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\begin{document}
\title{Curvaton Reheating in Non-oscillatory Inflationary Models}
\author{Bo Feng}
\email{fengbo@mail.ihep.ac.cn}
\author{Mingzhe Li}
\email{limz@mail.ihep.ac.cn}
\affiliation{Institute of High Energy Physics, Chinese
Academy of Science, P.O. Box 918-4, Beijing 100039, People's Republic of China}

\begin{abstract}
In non-oscillatory (NO) inflationary models, the reheating
mechanism was usually based on gravitational particle production or the mechanism of instant preheating.
In this paper we introduce the curvaton mechanism
into NO models to reheat the universe and generate the curvature perturbation.
Specifically we consider the Peebles-Vilenkin quintessential
inflation model, where the reheating temperature can be extended
from $1$MeV to $10^{13}$GeV.
\end{abstract}

\maketitle

\hskip 1.6cm
PACS number(s): 98.80.Cq
\vskip 0.4cm

It is now widely accepted that the very early universe experienced an era of
accelerated expansion, inflation \cite{guth}. Inflarionary
universe has solved many problems of the Standard Hot Big-Bang
theory, e.g., flatness problem, horizon problem, etc.
In addition, it has provided a causal interpretation of the origin of
the observed anisotropy of the cosmic microwave background (CMB) and the distribution of
large scale structure (LSS).
In the usual inflation models, the acceleration is driven by the
potential of a scalar field $\phi$, named inflaton. The
quantum fluctuation of the inflaton field generates a spatial
curvature perturbation which is responsible for the CMB anisotropy
and LSS. Generally, after inflation the inflaton field will oscillate about the minimum
of its potential and eventually decay to produce almost all elementary particles populating the
universe. This process is called reheating.

However, there exist some inflationary models, named NO models \cite{no,vilenkin,ford,joyce} by
the authors of Ref. \cite{kofman}, in which
the inflaton field does not oscillate after inflation and the standard reheating mechanism would not work.
Conventionally one turns to gravitational particle
production\cite{ford}. However, the authors of Ref. \cite{kofman}
pointed out that this mechanism is very inefficient. They also
proposed to use "instant preheating" \cite{felder} by
introducing an interaction $g^2\phi^2\chi^2$ of the inflaon field
$\phi$ with another scalar field $\chi$.

In this paper we introduce the curvaton mechanism \cite{lyth} which received many
attentions recently in the literature
\cite{enqv,liddle,wands,moroi,enqvist,dimo} into the NO inflationary models.
In our scenario, the reheating mechanism is based on the
oscillation and decays of another scalar field $\sigma$ (the
curvaton field) which is sub-dominant during inflation and has no interactions with the inflaton except
the gravitational coupling. We will show by an example that the reheating temperature
in our case can be as high as $10^{13}$GeV.
In addition, the almost scale-invariant curvature perturbation responsible for the CMB anisotropy and
the distribution of LSS is not originated
from the fluctuation of the inflton field during
inflation, but instead from the isocurvature perturbation of curvaton.
After inflation when the energy density of the curvaton
field becomes significant, the isocurvature perturbation in it is
converted into a sizable curvature one \cite{mollerach}. The
curvature perturbation becomes pure when the curvaton field
becomes to dominate the universe. The primordial density
perturbation in curvaton scenario has been studied specifically in
Ref. \cite{wands} and its effects on CMB analyzed by the authors
of Ref. \cite{moroi}. Also, the curvaton mechanism has been used
to liberate some inflation models \cite{liddle,dimo}. In addition, there were some motivations for
the curvaton model from particle physics \cite{lyth,enqvist}.


For a specific present, we consider the quintessential inflation model \cite{vilenkin}, one of the NO
models.
In quintessential inflation, the inflaton field will survive to
now to drive the present accelerated expansion suggested by recent measurements of type Ia
supernova \cite{super}. So, it can not decay to other particles. In this paper, we adopt
the model given by Peebles and Vilenkin \cite{vilenkin}, in which the potential of the infaton field is:
\bea
V(\phi)&=&\lambda(\phi^4+M^4)~~~{\rm for}~~\phi<0\nonumber\\
&=&\frac{\lambda M^8}{\phi^4+M^4}~~~~~~{\rm for}~~\phi\geq 0~.
\eea
The energy scale $M$ is very small compared with the Planck mass
$M_{pl}=1.22\times 10^{19}$GeV and it can be neglected during
inflation. Inflation begins with $-\phi\geq 4M_{pl}$ in order
for enough e-folding number of expansion to solve the flatness
problem. The CMB limit on the primordial gravitational waves
requires that the Hubble constant during inflation satisfies $H_{in}\leq 10^{-6}M_{pl}$, the
subscript $in$ denotes the value in inflation epoch. This leads to
\be
\lambda\leq 10^{-14}~.
\ee
In addition, the requirement of $\phi$ driving the present accelerated expansion
gives that $\phi_0>M_{pl}$ and $M>10^5$GeV, the
subscript $0$ represents present value.
We choose the quadratic potential for the curvaton field,
$V(\sigma)=m^2\sigma^2/2$, it has a minimum in the zero point of
$\sigma$ and $m\ll H_{in}$.
\begin{figure*}
\includegraphics[scale=.4]{curvaton}
\caption{\label{fig1} Evolution of the energy densities in the inflaton field (dotted line)
and the curvaton field (solid line). The subscripts 'end', 'osc',
'eq', 'rh', and 'f' respectively refer to the end of inflation,
the onset of curvaton oscillation, the equality of the energy
densities of $\phi$ and $\sigma$, reheating and freezing of
$\phi$.}
\end{figure*}



We numerically studied the evolution of the energy densities in the inflton
field and in the curvaton field. As demonstrated in Fig.
\ref{fig1}, in the era of inflation, the curvaton field slowly
rolled towards the minimum along its potential and its energy density changed little. Inflation ended at
$a=a_{end}$ when $\phi\simeq -0.5M_{pl}$, $a$ is the scale factor.
Then the universe entered into a period dominated by the kinetic
energy of inflaton, this phase was called "kination" by the authors of Re. \cite{joyce},
during which:
\bea
\rho_{\phi}\propto a^{-6}~,~~H\propto a^{-3}~.
\eea
The curvaton field held slow-rolling at the beginning of
kination because of $m\ll H$, and its energy density
$\rho_{\sigma}$ decreased very slowly. Then it began to
oscillate about its minimum at $a=a_{osc}$ when $H$ decreased below $m$.
To avoid another inflation stage driven by the curvaton field,
the universe must still be dominated by $\rho_{\phi}$ at this point, this
requires
\be
\rho_{\sigma osc}<\rho_{\phi osc}=\frac{3H_{osc}^2 M_{pl}^2}{8\pi}~.
\ee
From
\be\label{osc}
H_{osc}\simeq m~, ~~ \rho_{\sigma osc}\simeq m^2\sigma_{osc}^2~,
\ee
we can estimate that there must have
\be\label{sigma}
\sigma_{osc}^2<\frac{3M_{pl}^2}{8\pi}.
\ee
After the curvaton field began oscillating, $\rho_{\phi}$
continued to decrease as $a^{-6}$ towards the value of present
energy density of dark energy and subsequently froze almost like a
cosmological constant when $a\geq a_f$.
$\rho_{\sigma}$ red-shifted as $a^{-3}$ like that of matter when averaged over many oscillations
until it decayed into radiation at $a=a_{rh}$, the subscript $rh$
means reheating. This happened when $H=H_{rh}\sim \Gamma$ and
$\Gamma$ is the decay rate of the curvaton field (we have assumed
instant reheating). It can occur after the moment when the
curvaton field dominated the universe ($a_{rh}\geq a_{eq}$) or at
the epoch when the curvaton field was still sub-dominant
($a_{rh}<a_{eq}$).

In the first case, as shown in Fig. \ref{fig1}, this requires:
\be
\rho_{\phi rh}\leq \rho_{\sigma rh}~.
\ee
Since $\rho_{\phi}\propto a^{-6}$, $\rho_{\sigma}\propto a^{-3}$
and
\be
H_{osc}/H_{eq}=(a_{eq}/a_{osc})^3~,
~~~~H_{eq}^2/H_{rh}^2=(a_{rh}/a_{eq})^3~,
\ee
we get
\be
\frac{\Gamma^2}{mH_{eq}}\leq \frac{8\pi
\sigma_{osc}^2}{3M_{pl}^2}~.
\ee
In above we have used Eq. (\ref{osc}). From $\rho_{\sigma
eq}=\rho_{\phi eq}$
we have
\be
H_{eq}=\frac{8\pi m \sigma_{osc}^2}{3M_{pl}^2}~,
\ee
so the constraint is (with Eq. (\ref{sigma})):
\be
\frac{\Gamma}{m}\leq \frac{8\pi\sigma_{osc}^2}{3M_{pl}^2}<1~.
\ee
The reheating temperature can be estimated from:
\be
\rho_{\sigma rh}=\frac{3M_{pl}^2}{8\pi}\Gamma^2\simeq
\rho_{rad}=\frac{\pi^2}{30}g_{\star}T_{rh}^4~,
\ee
where $g_{\star}$ counts the total number of
degrees of freedom of the relativistic particles, $g_{\star}=10.75$ at temperature $T\sim
1$MeV, and for $T\geq 100$GeV, $g_{\star}\sim {\cal O}(100)$.
The reheating process must completed before BBN,
so in this case, the reheating temperature is approximately
\be\label{reheat1}
T_{rh}\simeq 0.78g_{\star}^{-1/4}\sqrt{M_{pl}\Gamma}\leq
\sqrt{m\sigma_{osc}^2/M_{pl}}~.
\ee

Similarly, in the latter case, when the curvaton field decayed
when it was still sub-dominant but later than oscillation
($a_{osc}\leq a_{rh}<a_{eq}$), we have (for general, we assume
$\sigma_{osc}>0$
\bea\label{reheat2}
& &\frac{8\pi\sigma_{osc}^2}{3M_{pl}^2}<\frac{\Gamma}{m}<1~,\nonumber\\
&
&\sqrt{m\sigma_{osc}^2/M_{pl}}<T_{rh}<0.4\sqrt{m\sigma_{osc}}~.
\eea
In this case, the radiation dominated era began with $T_{eq}\sim
(m\sigma^2)^{3/4}/(\Gamma^{1/4}M_{pl})$ which is within the range
\be
\sqrt{m\sigma_{osc}^3}/M_{pl}<T_{eq}<0.6 \sqrt{m\sigma_{osc}^2/M_{pl}}~.
\ee


For evaluation of $T_{rh}$, one has to consider the constraints on the mass of $\sigma$ field
and its initial value when it began oscillating, $\sigma_{osc}$.
Based on the curvaton mechanism \cite{lyth}, the curvature perturbation responsible for
the CMB anisotropy was originated, during inflation,
from the quantum fluctuation of the curvaton field rather than that of the inflaton.
This happened in two steps. First, the quantum fluctuation in the
curvaton field during inflation was converted into a classical
perturbation, corresponding to an isocurvature perturbation when
cosmological scales left the horizon. Then, in the post-inflation epoch, the
perturbation in the curvaton field was converted into a curvature
perturbation on large scales. In our case, the resulted power
spectrum of the curvature perturbation is given by\cite{lyth}
\be
{\cal P}^{1/2}_{\zeta}=\frac{H_{\star}}{3\pi\sigma_{\star}}\simeq
0.1\frac{H_{\star}}{\sigma_{\star}}~,
\ee
where the star in above equations denotes the time of horizon exit during inflation,
$k=a_{\star}H_{\star}$. In this era, the slow-roll parameter
satisfies
\be
\epsilon\equiv -{\dot H_{in}\over H_{in}^2}\ll 1~,
\ee
so $H_{in}\simeq H_{\star}$ and ${\cal P}^{1/2}_{\zeta}$ is almost
flat. The curvaton field rolled very slowly before oscillation, we
have $\sigma_{osc}\sim \sigma_{\star}$.
Combined these results with the COBE data of the CMB anisotropy, ${\cal P}^{1/2}_{\zeta}=4.8\times 10^{-5}$
and the gravitational waves limit $H_{in}\leq 10^{-6}M_{pl}$, one
has
\be\label{sigma2}
\sigma_{osc}\leq 10^{-3}M_{pl}\sim 10^{16}{\rm GeV}~.
\ee
Substituting Eq. (\ref{sigma2}) into Eq. (\ref{reheat1}) and
(\ref{reheat2}), and considering $m\ll H_{in}$ (we choose $m\leq
10^{-7}M_{pl}\sim 10^{12}$GeV for an evaluation), in both cases we
have approximately
\be
T_{rh}\leq 10^{13}{\rm GeV}~.
\ee


To ensure above analytical estimations  we have made a simple
numerical calculation. We set $\lambda=10^{-15}$, $M=4.7\times 10^{-14} M_{pl}$,
$m_{\sigma}=10^{-8}M_{pl}$ and $\sigma_{\star}=3.27\times 10^{-3}
M_{pl}$.
For curvaton decays at $a=a_{eq}$, one gets $T_{rh}\sim 1.8\times 10^{12}$GeV. Assuming that
 curvaton decays into two fermions $\sigma\rightarrow f \overline{f}$, the decay width takes
the form $\Gamma= g^2 m_{\sigma}/8 \pi$. We find for $\sigma$ decays at $a=a_{eq}$ one requires
$g\sim 0.05 $, which is quite resonable. When the curvaton decays at $a<a_{eq}$, $g<1$ gives
$T_{rh}< 8\times 10^{12}$GeV. The BBN constraint that $T_{rh}>1$MeV
can also be easily satisfied, as from Fig. \ref{fig1}.





In summary, we have introduced the curvaton mechanism into the
non-oscillatory inflationary models as a new reheating mechanism and did specific
studies in the quintessential inflation model proposed by Peebles and
Vilenkin. Our calculations showed that the reheating temperature
can be as high as $10^{13}$GeV, it is high enough for the
quintessential baryogenesis \cite{li,trodden}. In our scenario the
particle production and the curvature perturbation in the universe
are both due to the curvaton field. Our work also provided a new
way to the reheating problem of tachyon inflation \cite{tachyon}.

While this work was in progress a related paper \cite{dimo} appeared in the airchives.
In Ref. \cite{dimo} the author mainly dealt with qunintessential inflation model building and parameter
restriction and our paper is focusing on the curvaton reheating mechanism.





{\bf{Acknowledgments:}} We would like to thank Prof. Xinmin Zhang and Dr. Yun-Song Piao for
useful discussions. This work is supported in part by National Natural
Science Foundation of China under Grant No. 10047004 and by Ministry of
Science and Technology of China under Grant No. NKBRSF 4.

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\end{document}

