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%%  T. Asaka, M. Kawasaki and T. Yanagida
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\title{Superheavy Dark Matter and Thermal Inflation}
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\author{T. Asaka$^1$, M. Kawasaki$^2$, and T. Yanagida$^{1,2}$}
\address{$^1$Department of Physics,  University of
  Tokyo, Tokyo 113-0033, Japan}
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\address{$^2$Research Center for the Early Universe,
  University of Tokyo, Tokyo 113-0033, Japan}
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\date{April 23, 1999}
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\begin{abstract}
The thermal inflation is the most plausible mechanism
that solves the cosmological moduli problem naturally.
We discuss relic abundance of superheavy particle $X$
in the presence of the thermal inflation
assuming that its lifetime is longer than the age of the universe,
and show that the long-lived particle $X$ of mass 
$10^{12}$--$10^{14}$ GeV may form a part of the dark matter 
in the present universe
in a wide region of parameter space of the thermal inflation model.
The superheavy dark matter of mass $\sim 10^{13}$ GeV
may be interesting in particular,
since its decay may account for the observed ultra high-energy
cosmic rays if the lifetime of the $X$ particle is sufficiently long.
\end{abstract}
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%
A large class of string theories predicts a number of flat directions,
called moduli fields $\phi$ \cite{Green-Schwartz-Witten}.
They are expected to acquire their masses of order of the 
gravitino mass $m_{3/2}$ from some nonperturbative effects
of supersymmetry (SUSY) breaking \cite{Carlos-Casas-Quevedo-Roulet}.
The gravitino mass lies in a range of $10^{-2}$ keV--1 GeV
for gauge-mediated SUSY breaking models \cite{GMSB}
and in a range of 100 GeV--1 TeV for hidden-sector SUSY breaking
models \cite{HSSB}.
It is well known \cite{Moduli-Problem,Carlos-Casas-Quevedo-Roulet} 
that such moduli fields are produced too much as 
coherent oscillations in the early universe
and conflict with various cosmological observations.
Therefore, we must invoke a mechanism such as late-time 
entropy production to dilute the moduli density substantially.

The thermal inflation \cite{Lyth-Stewart} is the most plausible
mechanism to produce an enormous amount of entropy at the late time
of the universe's evolution.
In recent articles \cite{Asaka-Hashiba-Kawasaki-Yanagida} 
we have shown that
the thermal inflation is very successful in solving the above 
cosmological moduli problem.
Since it produces a tremendous amount of entropy to dilute the moduli 
density, abundances of any relic particles
are substantially diluted simultaneously,
which may provide a new possibility for a superheavy $X$ particle
to be a part of the dark matter in the universe.%
\footnote{
Other candidates for the dark matter 
in the presence of the thermal inflation are
the moduli themselves whose masses are less than about 100 keV 
\cite{Asaka-Hashiba-Kawasaki-Yanagida,%
Asaka-Hashiba-Kawasaki-Yanagida2},
and the axion with a relatively high decay constant 
$f_a \sim 10^{15}$--$10^{16}$ GeV
\cite{Kawasaki-Yanagida-axion,axionDM}.
}
In this paper we show that it is indeed the case
if the mass of $X$ particle is of order of $10^{12}$--$10^{14}$ GeV
and its lifetime is longer than the age of the universe.
Such a long-lived $X$ particle is particularly interesting,
since its decay may naturally explain 
\cite{Berezinsky-Kachelriess-Vilenkin,Kuzmin-Ruvakov,%
Kuzmin-Tkachev,Benakli-Ellis-Nanopoulos,Birkel-Sarkar} 
the observed ultra high-energy cosmic
rays \cite{UHCR1,UHCR2,UHCR3} beyond the Greisen-Zatsepin-Kuzmin cutoff
\cite{GZKcutoff} when the lifetime is sufficiently long.

In this paper we consider that the particle $X$ was primordially 
in the thermal equilibrium%
\footnote{
For the superheavy particle $X$ to be in the thermal equilibrium,
the reheating temperature after the primordial inflation
should be higher than $m_X \sim 10^{13}$ GeV.
Such a high reheating temperature is realized in e.g.
hybrid inflation models \cite{Hybrid-Inf}.
Although a large number of gravitinos are also produced in
this case,
they are sufficiently diluted by the thermal inflation
and become harmless.
}
and froze out at the cosmic temperature $T = T_f$ 
when it was nonrelativistic $x_f = m_X/T_f > 1$
(i.e., the $X$ is left as a cold relic.).%
\footnote{
Nonthermal productions of superheavy particles
were discussed in Refs.
\cite{Kuzmin-Tkachev,%
Chung-Kolb-Riotto}.
}
Then, the present relic abundance of $X$ 
(the ratio of the present energy density of $X$
to the critical density $\rho_{cr}$) is estimated by using the 
thermally-averaged annihilation cross section of the $X$,
$\langle \sigma_X \left| v \right| \rangle$, as\cite{Kolb-Turner}
\begin{eqnarray}
    \Omega_X^0 h^2 
    = 
    \frac{ 0.76 (n_f+1) x_f }
    { g_\ast (T_f)^{\frac 12}  M_G ( h^{-2} \rho_{cr} / s_0 )
      \langle \sigma_X \left| v \right| \rangle }
\end{eqnarray}
where $g_{\ast} (T_f) \simeq 200$ 
counts the effective degrees of freedom at $T=T_f$,
$M_G \simeq 2.4 \times 10^{18}$ GeV denotes the reduced Planck scale,
and $s_0$ is the present entropy density 
$(\rho_{cr}/s_0) \simeq 3.6 \times 10^{-9} h^2$ GeV
with the present Hubble parameter $h$ in units of 
100 km/sec$\cdot$Mpc$^{-1}$. 
Here $n_f$ parametrizes the dependence on $T$ of 
the annihilation cross section
and $n_f=0$ for $s$-wave annihilation, 
$n_f=1$ for $p$-wave annihilation, etc.
Assuming the $s$-wave annihilation with
$\langle \sigma_X \left| v \right| \rangle \simeq m_X^{-2}$
($x_f \simeq 7$ for $m_X \sim 10^{13}$ GeV) we obtain
\begin{eqnarray}
    \label{OmX}
    \Omega_X^0 h^2 \simeq
    \frac{ 0.4 m_X^2 }{ M_G ( h^{-2} \rho_{cr} / s_0 ) }
    =
    4.3 \times 10^{15} \left( \frac{ m_X }{ 10^{13} \GeV } \right)^{2}.
\end{eqnarray}
Therefore, the superheavy particle $X$ of $m_X \gtrsim 10^{5}$ GeV 
leads to overclosure 
of the universe if its lifetime is longer than the age of 
the universe.
In order to realize $\Omega_X h^2 \lesssim 1$
the dilution factor more than about $10^{16}$
is required for $m_X \sim 10^{13}$ GeV for example.
However, such a huge dilution 
may be naturally provided by the thermal inflation. 
In this paper, 
we examine whether 
the thermal inflation could sufficiently
reduce the energy density $\Omega_X$ 
of the superheavy particle even if it was in the thermal 
equilibrium, as well as that of the string moduli.

Let us start with reviewing briefly the thermal inflation model
\cite{Lyth-Stewart,Asaka-Hashiba-Kawasaki-Yanagida}.
The potential of the inflaton field $S$ is given by
\begin{eqnarray}
    \label{V_S}
    V = V_0 - m_0^2 \left| S \right|^2
    + \frac{ \left| S \right|^{2n+4} }{ M_\ast^{2n} },
\end{eqnarray}
where $-m_0^2 $ denotes a soft SUSY breaking negative mass 
squared which is expected to be of order of the electroweak scale.
$M_\ast$ denotes the cutoff scale 
and $V_0$ is the vacuum energy.
Then the vacuum expectation value of $S$ is estimated as 
\begin{eqnarray}
    \label{VEV-S}
    \langle S \rangle \equiv M
    =
    \left( n+2 \right)^{ - \frac{1}{2(n+1)} }
    \left( m_0 M_\ast^n \right)^{ \frac{ 1 }{ n+1 } },
\end{eqnarray}
and $V_0$ is fixed as
\begin{eqnarray}
    V_0 = \frac{ n+1 }{ n+2 } m_0^2 M^2,
\end{eqnarray}
so that the cosmological constant vanishes at the true vacuum.
The inflaton $\sigma$ ($\equiv$Re$S$), which we call it a flaton,
obtains a mass $m_\sigma^2 = 2 (n+1) m_0^2$.

After the thermal inflation ends, 
the vacuum energy is transferred into the thermal bath
through flaton decay and increases the entropy of the universe
by a factor \cite{Lyth-Stewart,Asaka-Hashiba-Kawasaki-Yanagida}%
\footnote{
Here we have assumed that 
the flaton can not decay into two $R$-axions which is 
an imaginary part of $S$ to obtain a successful dilution.
See Ref. \cite{Asaka-Hashiba-Kawasaki-Yanagida}.
}
\begin{eqnarray}
    \Delta 
    \simeq 
    \frac{ V_0 }{ 66 m_\sigma^3 T_R }.
\end{eqnarray}
Here $T_R$ is the reheating temperature after the thermal inflation
and determined by the flaton decay width $\Gamma_\sigma$
which can be written as
\begin{eqnarray}
    \Gamma_\sigma = C_\sigma \frac{ m_\sigma^3 }{ M^2 },
\end{eqnarray}
where $C_\sigma$ is a dimension-less parameter depending on 
the decay modes.
We consider that the flaton $\sigma$ dominantly decays into 
two photons or two gluons, and 
$C_\sigma$ is given by%
\footnote{
We assume that the decay amplitudes are obtained from one-loop
diagrams of some heavy particles. 
See also Ref. \cite{Asaka-Hashiba-Kawasaki-Yanagida}.
}
\begin{eqnarray}
    C_\sigma \simeq
    \left\{
        \begin{array}[]{ll}
        \frac{1}{4\pi} 
        \left( \frac{ \alpha }{ 4 \pi } \right)^2 
        &
        \mbox{for} ~~\sigma \rightarrow \gamma \gamma
        \\
        \frac{1}{4\pi} 
        \left( \frac{ \alpha_s }{ 4 \pi } \right)^2 
        &
        \mbox{for} ~~\sigma \rightarrow g g
    \end{array}
    \right..
\end{eqnarray}
Then the reheating temperature is estimated as
\begin{eqnarray}
    \label{TR}
    T_R 
    =
    \left( \frac{ 90 }{ \pi^2 g_\ast (T_R) } \right)^{1 \over 4 }
    \sqrt{ \Gamma_\sigma M_G }
    \simeq
    0.96 ~ C_\sigma^{1 \over 2 }
    \frac{ m_\sigma^{3 \over 2} M_G^{ 1 \over 2} }{ M }.
\end{eqnarray}
The entropy-production factor $\Delta$, therefore, can be written as
\begin{eqnarray}
    \Delta 
    &=&
    \frac{ M^3 }
         { 130 (n+2) C_\sigma^{ 1 \over 2} 
           m_\sigma^{ 5 \over 2 } M_G^{ 1 \over 2 } },
\end{eqnarray}
and for $n=1$ we obtain 
\begin{eqnarray}
    \label{Delta}
    \Delta 
    &\simeq&
    \left\{
        \begin{array}[]{ll}
            1.0 \times 10^{17} ~
            \displaystyle{
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ - \frac 52}
            \left( \frac{ M }{ 10^{10} \GeV } \right)^3
            }
            &
            \mbox{for} ~~\sigma \rightarrow \gamma \gamma
            \\
            6.4 \times 10^{15} ~
            \displaystyle{
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ - \frac 52}
            \left( \frac{ M }{ 10^{10} \GeV } \right)^3
            }
            &
            \mbox{for} ~~\sigma \rightarrow g g
        \end{array}
    \right..
\end{eqnarray}
Here the values $m_\sigma = 100$ GeV and $M = 10^{10}$ GeV 
correspond to  $M_\ast = 3.5 \times 10^{18}$ GeV for $n=1$.
In the following analysis, we take $n=1$ for simplicity.
From Eq. (\ref{Delta}) we see that the thermal inflation 
can dilute relic particle density extensively by producing 
an enormous entropy.
Here it should be noted that the reheating temperature 
should be $T_R \gtrsim 1$ MeV to keep 
the big bang nucleosynthesis successful \cite{Kawasaki-Kohri-Sugiyama}%
\footnote{
In Ref. \cite{Kawasaki-Kohri-Sugiyama}
the lower bound on the reheating temperature is 
determined as about 0.5 MeV.
Since our definition of the reheating temperature is different
by a factor $\sqrt{3}$, 
it leads to $T_R \gtrsim 1$ MeV in our case.
},
which leads to the upper bounds on $M$ from Eq. (\ref{TR}) as
\begin{eqnarray}
    M 
    &=&
    0.96 ~ C_\sigma^{\frac 12} 
    \frac{ m_\sigma^{\frac 32} M_G^{\frac 12} }{ T_R }
    \lesssim
    \left\{
        \begin{array}[]{ll}
            2.4 \times 10^{11} \GeV~
            \displaystyle{
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ \frac 32}
            }
            &
            \mbox{for} ~~\sigma \rightarrow \gamma \gamma
            \\
            3.9 \times 10^{12} \GeV~
            \displaystyle{
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ \frac 32}
            }
            &
            \mbox{for} ~~\sigma \rightarrow g g
        \end{array}
    \right..
\end{eqnarray}
These are translated into the upper bounds on $M_\ast$ as
\begin{eqnarray}
    \label{UB-Mst}
    M_\ast
    &=&
    3.2 ~ C_\sigma
    \frac{ m_\sigma^2 M_G }{ T_R^2 }
    \lesssim
    \left\{
        \begin{array}[]{ll}
            2.1 \times 10^{21} \GeV~
            \displaystyle{
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ 2 }
            }
            &
            \mbox{for} ~~\sigma \rightarrow \gamma \gamma
            \\
            5.4 \times 10^{23} \GeV~
            \displaystyle{
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ 2 }
            }
            &
            \mbox{for} ~~\sigma \rightarrow g g
        \end{array}
    \right..
\end{eqnarray}

In the presence of the thermal inflation 
the relic abundance of the superheavy particle $X$ 
[Eq. (\ref{OmX})] is reduced by the factor $\Delta$ 
in Eq. (\ref{Delta}) as
\begin{eqnarray}
    \label{OmX-TI}
    \Omega_X h^2 
    &=&
    \Omega_X^0 h^2 \times \frac{ 1 }{ \Delta }
    =
    140 ~C_\sigma^{ \frac 12}
     \frac{ m_X^2 m_\sigma^{\frac 52} }
          { M^3 M_G^{\frac 12} ( h^{-2} \rho_{cr}/s_0) }
    \nonumber \\
    &\simeq&
    \left\{
        \begin{array}[]{ll}
            0.04 ~
            \displaystyle{
            \left( \frac{ m_X }{ 10^{13} \GeV } \right)^2
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ \frac 52 }
            \left( \frac{ M }{10^{10} \GeV} \right)^{ -3 }
            }
            &
            \mbox{for} ~~\sigma \rightarrow \gamma \gamma
            \\
            0.68 ~
            \displaystyle{
            \left( \frac{ m_X }{ 10^{13} \GeV } \right)^2
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ \frac 52 }
            \left( \frac{ M }{10^{10} \GeV} \right)^{ -3 }
            }
            &
            \mbox{for} ~~\sigma \rightarrow g g
        \end{array}
    \right.,
\end{eqnarray}
In Fig. \ref{fig:OmX} we show the contour of $\Omega_X$
in the parameter space of the thermal inflation model
(in $m_\sigma$-$M_\ast$ plane).
We find that 
the thermal inflation can 
naturally realize $\Omega_X h^2 \lesssim 1$ 
keeping the constraint $T_R \gtrsim 1$ MeV
in a large region of the parameter space.

As mentioned in the introduction, 
the thermal inflation was originally proposed 
as a solution to the cosmological moduli problem.
Therefore, we turn to examine whether
the thermal inflation could sufficiently dilute
not only the density of the superheavy particle $X$ 
but also that of the string moduli, simultaneously.

When the Hubble parameter becomes comparable to the moduli masses,
the moduli $\phi$ start coherent oscillations and 
the corresponding cosmic temperature is estimated as
\begin{eqnarray}
    T_\phi \simeq 7.2 \times 10^{6} \GeV
    \left( \frac{ m_\phi }{ 100 \keV } \right)^{\frac 12}.
\end{eqnarray}
Here notice that the moduli oscillations always begin 
after the $X$ freezes out since $T_f \sim m_X \gg T_\phi$ 
even for the heavy moduli $m_\phi \simeq m_{3/2} \simeq
100$ GeV--1 TeV predicted in hidden sector SUSY breaking models.
Because the initial amplitudes of the oscillations, $\phi_0$,
are expected to be $\phi_0 \sim M_G$,
the present abundances%
\footnote{
If the moduli masses are less than about 100 MeV,
the moduli become stable until the present.
On the other hand,
for the heavier moduli mass region,
$\Omega_\phi$ is
regarded as the ratio, $(\rho_\phi/s)_D/ (\rho_{cr}/s_0)$,
where $(\rho_\phi/s)_D$ denotes the ratio of 
the energy density of the moduli oscillations 
to the entropy density when the moduli decay.
}
of the moduli oscillations are given by
\begin{eqnarray}
    \label{OmBB0}
    \Omega_\phi h^2 = 2.5 \times 10^{14}
    \left( \frac{ m_\phi }{ 100 \keV} \right)^{\frac 12}
    \left( \frac{ \phi_0 }{ M_G } \right)^2.
\end{eqnarray}
Such a huge energy density of the moduli leads to cosmological
difficulties for the typical moduli mass regions 
predicted in both gauge-mediated SUSY breaking and hidden-sector
SUSY breaking scenarios.

However, if the universe experienced the thermal inflation,
the moduli abundances are reduced by the factor $\Delta$ 
[Eq. (\ref{Delta})] as \cite{Lyth-Stewart,Asaka-Hashiba-Kawasaki-Yanagida}
\begin{eqnarray}
    \label{OmBB}
    \left( \Omega_\phi \right)_{BB} h^2 
    &=&
    22 C_\sigma^{ \frac 12 }
    \frac{ m_\phi^{ \frac 12 } m_\sigma^{\frac 52} M_G }
         { M^3 ( h^{-2} \rho_{cr} / s_0 ) }
    \left( \frac{ \phi_0 }{ M_G } \right)^2
    \nonumber \\
    &\simeq&
    \left\{
        \begin{array}[]{ll}
            2.4 \times 10^{-3}~
            \displaystyle{
            \left( \frac{ m_\phi }{ 100 \keV } \right)^{ \frac 12 }
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ \frac 52 }
            \left( \frac{ M }{10^{10} \GeV} \right)^{ -3 }
            \left( \frac{ \phi_0 }{ M_G } \right)^2
            }
            &
            \mbox{for} ~~\sigma \rightarrow \gamma \gamma
            \\
            3.9 \times 10^{-2}~
            \displaystyle{
            \left( \frac{ m_\phi }{ 100 \keV } \right)^{ \frac 12 }
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ \frac 52 }
            \left( \frac{ M }{10^{10} \GeV} \right)^{ -3 }
            \left( \frac{ \phi_0 }{ M_G } \right)^2
            }
            &
            \mbox{for} ~~\sigma \rightarrow g g
        \end{array}
    \right..
\end{eqnarray}
We call these moduli produced at $T = T_\phi$ as ``big-bang'' moduli.
%It is found from Eq. (\ref{OmX-TI})
%that $(\Omega_\phi)_{BB}$ is always smaller than
%$\Omega_X$ with $m_X \sim 10^{13}$ GeV 
%for the moduli mass region $m_\phi \lesssim$ 30 MeV.
In deriving Eqs. (\ref{OmBB0}) and (\ref{OmBB}) we have assumed
that the energy of the universe is radiation-dominated
when the big-bang modulus starts to oscillate at $H \simeq m_{\phi}$.
This assumption is justified for
$\left( m_\phi / 100 \keV \right)^{1/2}
\left( m_X / 10^{13} \GeV \right)^{-2} \gtrsim 2.9$.
On the other hand, when the energy at $H \simeq m_\phi$ is dominated by 
the superheavy particle $X$,
the present abundance of the big-bang modulus is related to 
the abundance of $X$ (\ref{OmX-TI}) as
\begin{eqnarray}
    \left( \Omega_\phi \right)_{BB} h^2
    = 
    \frac{ 1 }{ 6} \Omega_X h^2 
    \left( \frac{ \phi_0 }{ M_G } \right)^2.
\end{eqnarray}
Therefore, both abundances are comparable for $\phi_0 \sim M_G$.

Furthermore, it should be noticed that 
the secondary oscillations of the moduli 
start just after the thermal inflation ends \cite{Lyth-Stewart}.
We call the moduli produced by these secondary oscillations
as ``thermal inflation'' moduli.
The present abundances of the thermal inflation moduli 
are estimated as \cite{Lyth-Stewart,Asaka-Hashiba-Kawasaki-Yanagida}
\begin{eqnarray}
    \label{OmTI}
    \left( \Omega_\phi \right)_{TI} h^2
    &=&
    6.0 \times 10^{-2} ~ C_\sigma^{ \frac 12}
    \frac{ m_\sigma^{ \frac 72 } M }
         { m_\phi^2 M_G^{ \frac 32} ( h^{-2} \rho_{cr} / s_0 ) }
    \left( \frac{ \phi_0 }{ M_G } \right)^2
    \nonumber \\
    &\simeq&
    \left\{
        \begin{array}[]{ll}
            7.4 ~
            \displaystyle{
            \left( \frac{ m_\phi }{ 100 \keV } \right)^{ - 2 }
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ \frac 72 }
            \left( \frac{ M }{10^{10} \GeV} \right)
            \left( \frac{ \phi_0 }{ M_G } \right)^2
            }
            &
            \mbox{for} ~~\sigma \rightarrow \gamma \gamma
            \\
            1.2 \times 10^{2}~
            \displaystyle{
            \left( \frac{ m_\phi }{ 100 \keV } \right)^{ - 2 }
            \left( \frac{ m_\sigma }{100 \GeV} \right)^{ \frac 72 }
            \left( \frac{ M }{10^{10} \GeV} \right)
            \left( \frac{ \phi_0 }{ M_G } \right)^2
            }
            &
            \mbox{for} ~~\sigma \rightarrow g g
        \end{array}
    \right..
\end{eqnarray}

We see from Eqs. (\ref{OmBB}) and (\ref{OmTI})
that the thermal inflation dilutes the moduli density substantially.
In fact, it has been shown in Refs. 
\cite{Lyth-Stewart,Asaka-Hashiba-Kawasaki-Yanagida,Asaka-Kawasaki}
that two moduli mass regions: (i) $m_\phi \lesssim 1$ MeV and
(ii) $m_\phi \gtrsim 10$ GeV
survive various cosmological constraints 
in the presence of the thermal inflation.

We are now at the point to examine whether the thermal inflation can
solve the moduli problem and 
also dilute sufficiently the superheavy particle $X$ at the same time.
%
First, we consider the lighter allowed region of the moduli masses
predicted in gauge-mediated SUSY breaking models.
In Fig. \ref{fig:OmXmp100keV} 
we show the contour plot of the abundance of $X$ 
of mass $m_X = 10^{13}$ GeV as well as the various constraints
in the $m_\sigma$-$M_\ast$ plane for $m_\phi = 100$ keV.
Such light moduli are constrained from the overclosure limit.%
%%
\footnote{
Notice that for the modulus of mass $m_\phi = 100$ keV 
the constraint from the diffuse x$(\gamma)$-ray backgrounds
is not so severe as the overclosure limit.
It gives a more stringent upper bound on the modulus abundance
for 100 MeV $\gtrsim m_\phi \gtrsim 200$ keV
\cite{Kawasaki-Yanagida-xray}.
}
%%
Then the requirements $(\Omega_\phi)_{BB} h^2 \lesssim$ 1
and $(\Omega_\phi)_{TI} h^2 \lesssim$ 1 put 
the lower and upper bounds on $M_\ast$ respectively.
Furthermore, the condition $T_R \gtrsim 1$ MeV leads to 
the upper bound on $M_\ast$ as represented in Eq. (\ref{UB-Mst}).
We see from the figures that
in the parameter space which survives above constraints
one can obtain $\Omega_X h^2 \sim 10^{-6}$--$1$ for $m_X = 10^{13}$ GeV:
i.e.,
the thermal inflation can dilute the abundance of the superheavy 
particle $X$ sufficiently.
Similar results are obtained in the allowed moduli mass 
region $m_\phi \simeq 10^{-2}$ keV--$1$ MeV.
For example, when $m_\phi = 10^{-2}$ keV,
the thermal inflation solves the moduli problem
in the parameter regions $m_\sigma \simeq 10^{-2}$ GeV
--$1$ GeV and $M_\ast \simeq 10^{13}$ GeV--$10^{17}$ GeV
and in this region we obtain
$\Omega_X h^2 \sim 10^{-1}$--$1$ for $m_X = 10^{13}$ GeV.
We find that the thermal inflation is very successful,
in a whole range of $m_\phi \simeq 10^{-2}$ keV--1 MeV,
to solve the moduli problem as well as 
to reduce the density of 
the superheavy particle whose mass is $m_X \sim 10^{13}$ GeV
even if it was in the thermal bath in the early universe.


Next, we turn to discuss the case of the heavier moduli of masses
$m_\phi \simeq 100$ GeV--$1$ TeV predicted in hidden-sector SUSY
breaking models.
In Fig. \ref{fig:OmXmp100GeV} we show 
the contour plot of $\Omega_X$ with various constraints
for $m_X = 10^{13}$ GeV and $m_\phi = 100$ GeV.
Such heavy moduli are severely constrained not to destroy 
or overproduce light elements synthesized by the big bang
nucleosynthesis. It has been found that 
$(\Omega_\phi)_{BB} h^2 $ and $(\Omega_\phi)_{TI} h^2$ should 
be less than about $10^{-5}$ \cite{Holtman-Kawasaki-Kohri-Moroi}.
Fig. \ref{fig:OmXmp100GeV} shows that in order to solve the moduli problem,
the thermal inflation requires $M_\ast$ higher than in the previous
case,  which results in more dilution of $X$,
and hence we obtain $\Omega_X h^2 \sim 10^{-9}$--$10^{-7}$ for 
$m_X = 10^{13}$ GeV.


Although we have only considered the case $n=1$,
similar discussions also hold in the higher $n$ case,
except that the scale of $M_\ast$ becomes higher.

In the present analysis 
we have taken the cutoff scale of the thermal inflation model
$M_\ast$ as a free parameter.
However,
it is natural to choose it as the gravitational scale, i.e.,
$M_\ast \sim M_G$.
If it is the case,
the thermal inflation dilutes the string moduli
sufficiently only if their masses are $m_\phi \simeq 10^{-1}$keV
--$1$ MeV.
Moreover, as shown in Fig. \ref{fig:OmXmp100keVMst1D19},
the dark matter density of the superheavy particle of mass 
$m_X \simeq 10^{12}$--$10^{14}$ GeV
becomes $\Omega_X h^2 \sim10^{-4}$--1, 
which is just the mass region required to explain 
the observed ultra high-energy cosmic rays.
The required long lifetime of the $X$ particle
may be explained by discrete gauge symmetries
\cite{Hamaguchi-Nomura-Yanagida}
or by compositeness of the $X$ particle
\cite{Long-Lived}.

In this paper we have shown that 
the thermal inflation provides indeed
a new possibility that 
the superheavy $X$ particle may form a part of 
the dark matter in the present universe.
However, it gives rise to a
new problem, since it also dilutes the primordial baryon asymmetry
significantly.
Because the reheating temperature of the thermal inflation 
should be quite low $T_R \sim 1$--10 MeV 
to dilute $\phi$ and $X$ sufficiently,
the electroweak baryogenesis does not work.
However, as shown in Ref. \cite{Gouvea-Moroi-Murayama},
the Affleck-Dine mechanism \cite{Affleck-Dine} may produce 
enough baryon asymmetry even with tremendous entropy production 
due to the thermal inflation,
if the moduli are light ($m_\phi \lesssim 1$ MeV)
\cite{Kawasaki-Yanagida-xray}.
Therefore, in the present scenario, 
the light moduli predicted in gauge-mediated SUSY breaking
models are favored.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgments
%We would like to thank anonymous referee for pointing out
%a possibility of the $X$ dominance when the big-bang modulus
%starts to oscillate.
This work was partially supported by the Japan Society for
the Promotion of Science (TA) and
``Priority Area: Supersymmetry and Unified Theory
of Elementary Particles($\sharp$707)''(MK,TY).

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%%%%% ** Reference ** %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{references}
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Green-Schwartz-Witten}
M.B. Green, J.H. Schwartz, and E. Witten,
{\it Superstring Theory}, Cambridge University Press (1986).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Carlos-Casas-Quevedo-Roulet}
B. de Carlos, J.A. Casas, F. Quevedo and E. Roulet,
{Phys. Lett.} {\bf B318}, 447 (1993).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{GMSB}
For a review see,
G.F. Giudice and R. Rattazzi, .
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{HSSB}
For a review see,
H.P. Nilles, 
{Phys. Rep.} {\bf 110}, 1 (1984).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Moduli-Problem}
G.D. Coughlan, W. Fischler, E.W. Kolb, S. Raby and G.G. Ross,
{Phys. Lett.} {\bf B131}, 59 (1983);
T. Banks, D.B. Kaplan and A.E. Nelson,
{Phys. Rev.} {\bf D49}, 779 (1994).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Lyth-Stewart}
D.H. Lyth and E.D. Stewart, 
{Phys. Rev. Lett.}, {\bf 75}, {201}(1995);
{Phys. Rev.}, {\bf D53}, {1784}(1996).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Asaka-Hashiba-Kawasaki-Yanagida}
J. Hashiba, M. Kasasaki, and T. Yanagida,
Phys. Rev. Lett. {\bf 79}, 4525 (1997);
T. Asaka, J. Hashiba, M. Kasasaki, and T. Yanagida,
Phys. Rev. {\bf D58}, 083509 (1998).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Asaka-Hashiba-Kawasaki-Yanagida2}
T. Asaka, J. Hashiba, M. Kasasaki, and T. Yanagida,
Phys. Rev. {\bf D58}, 023507 (1998).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Kawasaki-Yanagida-axion}
M. Kawasaki and T. Yanagida,
{Prog. Theor. Phys.} {\bf 97}, 809 (1997).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{axionDM}
M. Kawasaki, T. Moroi, and T. Yanagida,
Phys. Lett. {\bf B383}, 313 (1996);
G. Lazarides, R. Shaefer, D. Seckel, and Q. Shafi,
Nucl. Phys. {\bf B346}, 193 (1990).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Berezinsky-Kachelriess-Vilenkin}
V. Berezinsky, M. Kachelrie{\ss},
and A. Vilenkin,
Phys. Rev. Lett. {\bf 79}, 4302 (1997).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Kuzmin-Ruvakov}
V.A. Kuzmin and V.A. Rubakov,
Phys. Atom. Nucl. {\bf 61}, 1028 (1998)
.
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Kuzmin-Tkachev}
V.A. Kuzmin and I.I. Tkachev,
JETP Lett. {\bf 68}, 271 (1998),
Phys. Rev. {\bf D59}, 123006 (1999).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Benakli-Ellis-Nanopoulos}
K. Benakli, J. Ellis, and D.V. Nanopoulos,
Phys. Rev. {\bf D59}, 047301 (1999).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Birkel-Sarkar}
M. Birkel and S. Sarkar,
Astropart. Phys. {\bf 9}, 297 (1998).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{UHCR1}
N. Hayashida {\it et al},
{Phys. Rev. Lett.} {\bf 73}, 3491 (1994);
S. Yoshida {\it et al},
{Astropart. Phys.} {\bf 3}, 105 (1995);
M. Takeda {\it et al},
Phys. Rev. Lett. {\bf 81}, 1163 (1998).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{UHCR2}
D.J. Bird {\it et al},
Phys. Rev. Lett. {\bf 71}, 3401 (1993);
Astrophys. J. {\bf 424}, 491 (1994);
Astrophys. J. {\bf 441}, 144 (1995).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{UHCR3}
N.N. Efimov {\it et al},
in {\it Proc. ICRR symposium on Astrophysical Aspects of 
the Most Energetic Cosmic Rays},
ed. M. Nagano and F. Takahara
(World Scientific, Singapore, 1991), p. 20;
T.A. Egorov {\it et al},
in {\it Proc. Tokyo Workshop on Techniques for the Study
of Extermely High Energy Cosmic Rays},
ed. M. Nagano
(ICRR, Tokyo, 1993);
B.N. Afnasiev,
in {\it Proc. of International Symposium on 
Extremely High Energy Cosmic Rays:
Astrophysics and Future Observatories},
ed. M. Nagano (ICRR, Tokyo, 1996), p. 32.
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{GZKcutoff}
K. Greisen,
{Phys. Rev. Lett.} {\bf 16}, {748} (1966);
G.T. Zatsepin and V.A. Kuzmin,
{Pis'ma Zh. Eksp. Teor. Fiz.} {\bf 4}, {114} (1966)
[JETP Lett. {\bf 4}, 78, (1966)].
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Hybrid-Inf}
A.D. Linde,
Phys. Lett. {\bf B259}, 38 (1991).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Chung-Kolb-Riotto}
D.J.H. Chung, E.W. Kolb, and 
A. Riotto,
Phys. Rev. {\bf D59}, 023501 (1999),
Phys. Rev. Lett. {\bf 81}, 4048 (1998),
Phys. Rev. {\bf D60}, 063504 (1999).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Kolb-Turner}
E.W. Kolb and M.S. Turner,
{\it The Early Universe} (Addison-Wesley, Redwood City, 1989).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Kawasaki-Kohri-Sugiyama}
M. Kawasaki, K. Kohri, and N Sugiyama,
Phys. Rev. Lett. {\bf 82}, 4168 (1999).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Asaka-Kawasaki}
T. Asaka and M. Kawasaki,
Phys. Rev. {\bf D} (to be published),
.
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Kawasaki-Yanagida-xray}
M. Kawasaki and T. Yanagida,
Phys. Lett. {\bf B399}, 45 (1997).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Holtman-Kawasaki-Kohri-Moroi}
For example, see the resent analysis,
E. Holtman, M. Kawasaki, K. Kohri, and T. Moroi,
Phys. Rev. {\bf D60}, 023506 (1999).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Hamaguchi-Nomura-Yanagida}
K. Hamaguchi, Y. Nomura, and T. Yanagida,
Phys. Rev. {\bf D58}, 103503 (1998);
Phys. Rev. {\bf D59}, 063507 (1999).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Long-Lived}
J. Ellis, G.B. Gelmini, J.L. Lopez, D.V. Nanopoulos, and S. Sarkar,
Nucl. Phys. {\bf B373} 399 (1992);
K. Benakli, J. Ellis, and D.V. Nanopoulos,
Phys. Rev. {\bf D59} 047301 (1999);
K. Hamaguchi, K.I. Izawa, Y. Nomura, and T. Yanagida,
.
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Gouvea-Moroi-Murayama}
A.D. Gouvea, T. Moroi, and H. Murayama,
{Phys. Rev.} {\bf D56}, 1281 (1997).
%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Affleck-Dine}
I. Affleck and M. Dine,
{Nucl. Phys.} {\bf B249}, 361 (1985).
%%%%%%%%%%%%%%%%%%%%%%%
%\bibitem{Kawasaki-Yanagida}
%M. Kawasaki and T. Yanagida,
%{Phys. Lett.} {\bf B399}, 45 (1997).

%

\end{references}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% ** Figure Caption ** %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% ** Figure ** %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%
%%% Fig. 
\begin{figure}[t]
    \centerline{\psfig{figure=mFig1a.eps,width=10cm}}
    \centerline{\psfig{figure=mFig1b.eps,width=10cm}}
%    \vspace{0.5cm}
\caption{%%
The contour lines of the relic abundance ($\Omega_X h^2$)
of the superheavy particle $X$ of mass $m_X = 10^{13}$ GeV
in the presence of the thermal inflation with $n=1$ 
for the cases that a flaton $\sigma$ decays mainly into two gluons (A)
and that a flaton $\sigma$ decays mainly into two photons (B).
The contour lines are represented by the dotted lines.
(The contour line of $\Omega_X h^2$=1 is represented by the thick
dotted line.)
We show the upper bound on the cutoff scale $M_\ast$ obtained
from $T_R >$ 1 MeV by the thick solid line.
}
    \label{fig:OmX}
\end{figure}
%%%%%%%%%%%%%%%%%
%%% Fig. 
\begin{figure}[t]
    \centerline{\psfig{figure=mFig2a_new.eps,width=10cm}}
    \centerline{\psfig{figure=mFig2b_new.eps,width=10cm}}
%    \vspace{0.5cm}
\caption{%%
The contour lines of the relic abundance ($\Omega_X h^2$)
of the superheavy particle $X$ of mass $m_X = 10^{13}$ GeV
in the presence of the thermal inflation 
with $n=1$ for the cases that 
a flaton $\sigma$ decays mainly into two gluons (A)
and that a flaton $\sigma$ decays mainly into two photons (B).
The contour lines are represented by the dotted lines.
(The contour line of $\Omega_X h^2$=1 is represented by the thick
dotted line.)
We show the upper bound on the cutoff scale $M_\ast$ obtained
from $T_R >$ 1 MeV by the thick solid line.
The lower bound on $M_\ast$ from $(\Omega_\phi)_{BB} h^2 < 1$
and the upper bound on $M_\ast$ from $(\Omega_\phi)_{TI} h^2 < 1$
for the case $m_\phi = 100$ keV
are also shown by the thick solid lines.
}
    \label{fig:OmXmp100keV}
\end{figure}
%%%%%%%%%%%%%%%%%
%%% Fig. 
\begin{figure}[t]
    \centerline{\psfig{figure=mFig3a.eps,width=10cm}}
    \centerline{\psfig{figure=mFig3b.eps,width=10cm}}
%    \vspace{0.5cm}
\caption{%%
The contour lines of the relic abundance ($\Omega_X h^2$)
of the superheavy particle $X$ of mass $m_X = 10^{13}$ GeV
in the presence of the thermal inflation
with $n=1$ for the cases that a flaton $\sigma$ 
decays mainly into two gluons (A) and that
a flaton $\sigma$ decays mainly into two photons (B).
The contour lines are represented by the dotted lines.
We show the upper bound on the cutoff scale $M_\ast$ 
obtained from $T_R >$ 1 MeV by the thick solid line.
The lower bound on $M_\ast$ from $(\Omega_\phi)_{BB} h^2< 10^{-5}$
and the upper bound on $M_\ast$ from $(\Omega_\phi)_{TI} h^2< 10^{-5}$
for the case $m_\phi = 100$ GeV
are also shown by the thick solid lines.
}
    \label{fig:OmXmp100GeV}
\end{figure}
%%%%%%%%%%%%%%%%%
%%% Fig. 
\begin{figure}[t]
    \centerline{\psfig{figure=mFig4a.eps,width=10cm}}
    \centerline{\psfig{figure=mFig4b.eps,width=10cm}}
%    \vspace{0.5cm}
\caption{%%
The contour lines of the relic abundance ($\Omega_X h^2$)
of the superheavy particle $X$ 
in the presence of the thermal inflation 
with $n=1$ for the cases that a flaton $\sigma$ 
decays mainly into two gluons (A)
and that a flaton $\sigma$ decays mainly into two photons (B).
We take the cutoff scale as 
$M_\ast = M_G \simeq 2.4 \times 10^{18}$ GeV and
$m_\phi = 100$ keV.
The contour lines are represented by the dotted lines.
(The contour line of $\Omega_X h^2$=1 is represented by the thick
dotted line.)
We show the lower bound on the flaton mass 
$m_\sigma > 1$ GeV to open the gluon decay channel (A) 
and $m_\sigma > 3.4$ GeV from $T_R >$ 1 MeV (B),
and also the upper bound on $m_\sigma$ 
from $(\Omega_\phi)_{TI} h^2 <1$
by the thick solid lines.
}
    \label{fig:OmXmp100keVMst1D19}
\end{figure}
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