2+
n(n-1) \left(\frac{\dot{r}}{r}\right)^2+ 6n
\left(\frac{\dot{r}\dot{R}}{rR}\right)\right) -V_{\rm eff}(r,R).
\end{equation}
Here $V_{\rm eff}$ includes the potential which stabilizes the radius of the
extra dimensions to a value $r_0$ and the potential of the
matter fields on the wall which in our case
will lead to the late inflation under discussion:
\begin{equation}
\label{eq:veff}
V_{\rm eff} (r,R) = R^3 \left( V_{bulk} (r) +V_{wall} (R) \right).
\end{equation}
Since we are interested in the epoch after the initial inflation,
at times when the radion is already stabilized close to its actual
minimum $r_0$, a good approximation for $V_{bulk}$ is to
take it to be quadratic. The mass of the radion has been calculated
in Ref.~\cite{ADM} to be
\begin{equation}
\label{eq:radionmass}
m_n ^2=\frac{V_{bulk}''(r_0)}{n(n+2) M_*^{n+2}r_0^{n-2}},
\end{equation}
where $m_n$ is the radion mass, which was found to be \cite{ADM} between
the $n$ independent lower bound of $10^{-3}$ eV, and an upper bound
of $10^{-2}$ eV for $n=2$ or an upper bound of $20$ MeV for $n=6$.
Thus our approximation for $V_{bulk}$ is
\begin{equation}
\label{eq:potential}
V_{bulk} (r)=\frac{1}{2} n(n+2) m_n ^2 M_*^{n+2}r_0^{n-2} (r-r_0)^2.
\end{equation}
Introducing dimensionless variables (which we will use from here on)
$r\to r_0 r, R\to r_0 R$,
the equations of motion for the new variables obtained from the effective
Lagrangian (\ref{eq:effaction}) are:
\begin{eqnarray}
\label{theeq}
0&=& -(n-1)r^{n-2}
\left( 6 \frac{ \dot{r}\dot{R}}{R}+2 \ddot{r}+(n-2)
\frac{\dot{r}^2}{r}\right)-
6 r^{n-1} \left (\frac{\dot{R}^2}{R^2}+
\frac{\ddot{R}}{R}\right)
+(n+2) m_n^2 (r-1) \nonumber \\
0&=&-3n r^{n-1}\left( (n-1)\frac{\dot{r}^2}{r}+2 \ddot{r}\right)
-12 n r^{n-1} \frac{\dot{r}\dot{R}}{R}
-6r^n \left( \left( \frac{\dot{R}}{R}\right)^2
+2 \frac{\ddot{R}}{R} \right) \nonumber \\
&&+3 m_n^2 \frac{n (n+2)}{2} (r-1)^2
+\frac{1}{M_{\rm Pl}^2}\left( 3 V_{wall}+R V'_{wall}\right) ~,
\end{eqnarray}
where the reduced Planck scale is given by
$M_{\rm Pl}^2 \equiv M_*^{n+2} r_0^n= (2\times 10^{18}\ {\rm GeV})^2$.
Next we assume that the theory contains an inflaton field which produces a
second period
of inflation (a late inflation) after the initial inflation due to the radion,
and explore the consequences. We assume that the vacuum energy of the
Universe is dominated
for a brief time by the vacuum energy of this field, which we parameterize as:
\beq
V_{wall} \approx V_I=\lambda^2 M_{\rm Pl}^2 +\cdots
\label{vac}
\eeq
where the ellipsis indicates other field dependent terms.
This vacuum energy will force the scale factor of the large dimensions,
$R$, to grow as
\beq
R \propto e^{H t}, \ \ \ H \approx {{\lambda}\over{\sqrt{6}}}.
\label{exponential}
\eeq
During this inflationary period the oscillations in the radion are rapidly
damped.
One can easily see that the coupling of the radion field to the scale factor
of the
large dimensions introduces a shift in the effective potential experienced by
the
oscillating radion field. From Eq. (\ref{theeq})
we can read off the derivative of this
potential:
\beq
V^\prime_{{\rm eff},bulk}(r) = V_{bulk}^\prime(r) -
6nr^{n-1} \left( \frac{\dot R^2}{R^2} + \frac{\ddot R}{R}\right).
\label{veff}
\eeq
It is this shift in the minimum of the potential \cite{DRT} that will provide
the crucial constraint
on these late inflation models. The effect of inflation shifting
the effective potential of a modulus has been known for a long time, and in
fact can be viewed as the real origin of the moduli problem \cite{DRT}.
However, in generic models the size of the shift of the
modulus usually depends on unknown physics (for example on
higher order K\"ahler potential couplings in the case of
supersymmetric theories), and thus cannot be reliably estimated.
In this example however the shift is just given by solving Eq. (\ref{theeq}).
In order to obtain a model that gives the appropriate conditions to form the
observed
Universe, the energy density in the radion oscillations must be very small.
If, however, the late inflation damps the
oscillations around a minimum that is far
from the true
minimum after inflation, then the radion will again begin to oscillate around
its true
minimum once inflation has ended. Thus we find that this shift in the minimum
during
inflation must be quite small; this requires that $\lambda$ be much smaller
than $m$.
This can be seen by looking for the steady-state inflationary solution,
taking $r$ to be a constant $r=r_I$, and $R$ to grow exponentially, as in
Eq. (\ref{exponential}). These forms exactly
solve the equations of motion and give the minimum during inflation
as
\beq
r_I={{(2n-1)m_n \pm \sqrt{m_n^2-8 ( {{n-1}\over{n+2}} )
\lambda^2 }}\over{2 m_n (n-1)}}.
\label{exactshift}
\eeq
The negative sign gives the solution that approaches the true minimum as
$\lambda \rightarrow 0$. For small $\lambda$ this can be
approximated\footnote{For $\lambda \gg m_n$, Eq. (\ref{exactshift}) has no
real solutions, and no damping of the moduli oscillations occurs \cite{DRT}.}
as
\beq
\label{shift}
r_I \approx 1 + {{2 \lambda^2}\over{ (n+2) m_n^2}}.
\eeq
This result can also be easily obtained from Eq. (\ref{veff}) by setting
$V^{\prime}_{eff}=0$ and linearizing in $\lambda^2/m^2$.
The energy density stored in the radion field at the end of the second
stage of inflation is given by
Eq. (\ref{eq:potential})
\begin{equation}
V=\frac{1}{2} n(n+2) m_n^2 M_{\rm Pl}^2
\left( r_I-1\right) ^2.
\end{equation}
In order for this energy density to
not overclose the Universe today (i.e. to not reintroduce
the radion problem), it is bounded by
\begin{equation}
\frac{V}{T^3_{RH}} < \frac{3}{2} \times 10^{-9} \hbox{ GeV} \hbox{ ,}
\label{mod1}
\end{equation}
where $T_{RH}$ is the reheat temperature of the Universe at the end
of the late inflationary period, and we have set the current Hubble
parameter to $H_0=50 \hbox{ km s}^{-1} \hbox{ Mpc}^{-1}$.
This in turn implies an upper
bound on the shift in $r$ at the end of the late inflation:
\begin{equation}
\left(r_I-1\right) < 3 \times 10^{-14}
\frac{1}{\sqrt{n(n+2)}}
\left(\frac{10^{-3} \hbox{ eV}}{m_n} \right)
\left(\frac{T_{RH}}{10 \hbox{ MeV}} \right)^{3/2}.
\label{upperr}
\end{equation}
A model independent, and $n$ independent, upper limit is obtained by inserting
$n=2$ and
the lower bound $m_n> 10^{-3}$ eV on the radion mass, which is
determined
from short-distance force experiments \cite{raman}. This gives
\begin{equation}
\left(r_I-1\right) < 10^{-14}
\left( \frac{T_{RH}}{10 \hbox{ MeV}} \right)^{3/2}.
\label{upperlimit}
\end{equation}
We note that
the actual constraint on $r_I$ from Eq. (\ref{upperr}),
for a given $m_n$ and $n$,
can be much stronger.
Next, using Eq. (\ref{shift}), and the upper
limit on $r_I$ given in Eq. (\ref{upperr}), an upper
bound on the inflationary scale $\lambda$ is:
\begin{eqnarray}
\label{upperlambda2}
\lambda &<& 4 \times 10^{-16} \hbox{ MeV }
\left(\frac{T_{RH}}{10 \hbox{ MeV}} \right)^{3/4}
\left( \frac{m_{n=2}}{10^{-2} \hbox{ eV}}\right)^{1/2} \hbox{ , }
n=2 \hbox{ ,} \\
\label{upperlambda6}
\lambda &<& 8 \times10^{-12} \hbox{ MeV }
\left(\frac{T_{RH}}{10 \hbox{ MeV}} \right)^{3/4}
\left( \frac{m_{n=6}}{\hbox{5 MeV}} \right)^{1/2} \hbox{ , }
n=6.
\end{eqnarray}
Provided that the shift in the minimum of the potential is sufficiently
small, then 5 to 6
e-foldings of inflation are required in order to sufficiently damp the radion
oscillations.
The number of e-foldings is given by \cite{KT}
\beq
N= \int H dt = \int d \phi {{3 H^2}\over{V_I^\prime}} \approx {{\Delta \phi
\lambda^2}\over{ 2 V_I^\prime}} ~,
\eeq
where $\phi$ is the inflaton field, and $\Delta \phi$ is the distance in field
space that
it travels
during the course of inflation.
This equation simply constrains the inflaton potential to be
sufficiently flat during inflation. In addition the slow-roll condition
requires
that $|V_I^{\prime\prime}| \ll 9 H^2$. For a
natural potential (where there are no fine-tuned cancelations between terms)
each term
in the potential should have sufficiently small derivatives in
order for slow-roll inflation to occur. These constraints imply a
bound on the mass of the inflaton field:
\beq
\label{mIbound}
m_I < \lambda.
\eeq
%Using the previous upper limits on $\lambda$, Eq. (\ref{upperlambda2}) and
%Eq. (\ref{upperlambda6}),
%we obtain an upper limit
%on $m_I$:
%\begin{eqnarray}
%\label{minflaton2}
%m_I &<& 6 \times 10^{-20} \ \hbox{ GeV}\,
%\left( \frac{m_{n=2}}{10^{-2} \hbox{ eV}} \right)^{1/2}
%\left(\frac{T_{RH}}{10 \hbox{ MeV}} \right)^{3/4}, \ \ \ n=2 \\
%m_I &<& 4 \times 10^{-15} \ \hbox{ GeV}
%\left( \frac{m_{n=6}}{\hbox{5 MeV}} \right)^{1/2}
%\left(\frac{T_{RH}}{10 \hbox{ MeV}} \right)^{3/4}, \ \ \ n=6 .
%\label{minflaton6}
%\end{eqnarray}
In order to avoid significant cosmological difficulties, the
reheat temperature $T_{RH}$ must be less than the ``normalcy''
temperature $T_*$ \cite{ADD2} below which
the 4D Universe is radiation dominated
with the bulk essentially empty of energy.
Processes such as $\gamma \gamma \rightarrow$ bulk graviton occurring
in the early Universe can dump too much energy into the bulk if the
temperature is above $T_*$.
The over-production of bulk gravitons, for example, can significantly
affect the expansion rate of the Universe during BBN and also overclose the
Universe. Furthermore, the late decay of Kaluza-Klein gravitons to two photons
can
produce spikes in the background photon spectrum and is a very
significant cosmological constraint.
Since the late photon
constraint may be avoided in models where the bulk is populated with many
branes,
we instead use the (weaker) constraint that is obtained from
requiring that the energy density in the bulk gravitons is less
than about a tenth of the energy in radiation during BBN \cite{ADD2}.
This leads to
the constraint: $T_{RH}< 3 \times 10^{-6} \hbox{ }M_*$ for
$n=2$, and
$T_{RH} <2 \times 10^{-3} \hbox{ }M_*$ for $n=6$.
%As can be seen from
%our final results, Eq. (\ref{reheatagain2}) and
%Eq. (\ref{reheatagain6}), whether
%the reheat temperature must be below
%${\cal O}($10 MeV$)$ or ${\cal O}($GeV$)$ is not so important.
Inserting these upper bounds on $T_{RH}$
into Eq. (\ref{upperlambda2}) and Eq. (\ref{upperlambda6}) gives an
upper bound on the mass of the inflaton in terms of $M_*$:
\begin{eqnarray}
\label{m2inflaton2}
m_I &<& 2 \times 10^{-16} \ \hbox{ MeV}\,
\left( \frac{m_{n=2}}{10^{-2} \hbox{ eV}} \right)^{1/2}
\left(\frac{M_*}{\hbox{1 TeV}} \right)^{3/4}, \ \ \ n=2 \hbox{ ,} \\
m_I &<& 4 \times 10^{-10} \ \hbox{ MeV}
\left( \frac{m_{n=6}}{\hbox{5 MeV}} \right)^{1/2}
\left(\frac{M_*}{\hbox{1 TeV}} \right)^{3/4}, \ \ \ n=6 .
\label{m2inflaton6}
\end{eqnarray}
Since the initial round of inflation in this model is thought to reheat the
Universe to a
temperature around 10 -- 200 MeV, 5 or 6 e-foldings will result in a Universe
too cold for
the BBN scenario which requires a radiation dominated Universe
at temperatures of a few MeV.
Therefore the Universe must be
reheated again
after the
second inflationary period, and the inflaton must therefore decay.
This can be made more precise. Denote by $T_1$ and $V_1$ the
temperature of the radiation and the energy density in radion oscillations,
respectively, at the {\it onset} of the late (second) inflationary phase.
Then the temperature at the end of inflation,
but before reheating, is $T_2 =e^{-N} T_1$. The energy in radion
oscillations at the end of inflation is
$V =e^{-3 N} V_1+V_{shift} >e^{-3 N} V_1$.
Here $V_{shift}$ is the energy density due
to the shift in the minimum of the potential
during the late inflation.
Then the overclosure constraint implies
\begin{equation}
\frac{3}{2} \times 10^{-9} \hbox{ GeV } > \frac{V}{T^3 _{RH}}
> \left( \frac{T_2}{T_{RH}} \right)^3 \, \frac{V_1}{T^3 _1}
= \left( \frac{T_2}{T_{RH}} \right)^3 \, \frac{V_i}{T^3 _i} .
\label{diluteT}
\end{equation}
In the last equality we have used $V/T^3 =$ constant to
express the result in terms of $T_i$ and $V_i$, the
values of $T$ and $V$, respectively, at the end of the
reheating following the {\it first
inflationary phase}, rather than their
values at the start of the {\it second inflationary phase}.
If there is a moduli
problem then $V_i \sim T^4_i$.
In fact this condition holds at the end of the first reheating
in the model of Ref. \cite{inflate}. Assuming that
{\it no} secondary reheating
is required, then $T_{RH}=T_2