\documentstyle[aps,prl,twocolumn,eqsecnum,floats,axodraw,psfig]{revtex}
%\documentstyle[aps,prl,eqsecnum,floats,axodraw,psfig]{revtex}
%\usepackage{graphicx}
%\usepackage{graphics}
\setlength{\textwidth}{7.00in}
\setlength{\textheight}{9.0in}
\setlength{\evensidemargin}{-0.2in}
\setlength{\oddsidemargin}{-0.2in}
\setlength{\topmargin}{0.0in}

% \documentstyle[aps,prl,twocolumn,floats,axodraw,srcltx]{revtex}
%\documentstyle[preprint,aps,axodraw]{revtex}
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}
\def\AofP{Annns. of Phys.}
\def\APP{Astropart. Phys.}
\def\PTP{Orig. Theor. Phys.}
\def\CPC{Comput. Phys. Commun.}
\def\NCA{Nuovo Cimento}
\def\NIM{Nucl. Instrum. Methods}
\def\NIMA{Nucl. Instrum. Methods A}
\def\NPB{Nucl. Phys. {\bf B}$\!$}
\def\NPBproc{Nucl. Phys. B (Proc. Suppl.)}
\def\PLB{Phys. Lett. B}
\def\PR{Phys. Rev.}
\def\PRL{Phys. Rev. Lett.}
\def\PRD{Phys. Rev. D}
\def\ZPC{Z. Phys. C}
\def\RMP{Rev. Mod. Phys.}
\def\MPLA{Mod. Phys. Lett {\bf A}$\!$}
\def\EPC{Eur. Phys. J. {\bf C}$\!$} 
\def\PREP{Phys. Rept.}
\def\ZPC{Zeitschrift f\"ur Physik {\bf C}$\!$}
\def\JHEP{JHEP}
\def\etal{{\sl et al.}} 
\def\KeyWord#1{$\backslash$\IfColor{$\!\!$\textRed{#1}\textBlack}{#1}$\!\!$}

\def\ordere{$\cal{O}(\epsilon)$} 

\begin{document}
\wideabs{

\title{
%{\hfill \small hep-ph/?????????}\\
$~$\\ Supergravity Inflation Free from Harmful Relics }
\author{Patrick~B.~Greene$^{(1)}$, Kenji~Kadota$^{(2)}$,
Hitoshi~Murayama$^{(2)(3)}$ }

\address{$^{(1)}${\it NASA/Fermilab Astrophysics Group, Fermi National
        Accelerator Laboratory, Batavia, IL 60510, USA}} \address{$^{(2)}${\it
        Department of Physics, University of California, Berkeley, CA 94720,
        USA}} \address{$^{(3)}${\it Theory Group, Lawrence Berkeley National
        Laboratory, Berkeley, CA 94720, USA}}








% \date{\today}
\maketitle
\begin{abstract}

We present a realistic supergravity inflation model which is free from the
overproduction of potentially dangerous relics in cosmology, namely moduli and
gravitinos which can lead to the inconsistencies with the predictions of baryon
asymmetry and nucleosynthesis.  The radiative correction turns out to play a
crucial role in our analysis which raises the mass of supersymmetry breaking
field to intermediate scale. We pay a particular attention to the non-thermal
production of gravitinos using the non-minimal K\"ahler potential we obtained
from loop correction.  This non-thermal gravitino production however is
diminished because of the relatively small scale of inflaton mass and small
amplitudes of hidden sector fields.

\end{abstract}
}

\narrowtext

\setcounter{footnote}{0} \setcounter{page}{1} \setcounter{section}{0}
\setcounter{subsection}{0} \setcounter{subsubsection}{0}


\section{Introduction}
 
There exist generic problems in constructing supergravity inflation models with
broken local supersymmetry in its vacuum.  Firstly we need to carefully choose
the superpotential and K\"ahler potential so that the non-renormalizable terms
should not spoil the flatness of inflaton potential and they can produce the
observed CMB spectrum.  Secondly most of the supergravity inflation models
predict the possible cosmological problems due to the abundant moduli and
gravitinos, and we need to make sure that 
a model is free from these problems not to
upset the data of baryon asymmetry and nucleosynthesis predictions.  Notably,
the nonthermal production of gravitinos has been analyzed recently
\cite{marco,long,toni,dangerous,firstgrav} and we pay a particular attention to
this problem in this letter.  These problems on gravitino production during
preheating era have been analyzed using the minimal K\"ahler potential so far,
but there are cases when we should be more careful about the form of K\"ahler
potential in dealing with gravitino problems. This is particularly the case
when the supersymmetry breaking field has flat direction at tree level and
radiative correction has a significant effect on its potential, which requires
the modification to minimal K\"ahler potential.
%{\bf Point out distinction between renormalizable and non-renormalizable
%hidden sector.-PBG}

The detailed analysis for nonthermal production of gravitinos in a system of
coupled fields has been done so far only for a specific supergravity inflation
model where a superpotential consists of an inflaton sector responsible for 
the inflation and supersymmetry breaking hidden sector of Polonyi potential
\cite{marco,marco2}, and it was shown that dominant fermion fields which are created
efficiently through preheating mechanism are inflatinos rather than gravitinos
(thus free from gravitino problem). This type of supergravity inflation model
is a good toy model to investigate the nonthermal particle production in
preheating era, but is not a realistic inflation model in that it
 sill suffers from Polonyi problem which is as serious a
problem as gravitino problem.  We give the first realistic supergravity
inflation model in this sense where nonthermal production of gravitinos as well
as moduli problem were explicitly analyzed.

The paper is structured as follows.  In section \ref{setup}, we explain our
choice of superpotential, and discuss how inflation develops in our model. We
then discuss how supersymmetry breaking field evolves and calculate its
radiative correction and its modification to minimal K\"ahler potential arising
from this loop correction.  In section \ref{dangerousrelics}, we see if our
model leads to any cosmological crisis, namely, moduli and gravitino problems.
We give the conclusion and discussion at the end.
%
%{\bf Contrast results for poloni with this model.-PBG}


\section{Setup}
\label{setup}


We consider the superpotential \cite{holman} consisting of the inflaton sector
and hidden sector of O'Raifeartaigh type \cite{or} which are gravitationally
coupled to each other.
%
\begin{equation}
W=\Delta^2~
\frac{(\Sigma-M)^2}{M_p}+\Phi_1\left(\kappa\Phi_2^2-\mu^2\right)+\lambda\Phi_2\Phi_3+C,
% W=\frac{m_\phi}{2}\,\Phi^2+\mu^2\,\left(\beta+S\right)
\label{sup}
\end{equation}
where superfield $\Sigma$ includes inflaton scalar component $\sigma$ and
$\Phi_1$ includes O'Raifeartaigh scalar field $\phi_1$. $ \Delta \sim 
10^{-4} M_p$ from COBE normalization and $M$ is set to $M_p (= 2.436
\cdot 10^{18} \mathrm{GeV} ) $ so that the inflaton potential keeps the
flatness around the origin (i.e. for $ \frac{\partial}{\partial \sigma}V(0)
\simeq 0, \frac{\partial^2}{\partial \sigma^2}V(0) \simeq 0 $ ).  The
dimensionless parameter $\kappa$ is of order unity while the other mass
parameters $\mu$ and $\lambda$ are of intermediate scale ( $\sim 10^{-8} M_p $
).  $C$ is the constant term to cancel the cosmological constant at the vacuum.
We start with the discussion for the evolution of scalar fields in inflaton
sector and hidden sector which are only gravitationally coupled to each other
and, for the sake of clarity, we first treat each sector separately followed by
the discussion including the coupling with non-minimal K\"ahler potential
arising from the radiative correction. We assume gauge singlets in the
potential for simplicity in the following.
%
\subsection{Inflaton Sector}
%
The superpotential of inflaton sector is given as
\begin{equation}
W_{inflaton}=\Delta^2~ \frac{(\Sigma-M_p)^2}{M_p}.
% W=\frac{m_\phi}{2}\,\Phi^2+\mu^2\,\left(\beta+S\right)
\label{sup}
\end{equation}
The general expression for supergravity potential becomes, for K\"ahler
potential, $K$, and superpotential, $W$,
\begin{equation}
V = m_i~{K^{-1}}_j{}^i ~m^j- 3~M_p^{-2} \vert m \vert^2
\label{SUGRApot}
\end{equation}
with
\begin{eqnarray}
K^i{}_j\equiv \frac{\partial^2 K }{\partial \phi_i \partial\phi^j},~~ m \equiv
\mathrm{e}^{\frac{K}{2M_p^2}} W \\ m^i \equiv D^i m \equiv \partial^i m +
\frac{1}{2 M_p^2} ~(\partial^i K) ~m ~.
\label{eqn:m}
\end{eqnarray}
If we consider, for the moment, the minimal form of K\"ahler potential,
% ${\cal {K}}
$K=\Sigma^\dagger\,\Sigma$, the effective supergravity potential from
$W_{inflaton}$ for the real part of scalar component, $\sigma$, becomes ( in
natural units )
\begin{eqnarray}
V_{inflaton}=\mathrm{e}^K \left(\left \vert \frac{\partial W}{\partial \Sigma}+
\Sigma^{\dag}W \right\vert^2-3\vert W\vert^2 \right) \label{eq:minimal} \\
=\Delta^{4}~\mathrm{e}^{\sigma^2/2}~\left(1-\frac{\sigma^2}{2}
-\sqrt{2}\sigma^3+\frac74\sigma^4-\frac{1}{\sqrt{2}}\sigma^5+\frac{\sigma^6}{8}\right).
\label{eqn:Vinflaton}
\end{eqnarray}
We here point out the absence of linear and quadratic terms which enables the
potential to keep the flatness around the origin. We can obtain the 
value of $\Delta\sim10^{-4} M_p$
 from COBE normalization condition\cite{white} \footnote{ \cite{holman} gives 
an order of estimates
$10^{-4}M_p\leq \Delta \leq 10^{-3.5}M_p$ from the constraints on gravitino
abundance and proton decay.},
\begin{equation}
\left(\frac{V}{\epsilon}\right)^{\frac{1}{4}}\simeq0.027M_p ( 1-3.2 \epsilon +
0.5 \eta),
\end{equation}
which  should be evaluated at the horizon exit. Scale of inflaton field 
when the inflation ends and the cosmological scales 
leave the horizon are obtained from slow-roll conditions,
$\epsilon \lesssim 1$, $\eta \lesssim 1$, and 60 e-folding
 condition, 
\begin{equation}
N(\sigma_{exit}) \simeq
\int\limits_{\sigma_{end}}^{\sigma_{exit}}\frac{V}{V'}d\sigma \,
\simeq 60.
\end{equation}
  We also note the scale of inflation is of the
order $\Delta^{4}$ and the mass of the inflaton is of the order
$\Delta^{2}/M_p $ with its decay width $ \Gamma_{\sigma}\simeq
m_{\sigma}^3/M_p^2=\Delta^6/M_p^5 $ assuming gravitational strength coupling to
ordinary fields.
%
\subsection{Hidden Sector}
%
The supersymmetry breaking sector is that of O'Raifeartaigh model,
\begin{equation}
W_{hidden}=\Phi_1\left(\kappa\Phi_2^2-\mu^2\right)+\lambda\Phi_2\Phi_3+C ~~.
\end{equation}

This is a familiar example of supersymmetry breaking due to non-vanishing
 $F$-term from $\Phi_1$, $\vert F \vert=\mu^2 $, in the vacuum. Therefore we
 add $C=\frac{\mu^2}{\sqrt{3}}M_p$ (compare with $\frac{-3\vert W\vert^2}{M_P}$
 term in eqn(\ref{eqn:Vinflaton}) ) for the vanishing cosmological constant at
 the vacuum.\footnote{ This additional constant term $C$, strictly speaking,
 should be modified if we include the radiative correction and the coupling
 between inflaton and hidden sectors.  We, however, stick to this value of $C$
 for simplicity because this modification essentially doesn't change our
 discussion.} We should, however, expect that, when the fields are far away
 from the vacuum during the inflation, there are additional $F$-terms from
 other fields in the effective scalar potential and these $F$-terms lead to
 additional `cosmological constant' $ \Lambda^{4}$ \cite{dine}. Adding
 ${e}^{K}\Lambda^{4}$ in the potential indicates us that this cosmological
 constant term during the inflation gives an additional effective mass of order
 $\frac{\Lambda^{4}}{M_p^{2}}$ to each field in the model. Fields $\phi_2$ and
 $\phi_3$ do not posses the linear terms and these two fields quickly roll down
 to the origin ( i.e. to their minimum) during the inflation. The scalar field
 $\phi_1$, however, has a liner term and its minimum shifts according to the
 evolution of inflaton field as we shall see in the next section. Because we
 are interested in the particle production after the inflation, we focus on the
 evolution of O'Raifeartaigh field $\phi_1$ among the fields in this
 supersymmetry breaking sector.  Moreover, the $F$-term at the vacuum has the
 contribution only from the scalar field $\phi_1$ which turns out to have flat
 direction at the tree level. Because of this flat potential with respect to
 $\phi_1$ at the tree level, the global supersymmetry radiative corrections
 have a significant effect on the effective potential and consequently give
 non-negligible modification to the minimal K\"ahler potential. The radiative
 corrections of local supersymmetry are always Planck mass suppressed and we do
 not consider them here for simplicity. We calculated this non-minimal K\"ahler
 potential from the calculation of loop correction \cite{huq,jac}.  Setting the
 parameter range to be $2\kappa\mu^2 < \lambda^2$ to make $ \phi_2$ and $\phi_3
 $ stay at the origin in the vacuum leads to the following one-loop correction
 due to $\phi_1$,
\begin{eqnarray}
V_{one\,loop} = \frac{1}{ 64\pi^{2} } \left(
\sum_{i=1}^{4}
(M_{i}^{2})^2 \left( \log(\frac{M_{i}^{2}}{\lambda^{2}}) - 
\frac{3}{2}\right)  \right.\\
\left. -2\sum_{i=1}^{2}(N_{i}^{2})^{2} \left( \log(\frac{N_{i}^{2}}{\lambda^{2}}  -
\frac{3}{2}\right) \right ),
\nonumber \\ 
\end{eqnarray}
where we have defined
\begin{eqnarray}
& &M_{1}^2=\frac{1}{2}(A_{1}-A_{2}),
M_{2}^2=\frac{1}{2}(A_{1}+A_{2}), \nonumber \\ &
&M_{3}^2=\frac{1}{2}(A_{3}-A_{4}), M_{4}^2=\frac{1}{2}(A_{3}+A_{4}), \nonumber
\\ & &N_1^2=\frac{1}{2}(B_1-B_2), N_2^2=\frac{1}{2}(B_1+B_2), \nonumber \\ &
&A_1=2\lambda^{2}-2\kappa\mu^{2}+4\kappa^{2}\vert \phi_1 \vert ^{2},~~
A_3=2\lambda^{2}+2\kappa\mu^{2}+4\kappa^{2}\vert \phi_1\vert ^{2}, \nonumber \\
& &A_2=\sqrt{ 4\kappa^{2}\mu^{4}+16\lambda^{2}\kappa^{2}\vert \phi_1 \vert
^{2}- 16\kappa^{3}\mu^{2}\vert \phi_1 \vert ^{2}+16\kappa^{4} \vert \phi_1
\vert ^{4} }, \nonumber \\ & &A_4=\sqrt{
4\kappa^{2}\mu^{4}+16\lambda^{2}\kappa^{2} \vert \phi_1 \vert ^{2}
+16\kappa^{3}\mu^{2} \vert \phi_1 \vert ^{2}+16\kappa^{4} \vert \phi_1 \vert
^{4} }, \nonumber \\ & &B_1=2\lambda^{2}+4\kappa^{2} \vert \phi_1 \vert ^{2},~~
B_2=\sqrt{ 16\lambda^{2}\kappa^{2} \vert \phi_1 \vert ^{2} +16\kappa^{4} \vert
\phi_1 \vert ^{4} }\nonumber \, . \\
\end{eqnarray}
We have used the $\overline{MS}$ scheme and taken the renormalization
scale to be $\lambda$.
% {\bf Formating issue above.-PBG}
We are concerned with the regime $|\phi_1| \lesssim \lambda/\kappa$, as we
shall show in the next section.  In this case, and for $2\kappa\mu^2 \ll
\lambda^2$,  we can approximate the one loop potential as
%\begin{eqnarray}
% \lefteqn{ V_{one\,loop}= \nonumber } \\
%& & \frac{1}{16\pi^{2}}\kappa^{2}\mu^{4}\log\left(2\lambda^{2}+4\kappa^{2}\phi_1^{2}+2\kappa\phi_1\sqrt{ 4\lambda^{2}+4\kappa^{2}\phi_1^{2}}\right) \nonumber~. \\
%\end{eqnarray}
%This can be further put into the form of, in the asymptotic regime, 
\begin{equation}
V_{one\,loop} = C_1 + \frac{\kappa^{2}\mu^{4}}{8\pi^{2}}
 \left(\frac{\kappa^{2}\vert \phi_1 \vert
 ^{2}}{\lambda^2} + \ldots \right) \, ,
\label{Vloop}
\end{equation}
%{\bf Need to clarify the numerical factor here and where it propagates
%through the rest of the paper.  I've left it unchanged for now.-PBG \\ }
where $\ldots$ are terms of order $\frac{\kappa^4|\phi_1|^4}{\lambda^4}$ and
higher and $C_1$ is a small ($\ll \mu^4$) constant
term which can be absorbed into the constant part of the superpotential.
We can find the one loop correction to the K\"ahler potential 
($K=K_{minimal}+K_{correction}$), by comparing eqn(\ref{Vloop})
with eqn(\ref{SUGRApot}) to be 
\begin{equation}
K_{correction}=- \frac{\kappa^{2}}{32\pi^{2}} \left(\frac{\kappa^{2}\Phi_1^2
\Phi_1^{\dagger 2} }{\lambda^2} \right) .
\label{eqn:correction}
\end{equation}  
We shall use this non-minimal K\"ahler potential in our analysis for
 non-thermal production of gravitinos.  We note that this radiative
 correction enhances the coupling to the longitudinal component of gravitino
 and raises the mass of $\phi_1$ which was massless at the tree level to the
 intermediate scale
\begin{equation}
m_{\phi_1}^2= \frac{\kappa^4\mu^4}{4\pi^2\lambda^2}= \frac{
\alpha_{\kappa}^2\mu^4}{ \lambda^2} ~\mbox{with} ~ \alpha_{\kappa} \equiv
\frac{\kappa^2}{4\pi},
\label{eqn:ormass}
\end{equation}
which turns out to be crucial to evade the moduli problem.


\section{Harmful Relics}
\label{dangerousrelics}

We are now in a position to discuss the fate of inflaton and O'Raifeartaigh
fields in the coupled effective potential with non-minimal K\"ahler potential
to see if our model leads to any cosmological crisis.  We first briefly review
the resolutions of so-called Polonyi or moduli problem and we further discuss
the non-thermal production of gravitino. \\
\subsection{ Moduli Problem}
We here start with the discussion on well-known potentially dangerous problems,
Polonyi problem or, in general, moduli problem.  There are two aspects which we
should worry about before our discussion on decay of moduli into
gravitinos. One is the case when the moduli decay very late ( i.e. during or
after the nucleosynthesis) which can jeopardize nucleosynthesis predictions
because of ultra-relativistic decay products directly from moduli fields
destroying the light elements, in particular, ${}^4$He and D nuclei. The other
is when the entropy release due to its decay is so big that it can over-dilute
the baryon asymmetry well below its acceptable amounts (so-called `entropy
crisis').  Our model does not have either of these problems because the
radiative correction raises its mass to as much as intermediate scale. Its
decay width is indeed enhanced up to $\Gamma_{\phi_1}\simeq m_{\phi_1}^5/\vert
F \vert^2\simeq \frac{\left(\mu^2 \alpha_{\kappa}/ \lambda\right)^5}{\mu^4}
\simeq \alpha_{\kappa}^5\mu $ with $ \alpha_{\kappa} \equiv \kappa^2/4\pi
\simeq O(10^{-1})$ and this is of order$10^{-13} M_p$.  So O'Raifeartaigh field
decays around $ 10^{13} M_p^{-1}$ in our model which is much before the
nucleosynthesis starts around $\sim 10^{40}M_p^{-1} $ and even well before the
reheating starts due to the inflaton decay around $ 1/\Gamma_{\sigma}\sim
M_p^5/\Delta^6 \sim 10^{25} M_p^{-1} $.  We however need a great care about the
possibility of decay products with long life time, especially
gravitinos. Gravitino decay rate is of order $ \Gamma_{m_{3/2}} \sim
m_{3/2}^3/M_p^2 \sim \mu^6/M_p^5 \sim 10^{-48}M_p $ and its relativistic decay
products, especially ultra-relativistic photon/photino, can destroy the light
elements in nucleosynthesis (photo-dissociation process) as we just mentioned.
The possible abundant gravitino production from O'Raifeartaigh fields can be
caused by the energy release stored during the inflation by the shift of the
minimum of SUSY breaking field potential as inflaton evolves.  If this energy
release is too big, it could lead to large amount of gravitinos and upset the
nucleosynthesis predictions.  We can see this is not the case for our model as
follows \cite{holman,izumi,randall}.  During the inflation, due to the coupling
to the inflaton ($\sigma \sim M_p $), O'Raifeartaigh field amplitude is around
the intermediate scale of order $\phi_1 \simeq \frac{\mu^2}{\Delta^2}M_p $ at
the minimum of its potential. Therefore we can estimate the energy stored in
this O'Raifeartaigh field when it starts oscillation ( i.e. $t_{\phi_1}\simeq
m_{\phi_1}^{-1}$) to be at most of order
\begin{equation}
\rho(t_{\phi_1}) \simeq m_{\phi_1}^2 \frac{\mu^4}{\Delta^4}M_p^2
\end{equation}
and its number density $n_{\phi_1}$ in this coherently oscillating
O'Raifeartaigh field is at most
\begin{equation}
n_{\phi_1}(t_{\phi_1})\simeq m_{\phi_1} \frac{\mu^4}{\Delta^4}M_p^2 .
\end{equation}
We can now estimate its number density to entropy ratio at the time of
 reheating for the gravitinos through the decay of $\phi_1$ ( at $ t=t_r $ ,
 say ) in an adiabatically expanding universe.  Assuming, for the upper bound,
 $\phi_1$ solely decays into gravitinos with 100$\%$ branching ratio and using
% $1.66 \cdot T_{RH}^2 \sqrt{g_*} / M_p \sim H \sim t_r^{-1}$,
$ s \sim \frac{2 \pi^2}{45} g_* T^3 $ (with $g_*$ effective degree of freedom)
and $a^3\sim t^2$ for matter domination \cite{kolb} due to the coherently
oscillating inflaton field which dominates the energy in the universe,
\begin{eqnarray}
\frac{n_{3/2} (t_r)}{s(t_r)} \simeq\frac{n_{\phi_1}(t_{\phi_1})\left(
\frac{a(t_{\phi_1})}{a(t_r)} \right )^3}{0.44g_* T_{RH}^3} \nonumber \\ \simeq
\frac{n_{\phi_1}(t_{\phi_1})\left( \frac{t_{\phi_1}} {t_r} \right )^2}{0.44 g_*
T_{RH}^3} .
\end{eqnarray}
This can lead to the estimate of $n_{3/2}/s$ after reheating by substituting
$1.66 \cdot T_{RH}^2 \sqrt{g_*} / M_p \sim H \sim t_r^{-1}$ for $t_r$,
\begin{equation}
\frac{n_{3/2}}{s} \simeq \frac{n_{\phi_1}(t_{\phi_1}) t_{\phi_1}^2 T_{RH} (
1.66)^2} {0.44 M_p^2} \simeq \frac{ (1.66)^2 \mu^4 T_{RH} } { 0.44 m_{\phi_1}
\Delta^4}.  \label{eqn:orns}
\end{equation}
We can now compare this value with one corresponding to the gravitinos produced
by the scattering in the thermal bath in reheating era obtained in
MSSM\cite{kawasaki,moroi2},
\begin{equation}
n/n_{rad}(T \ll 1\mbox{MeV}) \simeq 1.1 \cdot 10^{-11} \left (
\frac{T_{RH}}{10^{10} \mbox{GeV} } \right) .
%\left ( 1-0.0232 \log \left 
%( \frac{T_{RH}}{10^{10} \mbox{GeV} } \right)\right) .
\nonumber \\
\label{eqn:whatever}
\end{equation}
% k&t page 64, 67
Using $s=1.8\cdot g_* n_{rad}$ and $g_*(\ll MeV)\simeq 3.36$, we obtain
\begin{equation}
 n/s \simeq 1.8 \cdot 10^{-12} \left( \frac{T_{RH}}{10^{10} \mbox{GeV } }
\right)
\label{eqn:ns} ~.
\end{equation}
%, where we ignore a small correction due to the renormalization group flow of the gauge coupling. 
%Substituting $T_{RH}=10^9 GeV$ for $m_{3/2}\simeq 100GeV$, 
We can now transform eqn(\ref{eqn:orns}) by substituting (\ref{eqn:ormass}) 
to the following form,
%we obtain the ratio of gravitino number density from $\phi_1$ decay
%to the entropy, 
\begin{equation}
n/s \simeq 3.5 \cdot 10^{-14}\left(\frac{T_{RH}}{10^{10}\mbox{GeV}}\right) ~~.
\end{equation}
 This is smaller than the thermal production of gravitino (\ref{eqn:ns}) by two
 orders of magnitude. The radiative correction therefore induces intermediate
 mass scale for O'Raifeartaigh field and it consequently makes the
 O'Raifeartaigh field energy released through the decay into gravitinos after
 the inflation small enough to evade the abundant gravitinos.  Hence our model
 does not suffer from moduli problem as far as the constraint from thermal
 gravitino production is satisfied.

\subsection{Non-thermal Production of Gravitino}
\label{sec:nonthermal}
It has been argued recently that parametric resonance mechanism in preheating
era for the creation of gravitino can be much more efficient than the thermal
one \cite{marco,long,toni,dangerous,firstgrav}.  In this nonperturbative
mechanism, the gravitinos can be created non-adiabatically through the
amplification of vacuum fluctuation via rapid energy transfer from coherently
oscillating inflaton field which still dominates the energy density in the
universe just after inflation and before the reheating era.

In analyzing the gravitino field equations in the following, we see 
that the equations for transverse and longitudinal components of
gravitino decouple. While transverse component equation has a Planck mass
suppressed coupling and thus gravitationally suppressed particle creation,
longitudinal component equation is free from Planck mass suppression and it
could lead to the abundant gravitino production well above the constraint from
thermal production of gravitino.
%As a matter of fact, it can 
%be shown that longitudinal components of gravitinos does not 
%decouple even in the limit of $M_p \rightarrow \infty $.
 Moreover, gravitino-goldstino equivalence theorem states that the equation for
gravitino longitudinal component can be reduced to the equation of goldstino in
global supersymmetry in the limit of weak gravitational coupling.  This warns
us that gravitino longitudinal components could lead to its efficient copious
production without Planck mass suppression.  Our model however has a Planck
scale amplitude for inflaton field after inflation, and it is not obvious if
this naive intuitive picture analogous to the goldstinos in global SUSY applies
here.
%We instead apply the formulation recently developed in [, lev,
%marco] taking into account all the supergravity effects with non-minimal
%K\''hler potential we derived in section [].
%The analysis of non-thermal production of gravitino is rather model dependent 
%and  
We here instead apply the formalism developed in \cite{marco,long} to
calculate the
number density of gravitinos created through the nonthermal process.

We first need to describe the evolution of scalar fields and fermion fields and
their interactions.  It is convenient to work with, among other possible
choices, the following rescaled quantities in our numerical analysis,
\begin{eqnarray}
\lefteqn{ {\hat \phi_1} \equiv\frac{\phi_1}{M_p}, ~~ {\hat \phi_2}
\equiv\frac{\phi_2}{M_p},~~ {\hat \phi_3} \equiv\frac{\phi_3}{\Delta},~~ {\hat
\sigma} \equiv \frac{\sigma}{M_p},~~ {\hat \mu} \equiv\frac{\mu}{\Delta},
\nonumber }\\ & & {\hat \lambda} \equiv \frac{\lambda}{\Delta},~~ {\hat t}
\equiv t~\frac{ \Delta^2}{M_P},~~ {\hat H} \equiv H \frac{M_p}{\Delta^2},~~
{\hat V} \equiv \frac{V}{\Delta^4} , \end{eqnarray} where $H$ is Hubble
constant, $V$ is a scalar potential from eqn (\ref{SUGRApot}) with non-minimal
K\"ahler potential obtained in (\ref{eqn:correction}),
\begin{eqnarray}
\!\!\!K=\Sigma\Sigma^{\dag}+\Phi_1\Phi_1^{\dag}+\Phi_2\Phi_2^{\dag}+
\Phi_3\Phi_3^{\dag} \nonumber \\ - \frac{\kappa^{2}}{32\pi^{2}}
\left(\frac{\kappa^{2} \Phi_1^2 \Phi_1^{\dag2 }}{\lambda^2} \right) .
\end{eqnarray}
 In terms of these rescaled quantities, the equations of motion for coherently
oscillating scalar fields $\phi$($= \sigma, \phi_i$) read
\begin{equation}
\frac{d^2 {\hat \phi}}{d {\hat t}^2} + 3 \, {\hat H} \, \frac{d {\hat \phi}}{d
{\hat t}} + \frac{d {\hat V}}{d {\hat \phi}} = 0 ~~.
\end{equation}
We omit $\hat{}$~ in the following discussion as long as it is clear from the
 context.  We can concentrate on the field equations for $\sigma$ and $\phi_1$
 because the other fields in supersymmetry breaking sector quickly roll down to
 the origin during the inflaton and stay there\footnote{ Once these fields roll
 down to the origin, they stay at the origin to any higher order because of
 R-symmetry.}. So we can let the amplitudes of $\phi_2$ and $\phi_3$ vanish
 after obtaining the equation of motion for $\sigma$ and $\phi_1$ to see the
 field evolutions after the inflation.

%The evolutions of inflaton and O'Raifeartaigh fields are given in 
%figure\ref{fig1}.








The Fermion equation follows from the supergravity Lagrangian
\begin{eqnarray}
e^{-1} L = -\frac{1}{2} \, M_p^2 \, R - K_i{}^j \left( \partial_\mu \, \phi^i
\right) \left( \partial^\mu \phi_j \right) - V \nonumber\\ -\,\frac12 \, M_p^2
\, \bar \psi_\mu \, R^\mu + \frac12 \, m \, \bar \psi_{\mu R} \, \gamma^{\mu
\nu} \, \psi_{\nu R} \nonumber\\ +\,\frac12 \, m^* \, \bar \psi_{\mu L} \,
\gamma^{\mu\nu } \, \psi_{\nu L} - K_i{}^j \left[ \bar \chi_j \, \not\!\! D
\bar \chi^i + \bar \chi^i \not\!\! D \bar \chi_j \right] \nonumber \\ - m^{ij}
\, \bar \chi_i \, \chi_j - m_{ij} \, \bar \chi^i \, \chi^j \nonumber \\ +
\left( 2 \, K_j{}^i \bar \psi_{\mu R} \, \gamma^{\nu \mu} \, \chi^j \,
\partial_\nu \phi_i + \bar \psi_R \cdot \gamma \upsilon_L + \mbox{h.c.} \right)
\nonumber \\ +\mbox{( four fermion and gauge interaction terms )}.
\label{lag}
\end{eqnarray}
This Lagrangian includes chiral complex multiplets $( \phi_i,\, \chi_i )$ and
the Ricci scalar $R$.  Subscript $L$ and $R$ denote its projection through
operators $P_L \equiv ( 1 + \gamma_5 )/2, ~P_R \equiv ( 1 - \gamma_5 )/2$.
Gravitino kinetic term shows up in the form of
\begin{equation}
R^\mu = e^{-1} \, \epsilon^{\mu \nu \rho \sigma} \, \gamma_5 \, \gamma_\nu \,
D_\rho \, \psi_\sigma\,,
\end{equation}
with covariant derivative
\begin{equation}
D_\mu \psi_\nu = \left( \left( \partial_\mu + \frac{1}{4} \omega_\mu^{m n}
\gamma_{m n} \right) \delta_\nu^\lambda - \Gamma_{\mu \, \nu}^\lambda \right)
\psi_\lambda\,.
\end{equation}
The kinetic term for chiral fermion is
\begin{eqnarray}
\lefteqn{D_\mu \chi_i \equiv \left( \partial_\mu + \frac{1}{4} \omega_\mu^{m n}
\gamma_{m n} \right) \chi_i \nonumber }\\ & &+ \frac{1}{4M_p^2} \left[
\partial_j K \partial_{\mu}\phi^j - \partial^j K \, \partial_\mu \phi_j \right]
\chi_i + \Gamma_i^{j\,k} \chi_j \partial_\mu \phi_k\,
\end{eqnarray}
with K\"ahler connection $\Gamma_i^{j\,k} \equiv {K^{-1}}_i{}^l \partial^j
K_l{}^k $ and $\gamma_{m\,n} \equiv [ \gamma_m, \gamma_n ] /2$ .  Its mass term
reads
\begin{equation}
m^{i j} \equiv D^i D^j m = \left( \partial^i + \frac{1}{2M_p^2} (\partial^i K )
\right) m^j - \Gamma_k^{i\,j} \, m^k\,.
\end{equation}
The combination of matter fields gives left-handed component of goldstino
\begin{equation}
\upsilon_L \equiv m^i \, \chi_i + \left( \not\!\partial \phi_i \right) \chi^j
\, K_j{}^i\,.\label{gold}
\end{equation} 
The supersymmetry transformation of goldstino \cite{long}
\begin{equation}
\delta\upsilon=-\frac{3M_p^2}{2}(m_{3/2}^2 + H^2)~\epsilon,~
m_{3/2}\equiv\frac{\vert m \vert }{M_p^2}
\end{equation}
indicates that gravitino mass and Hubble parameter signal supersymmetry
breaking.  We can obtain the gravitino equation from this Lagrangian,
\begin{equation}
\not\!\! D\psi_{\mu}+m\psi_{\mu}=\left(D_{\mu}-\frac{m}{2}\gamma_{\mu}\right)
\gamma^{\nu}\psi_{\nu} .
\end{equation}
In solving this gravitino equation of motion, we gauge away the goldstino(
unitary gauge ) and use plane-wave ansatz for the spatial dependence of
$\psi_{\mu}\sim {e}^{i \boldmath k \cdot \boldmath x }$.  Moreover it is
convenient to decompose the space component of gravitino field into the
transverse part $\psi_{i}{}^{T} $ and trace parts $\theta \equiv \gamma^i
\psi_i$ and $k_i \psi_i$ as
\begin{equation}
\psi_i = \psi_i^T + \left( P_\gamma \right)_i \, \theta + \left(P_k\right)_i
k_i \psi_i ,
\end{equation}
where
\begin{eqnarray}
\left( P_\gamma \right)_i &\equiv& \frac{1}{2} \left(\gamma^i -
\frac{1}{\vec{k}^2} \, k_i \left( k_j \, \gamma^j \right)\right), \nonumber\\
\left(P_k \right)_i &\equiv& \frac{1}{2 \, \vec{k}^2} \left( 3 \, k_i -
\gamma_i \left( k_j \, \gamma^j \right) \right).
\end{eqnarray}

This leads to the the following succinct form of dynamical field equations
which describe the degree of freedom corresponding to transverse and
longitudinal components,
\begin{eqnarray}
\left[ \gamma^0 \, \partial_0 + i \, \gamma^i \, k_i + \frac{\dot{a} \,
\gamma^0}{2} + \frac{a \, \underline{m}}{M_p^2} \right] \psi_{i}{}^{T} = 0\, ,
\label{eq:t} \\ \left(\partial_0 + \hat{B} + i \, \gamma^i \, k_i \, \gamma^0
\, \hat{A} \right) \theta - \frac{4}{\alpha \, a} \, k^2 \, \Upsilon = 0\,,
\label{eq:l}
\end{eqnarray}
where
\begin{eqnarray}
\Upsilon = K_j{}^i \left( \chi_i \, \partial_0 \, \phi^j + \chi^j \, \partial_0
\, \phi_i \right)\nonumber\\ \underline{m} = P_R \, m + P_L \, m^*\,,\qquad
\vert m \vert^2 = \underline{m}^\dag \, \underline{m}\, \nonumber\\ \hat{A} =
\frac{1}{\alpha} \left( \alpha_1 - \gamma^0 \, \alpha_2 \right) \nonumber \\
\hat{B} = - \frac{3}{2} \, \dot{a} \, \hat{A} + \frac{1}{2 \, M_p^2} \, a \,
\underline{m} \, \gamma^0 \left(1 + 3 \, \hat{A} \right) \nonumber \\ \alpha =
3 \, M_p^2\left(H^2+ \frac{\vert m \vert^2}{M_p^4} \right)\nonumber\\ \alpha_1
= - M_p^2 \left(3 \, H^2 + 2 \, \dot{H} \right) - \frac{3}{M_p^2} \, \vert m
\vert^2, ~\alpha_2 = 2 \dot{ \underline{m}^\dag }.
\label{manydef}
\end{eqnarray}
We can easily see, reducing the equation into this form, eqn(\ref{eq:t})
describing the transverse component of gravitino $\psi_i^T$ is decoupled from
the longitudinal gravitino component, and its coupling to the other fields are
Planck mass suppressed. So we hereafter pay our attention to the equation which
describes the longitudinal component of gravitino, eqn(\ref{eq:l}).
%\footnote{ This of course doesn't mean 
%we can always neglect the
%transverse component contribution of gravitino, but it would be 
%relatively small compared with the longitudinal component 
%due to Planck mass suppression in most of the inflation models}
%%
%% added
The form of $\Upsilon$ in (\ref{manydef}) tells us that, in the absence of
K\"ahler terms which mix the various left chiral superfields, we need only worry
about the fermionic partners of dynamical scalar fields.  Furthermore,
for our superpotential, there is
no mixing between the fermion associated with $\phi_1$ and those of
$\phi_2$ and $\phi_3$, as long as $\phi_2=\phi_3=0$ which is true 
because they stay at the origin due to R-symmetry once they 
roll down to the origin during the inflation.  Thus, even though
the effective masses of the fermions corresponding to $\phi_2$ and $\phi_3$
are changing, those fermions do not contribute to the goldstino and we can
 concentrate on the evolution of the other fields for our purpose.

Based on the form of equation of motion involving two chiral superfields,
 we can infer the terms in the Lagrangian which describe the interactions
of the
 two fields of our interests, namely, $ \theta$ ( longitudinal component of
 gravitino) and $\Upsilon$ ( combination of chiral fermions orthogonal to
 goldstino $\upsilon$).
Those interaction terms lead to the equation of motion
 in the following matrix form,

\begin{equation}
\left( \gamma^0 \, \partial_0 + i \, \gamma^i \, k_i N + M \right) X =
0\,,\label{eqn:fermioneom}
\end{equation}
 with vector $X \equiv\left (\frac {\tilde \theta} {\tilde \Upsilon} \right)$
consisting of canonically normalized fields
\begin{eqnarray}
\theta &=& \frac{2 \, i \, \gamma^i \, k_i}{\left(\alpha \, a^3 \right)^{1/2}}
\, {\tilde \theta}\,, \nonumber\\ \Upsilon &=& \frac{\Delta}{2} \left(
\frac{\alpha}{a} \right)^{1/2} {\tilde \Upsilon} ,
\end{eqnarray}
and diagonal mass matrix $M$ is given by
\begin{eqnarray}
M = \mbox{diag} \Big( ~~~ \frac{m a}{2 M_p^2} + \frac{3}{2} \left( \frac{m a}{
M_p^2 } \alpha_1 + \dot{a} \alpha_2 \right), \nonumber \\
-\frac{ma}{2M_p^2}+\frac{3}{2}\frac{ma}{M_p^2} \tilde{\alpha_1}+\dot{a}
\tilde{\alpha_2}+a(m_{11}+m_{22}) ~~~ \Big)
%-\frac{m  a}{2 \M_p^2} + \frac{3}{2} \left( \frac{m  a}
%{M_p^2}\alpha_1 + \dot{a} \alpha_2 \right) + a \left( m_{11} + m_{22} \rgroup 
\label{eqn:massmatrix}
\end{eqnarray}
and matrix $N$
\begin{equation}
N
%\equiv N_1 + \gamma^0 \, N_2 
\equiv \pmatrix{- \tilde{\alpha_1} & 0 \cr 0 & - \tilde{\alpha_1} } + \gamma^0
\pmatrix{- \tilde{\alpha_2} & - \Delta \cr - \Delta & \tilde{\alpha_2} }
\label{kaptilde}
\end{equation}
for $ {\tilde \alpha_i} \equiv \alpha_i / \alpha$ and $\Delta=\sqrt{1-{\tilde
\alpha_1}^2-{\tilde \alpha_2}^2}$.  The existence of off-diagonal terms warns
us the non-trivial mixing of fermion eigenstates.

Once we can reduce the Lagrangian into this form of matrix expression, we can
 obtain the evolution equations for the mode functions ( the function
 multiplying the creation/annihilation operator) $U_r^{ij}$ and $V_r^{ij}$ (
 i,j runs over 1 and 2 for two field case and $r$ for helicity $\pm$) and
 calculate the occupation number of gravitino created from vacuum through these
 mode functions.  What we are interested in are the physical eigenstates states
 ($\psi_1,\psi_2 $) which are in general non-trivial combinations ( with matrix
 coefficients) of gravitino $ \theta$ and matter chiral fermion $\Upsilon$. So
 we, strictly speaking, need to diagonalize the Hamiltonian at each moment of
 field evolution to keep track of the mass eigenstates and their abundance.
This diagonalization process is rather involved\footnote{ We refer the
readers to \cite{marco} for the general discussion.}, so  here we use a
further simplification for our numerical analysis.  Namely we keep track of
the fermion states which are mixed so that they always diagonalize the
 Hamiltonian\footnote{This approximation is exact only at the early and late
 times. We however expect this simplification would not change our conclusion 
as we shall show in the following.}, and this assumption simplifies the
 mode decomposition in the following familiar form,
\begin{eqnarray}
\lefteqn{ X^i \left( x \right) = \nonumber} \\ & & \int \frac{d^3
\mathbf{k}}{\left( 2 \pi \right)^{3/2}} e^{i \mathbf{k \cdot x}} \Big[ U_r^{ij}
\left( k,\, \eta \right) a_j^r \left( k \right) + V_r^{ij} \left( k,\, \eta
\right) b_j^{\dagger r} \left( - k \right) \Big]. \nonumber \\
\end{eqnarray}
We then define the spinor matrix $U_{-}$ and $U_{+}$
\begin{equation}
U_r^{ij} \equiv \left[ \frac{U_+^{ij}}{\sqrt{2}} \, \psi_r,
\frac{U_-^{ij}}{\sqrt{2}} \, \psi_r \right]^T,\qquad V_r^{ij} \equiv \left[
\frac{V_+^{ij}}{\sqrt{2}} \, \psi_r, \frac{V_-^{ij}}{\sqrt{2}} \, \psi_r
\right]^T \label{spin}
\end{equation}
with eigenvectors of the helicity operator $\mathbf{\sigma}\cdot \mathbf{v} /
\vert \mathbf{v} \vert$, $\psi_{+} =$ {\scriptsize{$\pmatrix{1 \cr 0}$}} and
$\psi_{-} =$ {\scriptsize{$\pmatrix{0 \cr 1}$}}.  Using these spinor matrix,
the field equation of motion (\ref{eqn:fermioneom}) has a following simple form
in terms of the matrices $U_+$ and $U_-$,
\begin{equation}
a(t)\dot{U_\pm} = - i \, k \, U_\mp \mp i \, M \, U_\pm .
\end{equation}

%WE SHOULD GIVE SOME ARGUMENTS WHY THIS ADIABATIC APPROXIMATION IS OK

We can then expand $U_\pm$ in terms of positive and negative frequency
solutions,

\begin{eqnarray}
U_+(t) \equiv \left( 1 + \frac{M}{\omega} \right)^{1/2} \mathrm{e}^{- i \int^t
\omega \, d t'} \, A \nonumber \\ -\left( 1 - \frac{M}{\omega} \right)^{1/2}
\mathrm{e}^{i \int^t \omega \, dt'} \, B \nonumber\\ \equiv \left( 1 +
\frac{M}{\omega} \right)^{1/2} \alpha - \left( 1 - \frac{M}{\omega}
\right)^{1/2} \beta\,, \nonumber\\ U_-(t) \equiv \left(1 - \frac{M}{\omega}
\right)^{1/2} \mathrm{e}^{- i \int^t \omega \, dt'} \, A \nonumber \\ +\left( 1
+ \frac{M}{\omega} \right)^{1/2} \mathrm{e}^{i \int^t \omega \, dt'} \,
B\nonumber\\ \equiv \left( 1 - \frac{M}{\omega} \right)^{1/2} \alpha + \left( 1
+ \frac{M}{\omega} \right)^{1/2} \beta \,\label{decoferm},
\end{eqnarray} 
where diagonal matrix $\omega \equiv \sqrt{k^2+M^2} $.  $\alpha$ and $\beta$
  are precisely the generalization of Bogolubov coefficients.  Indeed, in the
  same way as Bogolubov coefficients, it can be shown that the occupation
  number of i$^{th}$ fermion eigenstates can be obtained through $\beta$ as
\begin{equation}
N_i \left( t \right) = \left( \beta^* \beta^T \right)_{ii} \mbox{ ( no
summation for $i$ )}.
\label{numferm}
\end{equation}
We also keep in our mind that, because of nontrivial mixing of fermion mass
eigenstates for the case of coupled field system, we need an extra care about
the identification of inflatinos and gravitinos.

In our analysis, $N_1$ corresponds to the field abundance whose mass converges
 to gravitino mass $\vert m \vert /M_p^2 $ which shows up in the first element
 of the mass matrix $M$ in eqn(\ref{eqn:massmatrix}), and we denote its
 asymptotic value at later time as $N_{\theta}$ which gives an estimate of
 gravitino abundances produced in preheating era. Consequently, $N_2$ at the
 later time gives an estimate for inflatino. $N_{\theta}$ is plotted in
%(FIG.\ref{fig1}.) 
the figure below as a function of comoving momentum at time 1000 in the units
 of inflaton mass $m_{\sigma}=\frac{\Delta^2}{M_p}$ which gives a typical time
 scale of inflaton oscillation, where we used the typical parameter values
 $\hat{\mu}=0.0001, \hat{\lambda}=0.001$ with initial amplitude of inflaton $
 0.2 M_p$ and that of O'Raifeartaigh field $\hat{\mu}^2 M_p $ and we initially
 normalized $a(t)$ to be one.
% \footnote{We also obtained plots for 
%other sets of parameters and  that the overall behavior and the 
%general features discussed in the following .}. 
%We also 
%checked that the following qualitative discussion does not change for different 
%ranges of parameters.

\begin{figure}[h]
\centerline{\psfig{file=830.eps,width=0.45 \textwidth}}
\caption{ Gravitino abundance as a function of comoving momentum in
units of inflaton mass. }
\label{fig1}
\end{figure}




Around this time, we checked $N_1$ and $N_2$ do not change anymore and
represent a good asymptotic behavior.  The values for $N_1$ and $N_2$ at
intermediate time ( i.e. the time when $\sigma$ and $\phi_1$ are still far from
its settlement in the vacuum) does not represent either of the inflatino or
gravitino because there exist non-negligible contributions of supersymmetry
breaking from both inflaton and hidden sectors as we can easily see from the
supersymmetry transformation of chiral fermion, $f_{\chi}$( superpartner of
scalar $\chi$, say),
\begin{equation}
\delta f_{\chi}=
-\frac{1}{2}P_L\left[m_{\chi}-\frac{\gamma^0}{\sqrt{2}}\frac{d\chi}{dt}
\right]\epsilon ~~,
\label{eqn:breaking}
\end{equation}
where $m_{\chi}$ is defined in eqn(\ref{eqn:m}). We can see, in the presence of
time dependent background as usually the case in dealing with cosmological
problems, there arises kinetic term of scalar fields which can cause the
supersymmetry breaking.




In such a small parameter range with so small initial amplitude for $\phi_1$ as
in our model, however, this kinetic term of $\phi_1$ is negligible and $\phi_1$
almost `sits still' very close to the origin while $\sigma$ keeps rapid
oscillation around its minimum. This oscillation of $\sigma$ is big enough for
a long period to dominate the supersymmetry breaking compared with $F$-term
contribution from $\phi_1$ and almost negligible $\dot{\phi_1}$.  The time when
this contribution of $\dot{\sigma} $ and $m_{\sigma}$ becomes comparable with
or less than that of supersymmetry breaking sector is beyond the range of our
numerical integration.  We however found
% because of $\phi_1$ with its initial 
%position just after inflation very close to its true minimum and small 
%initial amplitude,
 $N_{\theta}$ and $N_{\Upsilon}$ converge to some value for a given set of
parameters and does not change anymore at relatively early stage even when
$\sigma$ still keeps its oscillation.  We thus expect that the significant
contributions of supersymmetry breaking both from the inflaton and hidden
sectors are required for the efficient gravitino production.  We also should
mention that, because of the couplings, hidden sector may not be the sole cause
for supersymmetry breaking even in the vacuum, and in fact, there could be
still supersymmetry breaking from inflaton sector in the vacuum at a later
time. It, however, can be shown that in the vacuum the supersymmetry
contribution from inflaton sector is at most of order $\vert F \vert^4$
compared with $\vert F \vert^ 2 ~( \sim \mu ^4 ) $ due to hidden sector
\cite{hitoshi,weinberg}, and we still observe the dominant contribution of
local supersymmetry breaking from the O'Raifeartaigh field at a later time. We
thus expect our plot of $N_{\theta}$ represents a fairly good overall behavior
of gravitino abundance in asymptotic regime.

For the comparison with the gravitino number density constraints from
photo-dissociation process in nucleosynthesis for the case of thermal
production of gravitinos in thermal bath (\ref{eqn:ns}), we need to integrate
$N_{\theta}(k)$ over the comoving momentum space.  As usually the case with the
preheating of fermions, our plot also indicates that occupation number as a
function of comoving momentum k can be as large as of order unity at most up to
the order of inflaton mass scale, $k_{cutoff} \simeq m_{\sigma} $ and decreases
exponentially for bigger k.  So the number density for longitudinal components,
\begin{equation}
n_{3/2}=\frac{1}{\pi^2}\frac{1}{a^3}\int_0^{k_{max}} \vert \beta_k \vert ^2 k^2
dk
\end{equation}
during the preheating is at most
\begin{equation}
n_{3/2}(t_{pre}) \lesssim k_{cutoff}^3 \simeq m_{\sigma}^3 \simeq
\frac{\Delta^6}{M_p^3}\simeq 10^{-25} M_p{}^3.
\label{eqn:npre}
\end{equation}
We can now estimate the upper bound of the ratio of gravitino number density
$n_{3/2}$ to entropy density in analogy with ( \ref{eqn:orns} ).
\begin{equation}
 \frac{n_{3/2} (t_r)}{s(t_r)} \simeq\frac{n_{3/2}(t_{pre})\left(
\frac{a(t_{pre})}{a(t_r)} \right )^3}{0.44 g_* T_{RH}^3} \simeq
\frac{n_{3/2}(t_{pre})\left( \frac{t_{pre}}{t_r} \right )^2}{0.44 g_* T_{RH}^3}
,
% & & \simeq  \frac{n_{3/2}(t_{pre}) t_{pre}^2 T_{RH}( 1.66)^2} {0.44 M_p^2} . 
%\nonumber \\
% \label{eqn:gravitinos}
\end{equation}   
and substitution of $1.66 \cdot T_{RH}^2 \sqrt{g_*} / M_p \sim H \sim t_r^{-1}$
for $t_r$ gives us the estimate of $n_{3/2}/s$ after reheating,
\begin{equation}
\frac{n_{3/2}}{s} \simeq \frac{n_{3/2}(t_{pre}) t_{pre}^2 T_{RH}( 1.66)^2}
{0.44 M_p^2} .
%\nonumber \\
\label{eqn:gravitinos}
\end{equation}
We expect this efficient gravitino production occurs well within the time range
of typical oscillation of supersymmetry breaking field and we can substitute
$t_{pre} \sim 1/m_{\phi_1} \sim (\alpha_{\kappa} \mu)^{-1} \sim 10^9 M_p^{-1}$
and eqn (\ref{eqn:npre}) in above equation to obtain the upper bound,
\begin{equation}
\frac{n_{3/2}}{s} \lesssim 6.3 \cdot 10^{-15}
\left(\frac{T_{RH}}{10^{10}\mbox{GeV}}\right).
\end{equation}
This upper bound of $n/s$ for the gravitinos from nonthermal production in our
model is thus smaller than eqn(\ref{eqn:ns}) of thermal scattering by at least
two orders of magnitude.

We therefore find that our model does not lead to the overproduction of
gravitinos due to nonthermal process in preheating period, and reheating
temperature constraint due to this effect is less severe than that of
gravitinos produced by the scattering in thermal bath during the reheating
period. We also point out that the expression given by
eqn(\ref{eqn:gravitinos}) was derived in a general setting and it can be used
to obtain, in combination with eqn(\ref{eqn:ns}), the estimate for the relative
significance of the gravitino production in thermal and non-thermal processes
once the model of supergravity inflation is given.

\section{Conclusion and Discussion}
We showed in this letter a realistic supergravity inflation model which breaks
local supersymmetry in the vacuum dominantly via $F$-term coming from
O'Raifeartaigh field in the hidden sector.  We emphasized the significance of
radiative correction in supersymmetry breaking sector to evade the moduli
problem, and subsequently obtained the non-minimal K\"ahler potential arising
from this loop correction. Using this non-minimal form of K\"ahler potential,
we analyzed the possible non-thermal production of gravitinos, which occurs
when {\it both} inflaton and supersymmetry breaking fields have non-negligible
amplitudes and kinetic energy. This corresponds to the period when there exist
the significant contributions of supersymmetry breaking both from the inflaton
and hidden sectors as hinted in (\ref{eqn:breaking}).  This `mixing' effect
however is diminished because of the small amplitude and kinetic energy of
O'Raifeartaigh field even though inflaton field has a relatively big amplitude
of order $M_p$ after inflation. Once they start settling down to their true
minimum we would not expect an effective non-thermal production of gravitinos
anymore.
%In our model, O'Raifeartaigh field has a small 
%amplitude of at most intermediate scale and start rolling down to its 
%vacuum from very close to its minimum.
% Moreover, because of the small mass 
%scale of inflaton fields, the typical momentum scale of produced gravitino
%turns out to be small as well.
We estimated the upper bound of number density of non-thermally produced
gravitinos by integrating out its comoving occupation number over momentum
space. Because of the small mass scale of inflaton field, the typical momentum
scale of produced gravitinos and consequently the momentum space over which
occupation number is integrated out turns out to be small as well. This leads
to the relatively small number density of gravitinos and we showed that it
gives less significant constraint than that of gravitinos which are produced by
thermal scattering.

We point out that the cases involving three and more superchiral fields are
rather involved. We can basically follow the formalism discussed in section
\ref{sec:nonthermal}, but we need additional care in interpreting the fermion
fields as a superposition of mass eigenstates because we cannot completely
gauge away one of fermion fields via unitary gauge as we can do in the two field case
\cite{marco,long}.  The case including the gauge interaction terms and the
model of other supersymmetry breaking mechanism besides hidden sector
supersymmetry breaking are also to be examined.
%We didn't discuss the preheating effects of moduli and 
%modulini fields for our model 
%{ \bf and leave it for future work. ??????? }

\acknowledgments We are pleased to thank J.~Cohn, L.~Kofman, M.~Peloso and
A.~Pierce for helpful discussions. H.M. was supported by NSF under grant
 and DOE contract DE-AC03-76SF00098. 
PBG was supported by the DOE and the NASA grant NAG 5-10842 at
Fermilab.

\begin{thebibliography}{99}
\frenchspacing 
\bibitem{marco}
H. P. Nilles, M. Peloso and L. Sorbo,
JHEP {\bf 0104}, 004 (2001), ; 
H. P. Nilles, M. Peloso and L. Sorbo, 
Phys. Rev. Lett. {\bf 87}, 051302 (2001), . 

\bibitem{long}
R. Kallosh, L. Kofman, A. Linde and A. Van Proeyen,
Class. Quant. Grav. {\bf 17}, 4269 (2000), ;
R. Kallosh, L. Kofman, A. Linde and A. Van Proeyen, Phys. Rev. D {\bf 61}, 
103503 (2000),  

\bibitem {toni}
% THERMAL AND NONTHERMAL PRODUCTION OF GRAVITINOS IN THE EARLY UNIVERSE. 
G.F. Giudice, I.I. Tkachev and A. Riotto, JHEP {\bf 9911}, 036 (1999), . 

\bibitem {dangerous}
% NONTHERMAL PRODUCTION OF DANGEROUS RELICS IN THE EARLY UNIVERSE.
G.F. Giudice, I. I. Tkachev and A. Riotto, JHEP { \bf 9908}, 009 (1999), . 

\bibitem{firstgrav}
D. H. Lyth, Phys. Lett. B {\bf 476}, 356 (2000), . 

\bibitem{marco2}
H. P. Nilles, K. A. Olive, M. Peloso, Phys. Lett. B {\bf 522}, 304 (2001), . 

\bibitem{holman}
 G.D. Coughlan, R. Holman, P. Ramond and G.G. Ross,  
 Phys. Lett. B {\bf 140}, 44 (1984). 

\bibitem{or}
L. O'Raifeartaigh, Nucl. Phys. B {\bf 96}, 331 (1975).

\bibitem{lyth} 
 A.R. Liddle and D.H. Lyth, {\it Cosmological Inflation and Large-Scale Structure}, (Cambridge Univ. Pr., Cambridge, 2000). 

\bibitem{white}
E.~F.~Bunn, A. R. Liddle and M.~White, Phys. Rev. D {\bf 54}, 5917 (1996), .

\bibitem{dine}
M.~Dine, W.~Fischler and D.~Nameschansky, Phys.\ Lett.\ B {\bf 136}, 169 (1984).
\bibitem{huq}
M. Huq, Phys. Rev. D {\bf 14}, 3548 (1976).

\bibitem{jac}
R. Jackiw, Phys. Rev. D {\bf 9}, 1686 (1974).

\bibitem{izumi}
J.~Izumi and M.~Yamaguchi, Phys.\ Lett.\ B {\bf 342}, 111 (1995), .

\bibitem{kawasaki}
M. Kawasaki and T. Moroi, Prog. Theor. Phys. {\bf 93}, 879 (1995), .   

\bibitem{moroi2}
M. Kawasaki, K. Kohri and T. Moroi, Phys. Rev. D {\bf 63}, 103502 (2001), .   

 \bibitem{randall}
J. Bagger, E. Poppitz and L. Randall, Nucl. Phys. B {\bf 426}, 3 (1994),
 .

\bibitem{kolb}
E.W. Kolb and M.S. Turner, {\it The Early Universe}, ( Addison-Wesley
Pub. Co., Redwood City, 1990). 

\bibitem{moroi}
T.~Moroi, Ph-D thesis, Tohoku, Japan, .

\bibitem{hitoshi}
Y.~Kawamura, H.~Murayama and M.~Yamaguchi, 
Phy. Rev. D {\bf 51}, 1337 (1995), .

\bibitem {weinberg}
L.~Hall, J~Lykken and S.~Weinberg, Phys.~Rev.~D {\bf 27}, 2359 (1983).

\bibitem { modulipreheating}
G.F. Giudice, A. Riotto and I.I. Tkachev, JHEP {\bf 0106}, 020 (2001), . 
% THE COSMOLOGICAL MODULI PROBLEM AND PREHEATING.


\end{thebibliography}

\end{document}







