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 \begin{flushright}{SIT-HEP/TM-13}
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\centerline{\large{\bf Thermal hybrid inflation in brane world}}
\vskip .75 truecm
\centerline{\bf Tomohiro Matsuda
\footnote{matsuda@sit.ac.jp}}
\vskip .4 truecm
\centerline {\it Laboratory of Physics, Saitama Institute of
 Technology,}
\centerline {\it Fusaiji, Okabe-machi, Saitama 369-0293, 
Japan}
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\begin{abstract}
\hspace*{\parindent}
In conventional scenario of thermal inflation, the requirement from the
reheating  temperature puts a lower bound on the mass of the inflaton field.  
At the same time, the mass of the inflaton field
and the height of the potential during thermal inflation is intimately
 related. 
With these conditions, the conventional models for thermal inflation are
 quite restricted. 
Naively, one may expect that the above constraints may be removed if
 thermal inflation is realized within the setups for hybrid 
inflation.
In this respect, constructing a realistic model for thermal hybrid 
inflation seems interesting.
We consider this problem in the brane world and show that thermal
inflation can become free of the above restrictions.
\end{abstract}

\newpage
\section{Introduction}
\hspace*{\parindent}
In conventional models for thermal inflation, inflation starts from the
state where the symmetry is restored by the thermal effect.
One may regard it as a modification of new inflation, where inflation
starts at the top of the potential.
On the other hand, hybrid inflation starts with the chaotic initial
condition.
During hybrid inflation, the inflaton field stabilizes the trigger field
at the false vacuum.
In this respect, hybrid inflaton is a modification of chaotic inflation.
Taking these things into consideration, it seems rather difficult to
construct thermal hybrid inflation without adding unnatural components 
by hand.
What we want to show in this paper is that the collaboration between
these different kinds of inflation can be realized in the natural
settings of brane inflation. 

\section{Thermal hybrid inflation}
\hspace*{\parindent}
In this section we first consider an old idea of inverted hybrid
inflation\cite{inverted} where the hybrid potential is used while the
initial condition is not chaotic.

\underline{Inverted hybrid inflation}

In the scenario for inverted inflation, the inflaton field 
slowly rolls away from the origin, and finally   
the trigger field terminates inflation.
Thus at first sight, it seems possible to construct thermal hybrid 
inflation, if it is realized in the setups of the inverted scenario.
Here we examine the above simplest idea and show that it is indeed quite
difficult to be realized without fine-tunings.

Perhaps the simplest way to explain the idea of inverted hybrid
inflation is to consider the potential 
\begin{equation}
V(\phi, \sigma)=\frac{1}{4}\lambda_{\sigma}\left(\sigma^2 + M^2 \right)^2 
-\frac{1}{2}\lambda_{int} \phi^2 \sigma^2
+\frac{1}{4}\lambda_{\phi}\left(\phi^2-M'^2\right)^2
\end{equation}
where $\phi$ is the inflation field and $\sigma$ is the trigger field.
Inverted hybrid inflation starts when the inflaton field ($\phi$) stays at
the top of the potential ($\phi\simeq 0$), where the trigger 
field is stabilized at $\sigma =0$.
Here we examine if the thermal initial condition works well for this model.
We assume that, because of the thermal effect, $\phi$ is held at the
origin during thermal inflation. 
Since the inflaton field ($\phi$) stays at $\phi=0$, the trigger field
($\sigma$) is also held at a false vacuum $\sigma=0$ until
$\phi$ reaches the critical value $\phi_c$.
Then the effective mass squared for the $\sigma$ field becomes negative,
allowing $\sigma$ to roll down to its true vacuum.
In this simplest model, however, the dimensionless coupling constant
$\lambda_\phi$ must be fine-tuned so that thermal inflation is allowed, 
which is the same situation as the original model for thermal
inflation\cite{thermal}.
Alternatively, one may consider a flat potential for $\phi$,
\begin{equation}
\label{flat}
V(\phi, \sigma)=\frac{1}{4}\lambda_{\sigma}\left(\sigma^2 + M^2 \right)^2 
-\frac{1}{2}\lambda_{int} \phi^2 \sigma^2
-\frac{1}{2}m_{\phi}^2\phi^2 + A_n \frac{\phi^{n+4}}{M^n}.
\end{equation}
We assume that the potential for the inflaton $\phi$ is flat in the
supersymmetric limit, and is
destabilized by the soft supersymmetry breaking term, which is denoted by
$-\frac{1}{2}m_{\phi}^2\phi^2$.
%1
In this case, the flatness of the potential is ensured by
supersymmetry.
However, a problem arises for the interaction term.
If the flat potential (\ref{flat}) is obtained from a superpotential,
it is hard to obtain the interaction term 
$-\frac{1}{2}\lambda_{int} \phi^2 \sigma^2$ with $\lambda_{int}\sim
O(1)$, while keeping the flatness of the field $\phi$.
For example, if one considers a superpotential of the form
\begin{equation}
W=S(\lambda_\sigma' \sigma^2 -\lambda_\phi' \phi^2 +M^2)
\end{equation}
and assume that $S=0$, the potential looks similar to eq.(\ref{flat}).
However, in order to obtain a flat potential for the inflaton $\phi$,
the constant $\lambda_{\phi}'$ must be fine-tuned.
One may think that this problem can be solved by nonrenormalizable
terms.
However, as is already discussed in ref.\cite{inverted_critical},
it is still hard to obtain a flat potential while making the required
coupling large enough to destabilize the trigger field at the end of
inflation.
Moreover, if one wants to construct ``strong'' thermal
inflation with the number of e-foldings $N_e >35$, 
one should consider another constraint for $\lambda_{\sigma}M^4$.
During thermal inflation, the inflaton field is stabilized at the
symmetric point by the thermal effect.
Thermal inflation starts at $T_{in} \simeq
\left(\frac{1}{4}\lambda_\sigma M^4 \right)^{1/4}$ and ends at
$T_c \simeq m_{\phi}$.
The expansion during this period is $N_e \simeq 
ln\left(\frac{T_{in}}{T_c}\right)$.
In order to obtain large number of e-foldings, the vacuum energy during 
thermal inflation must be as large as $V_0^{1/4}\simeq m_\phi e^{N_e}$.
Within the setups of the conventional models for supergravity,
it seems very hard to satisfy any of the above constraints without adding 
unnatural extra components.

Thus we conclude that the model is not suitable for our purposes.
We will show that these problems are solved in the setups of brane
inflation.

\underline{Non-tachyonic brane inflation due to the D-term}

Here we consider non-tachyonic brane inflation in ref.\cite{matsuda_nontach2},
and examine whether one can construct a model that is suitable for our
purposes.
The model should be similar to the model of inverted hybrid inflation 
that we have discussed above.
For example, we consider a potential of the form
\begin{equation}
\label{simple}
V(\phi, \sigma)=\frac{1}{4}\lambda_{\sigma}\left(\sigma^2 + M^2 \right)^2 
-\frac{1}{2}\lambda_{int} e^{-(M_* r)^2}\phi^2 \sigma^2
+\frac{1}{4}\lambda_{\phi}\left(\phi^2-M'^2\right)^2.
\end{equation}
Here we have considered two branes at a distance, which we denote by 
1 and 2. 
The field $\sigma$ and the field $\phi$ are localized on brane 1 and
brane 2, respectively. 
The interaction term is accompanied by an exponential factor, because 
the field $\phi$ and the field $\sigma$ are localized on
different branes at a distance $r$.
On can easily find that the above effective four-dimensional potential
is a simple modification of the potential for inverted hybrid inflation.
In this model, the inflaton field is the moduli for the brane distance
$r$, which we denote by $\psi=M_*^2 r$.
The most obvious difference is that the field $\phi$ is not required to
be placed at the unstable point.
Inflation starts because the exponential factor is nearly zero at the
beginning of inflation.
The field $\sigma$ is destabilized at the end of inflation, when two
branes come close.

It is possible to express the above idea in the explicit supersymmetric
form\cite{matsuda_nontach2}.
In general, F-term inflation suffers 
from the old serious difficulty even if it is extended to the models for
brane inflation. 
In the past, the idea of D-term inflation was invoked to solve the
problem of F-term inflation in conventional supergravity.
Thus we consider a localized Fayet-Iliopoulos term on a brane at
$\vec{r}=0$
 of the form
\begin{equation}
\label{FI}
\xi D \delta(\vec{r})
\end{equation}
where $D$ is an auxiliary field of the vector superfield.
We consider an additional abelian gauge group $U(1)_X$ in the bulk,
while the Fayet-Iliopoulos term for $U(1)_X$ is localized on a brane. 
We also include the field $\phi_X$ that has $U(1)_X$ charge and
localized on the other brane at $\vec{r}=\vec{r_1}$.
When two branes are located at a distance, $|\vec{r_1}| >>M_*^{-1}$,
the Fayet-Iliopoulos term (\ref{FI}) breaks supersymmetry on the brane
and inflation starts.
In this case, as in the conventional models for brane inflation, 
the inflaton field is the moduli that parametrizes the
brane distance.
The moduli is denoted by $\psi=M_*^2 r_1$, 
where $M_*$ denotes the fundamental scale of the model. 
As we are considering D-term inflation, the mass of the inflaton field
($m_{\psi}$) may be much smaller than the Hubble parameter.
Then a modest limit is $m_{\psi} \ge m_{3/2}$,
where  $m_{3/2}$ is the gravitino mass in the true vacuum.
The trigger field is the localized field $\phi_X$, which develops vacuum
expansion value to compensate the D-term (\ref{FI}) when two branes come
close.

The most significant difference from the original model for inverted
hybrid inflation is that the interaction that destabilizes the trigger
field is accompanied by an exponential factor $e^{-(M^{-1}_{*} \psi)^2}$.
The destabilization in the potential (\ref{simple}) is not due to the
variation of the field $\phi$, 
but due to the variation of the effective interaction constant
$\lambda_{int}e^{-(M^{-1}_{*} \psi)^2}$.

In this case, however, the initial condition is chaotic. 
In this respect, to collaborate with thermal inflation, we should modify
the initial 
condition for the above extended model for hybrid inflation.
To find the solution for the problem, we must first review the original
idea for thermal brane inflation that was advocated in
ref.\cite{thermal_brane}.
In ref.\cite{thermal_brane}, it is discussed that an open string model can be
thermalized to stabilize a brane on the top of the different brane.
Unlike the usual models for brane inflation that starts with a chaotic
initial condition, thermal brane inflation starts at the top of an
another brane.

\underline{Thermal brane inflation (original)}

Let us briefly review the idea of thermal brane
inflation proposed by Dvali\cite{thermal_brane}.
The following conditions are required so that the mechanism functions.

1) \,Exchange of the bulk modes such as graviton, dilaton or RR
fields govern the brane interaction at the large distance.

2)\, In the case when branes initially come close, bulk modes are in
equilibrium and their contribution to the free energy
can create a positive $T^{2}$ mass term for $\psi$ to stabilize the
branes on top of each other until the Universe cools down to a certain
critical temperature $T_{c}\sim m_{s}$.\footnote{
The author of ref.\cite{thermal_brane} 
considered open string modes stretched between different
branes.
When branes are on top of each other, these string modes 
are in equilibrium and their contribution to the free energy creates a
positive $T^2$ mass term. }
Here $m_s$ represents the negative curvature of $\psi$ at the origin,
 which is determined by the supersymmetry breaking.

The resultant scenario of thermal inflation is straightforward.
Assuming that there was a period of an early inflation with a reheat
temperature $T_{R}\sim M$, and at the end of inflation some of the
repelling branes sit on top of each other stabilized by the thermal
effects, one can obtain the number of e-foldings 
\begin{equation}
N_e=ln(\frac{T_{R}}{T_{c}}).
\end{equation}
Taking $T_{R}\sim 10 TeV$ and $T_{c}\sim 10^{3}- 10$ MeV, one finds 
$N_{e}\sim 10-15$, which is consistent with the original thermal
inflation\cite{thermal} and is enough to get rid of unwanted 
relics.
In the original model the crucial restriction appears in $m_s$, which
must be large enough to satisfy the lower limit for the reheating
temperature.
In our model, the above restriction is actually removed.

\underline{Hybrid alternative (Thermal hybrid inflation in the 
brane universe)}

Now it seems straightforward to improve the initial condition for
non-tachyonic brane inflation 
to fit the settings of thermal brane inflation.
To explain the idea, let us consider two branes (brane A and brane B)
fixed on some 
point in the extra dimensions, and a moving brane that is not fixed yet.
In the true vacuum, the moving brane stays on top of the brane A, while 
the thermal effects confine it on top of the brane
B during thermal inflation.
As we have discussed above, the initial condition for the non-tachyonic
brane inflation is satisfied within the above settings.
What we want to consider in this paper is a hybridization of thermal
brane inflation and non-tachyonic brane inflation.
Thermal hybrid inflation occurs if a moving brane, which is responsible
for the supersymmetry on the brane A attaches to an another brane at a
distance.
If the components on each brane are appropriate to meet the
requirement from non-tachyonic brane inflation, thermal inflation starts.
At the end of thermal inflation, the moving brane falls apart from 
the brane B and moves toward the brane A.
When the moving brane come near to the brane A, the spontaneously broken
supersymmetry is recovered and the inflation ends with the oscillation
of the field on the brane.

There are two unique characteristic features in this model.
In conventional models for thermal inflation, there is a lower limit for
the mass of the inflaton field that is derived from the requirement for
the reheating temperature.
This constraint is removed in our model because of the hybrid potential.
The second is that thermal inflation can become ``strong'' in our model.
To be more precise, we show why the conventional models for thermal
inflation were ``weak''.
In generic models for thermal inflation, inflation starts at 
$T_{in}\simeq \sqrt{m_{I}M}$, where $M$ and $m_I$ are the vacuum explain
value of the inflation field in the true vacuum and the mass of the
inflaton field, respectively.
Thermal inflaton ends at the temperature $T_{end}\simeq m_{I}$.
During this period, the Universe expands with the number of e-foldings of
\begin{equation}
N_e \simeq ln \left( \frac{T_{in}}{T_{end}} \right) \simeq
ln \left(\sqrt{\frac{M}{m_{I}}}\right).
\end{equation}
In this case, even if $M$ is as large as the GUT scale,
the number of e-foldings is at most $N_e\simeq 17$ for $m_I\simeq 1$GeV.
Thus we should conclude that the conventional model for thermal
inflation is a model for weak inflation, which cannot be used for the
first inflation. 

In our model, however, the situation is changed.
The energy density during inflation, which we denote by $V_0$, is
not related to the mass of the inflaton field.
The expect number of e-foldings is 
\begin{equation}
N_e \simeq ln \left(\frac{T_{in}}{T_{end}}\right) = 
ln\left(\frac{V_0^{1/4}}{m_{I}}\right).
\end{equation}
For $V_0^{1/4}\simeq M_*=10^{6}$GeV and $m_I\simeq m_{3/2}= 10^2$keV, 
one can obtain a large
number of e-foldings, $N_e \simeq 35$.

\section{Conclusions and Discussions}
\hspace*{\parindent}
In this paper we have constructed an example for thermal hybrid inflation.
In conventional thermal inflation, the requirement from the reheating
temperature puts a lower bound on the mass of the inflaton field.
The mass of the inflaton and the height of the potential during
inflation is intimately related to prevent ``strong'' inflation.
Our naive expectation is that one can remove the above constraints if 
 thermal inflation is realized within the setups for hybrid 
inflation.
With these things in mind, we have constructed a realistic model for
thermal hybrid inflation.
We have considered this problem in the scenarion of the brane world.
Our result drastically modifies the cosmological scenarios related to
thermal inflation.
For example, in ref.\cite{curvaton}, the curvaton hyposesis is 
discussed in the framework of thermal inflation.
In ref.\cite{curvaton}, they have concluded that the cosmological scales
cannot leave the horizon during ordinary thermal inflation, because the
constraint from the reheating temperature cannot meet the requirement.
However, as we have discussed above, such a constraint is removed
in our model.

We believe that our new model for thermal inflation opens up new
possibilities for the brane world.

\section{Acknowledgment}
We wish to thank K.Shima for encouragement, and our colleagues in
Tokyo University for their kind hospitality.

\begin{thebibliography}{1}
\bibitem{inverted}
D. H. Lyth, Ewan D. Stewart, Phys.Rev.D54:7186-7190,1996 
\bibitem{inverted_critical}
S.F. King and J. Sanderson, Phys.Lett.B412:19-27,1997 
\bibitem{matsuda_nontach2}
T.Matsuda, ``Non-tachyonic brane inflation'', ;
`` F-term, D-term and hybrid brane inflation'', 
\bibitem{thermal_brane}
G.R. Dvali, Phys.Lett.B459:489-496,1999 
\bibitem{thermal}
D.H. Lyth and E. D. Stewart, Phys.Rev.D53:1784-1798,1996 
\bibitem{curvaton}
K. Dimopoulos and D. H. Lyth, 
``Models of inflation liberated by the curvaton hypothesis'', 
 
\end{thebibliography}
\end{document}

