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\title{Spontaneous baryogenesis in warm inflation}
%
\author{Robert H. Brandenberger}
%
\affiliation{Physics Department, Brown University, Providence, RI
 02912, USA} 
%
\author{Masahide Yamaguchi}
%
\affiliation{Physics Department, Brown University, Providence, RI 02912, USA}
%
\date{\today}
%%
\begin{abstract}
  We discuss spontaneous baryogenesis in the warm inflation scenario.
  In contrast to standard inflation models, radiation always exists in
  the warm inflation scenario, and the inflaton must be directly
  coupled to it. Also, the transition to the post-inflationary
  radiation dominated phase is smooth and the entropy is not
  significantly diluted at the end of the period of inflation. In
  addition, after the period of warm inflation ends, the inflaton does
  not oscillate coherently but slowly rolls. We show that as a
  consequences of these features of warm inflation, the scenario can
  well accommodate the spontaneous baryogenesis mechanism, provided
  that the decoupling temperature $T_{D}$ of the baryon or the $B-L$
  violating interactions is higher than the temperature of radiation
  during the late stages of inflation.
\end{abstract}

\pacs{98.80.Cq \hspace{7.9cm} BROWN-HET-1345} \maketitle

%]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\label{sec:introduction}

Inflation gives the most natural solution to some of the problems of
standard big bang cosmology such as the horizon problem and the
flatness problem, and provides a causal mechanism for the origin of
the primordial density perturbations whose present state is being
mapped to high precision by observational cosmologists
\cite{inflation}.  There are two types of inflation models. The first
(to which most of the proposed inflation models belong) is isentropic:
any pre-existing radiation before the onset of inflation is completely
diluted away during inflation, and the radiation must then be
re-generated at the end of the phase of inflation during inflationary
reheating.  The other is non-isentropic and called warm inflation
\cite{warm}: here radiation is continuously produced by the decay of
the inflaton, the scalar field which generates inflation, and this
decay in turn supports the slow-roll behavior of the inflaton. In this
scenario, the temperature of radiation remains large during the period
of inflation, and no non-adiabatic radiation generation mechanism
needs to be postulated at the end of inflation.

There are significant differences between the warm inflation scenario
and standard isentropic inflation. Most importantly for the purpose
explored in this paper, the inflaton should be coupled to ordinary
matter, whereas in standard inflation it is usually assumed to be a
gauge singlet. Also, since radiation always exists, the transition
from the inflationary period to the radiation dominated period is
straightforward. In particular, after the end of the inflationary
phase the inflaton field will in general still be rolling slowly,
rather than oscillating about the minimum of its potential as happens
in the standard inflationary models. In addition, in warm inflation
the primordial density fluctuations originate from thermal
fluctuations rather than quantum fluctuations of the inflaton
\cite{thermal}.

A requirement for warm inflation is that the constant rate which
describes the decay of the inflaton into particles dominates over the
Hubble damping coefficient in the inflaton equation of motion. It is a
nontrivial problem to obtain such a large decay width and realize warm
inflation. The dynamics of warm inflation has been investigated in the
context of quantum field theory \cite{BGR,YL}, but the dissipative
dynamics is not yet understood fully. It has been suggested that, in
order to realize warm inflation, the inflaton must couple to a very
large number of particle species \cite{YL}. Some models which achieve
this from first principles were proposed \cite{principle}.

We will show that as a consequence of the abovementioned differences
between warm inflation and standard inflation, it is easier to obtain
spontaneous baryogenesis in the context of warm inflation than in the
context of standard inflation. Most importantly, the inflaton must be
coupled to ordinary matter in a warm inflation scenario, and thus it
is natural to assume that it is not a gauge singlet, whereas in
standard inflation it is usually assumed to be a gauge singlet.  In
addition, in the standard isentropic inflation scenario, the inflaton
oscillates coherently after inflation. Due to the friction term caused
by the current violating operator, this oscillation becomes asymmetric
so that a baryon charge or a $B-L$ charge is generated. This mechanism
also applies to the oscillation of a Nambu-Goldstone boson like an
axion and has been discussed in detail in Ref. \cite{DF}. As shown in
that reference, the oscillations lead to a suppression of the strength
of net baryogenesis over what would be obtained using the naive
classical analysis. However, in warm inflation, the inflaton does not
oscillate coherently but continues to slowly roll even after inflation
ends.  Thus, the analysis of spontaneous lepto/baryogenesis in the
warm inflation scenario will be different. As we will show, the
classical analysis is justified in this case and hence baryogenesis
will be more efficient.

We shall assume that among the many particles the inflaton $\phi$
couples to, it will also couple - albeit derivatively - to the baryon
current or to the $B-L$ current.  Following the arguments by Cohen and
Kaplan \cite{CK}, we will show that baryo/leptogenesis may be possible
if the inflaton has a derivative coupling to such a current given by
%
\beq
  \CL_{\rm eff} = \frac{c}{M}\,\del_{\mu}\phi\,J^{\mu},
\eeq
%
where $c$ is a coupling constant, $M$ is the cutoff scale which
describes the physics of baryon number violation and which is set to
be the reduced Plank mass $M_{G} \sim 10^{18}$ GeV in this paper. In
the above, $J^{\mu}$ is either the baryon current or the $B-L$
current.  Integrating this coupling by parts, we have an interaction
term given by
%
\beq
  \CL_{\rm eff} = - \frac{c}{M}\,\phi\,\del_{\mu}J^{\mu}.
\eeq
%
If baryon number conservation or $B-L$ number conservation is
violated, the divergence does not disappear and is replaced by a
current violating operator, which can cause baryo/leptogenesis like in
Affleck-Dine baryogenesis \cite{AD}.

In this paper, we explore in detail the spontaneous baryo/leptogenesis
mechanism in warm inflation. In the next section, we briefly review
the warm inflation scenario. In Sec III, we discuss the possibility of
spontaneous baryo/leptogenesis in warm inflation. In the final
section, we summarize our results.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Dynamics of warm inflation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\label{sec:warm}

In this section, we briefly review the dynamics of warm inflation.
First of all, we assume that the inflaton couples to a large number of
particles, generating a large and constant decay width $\Gamma$. Then,
the equation of motion of the inflaton and the time development of the
energy density of radiation $\rho_{\rm r}$ are given by
%
\bea
 \ddot{\phi} + (3H + \Gamma)\dot{\phi} + V'(\phi) = 0, && 
 \label{eq:eqn} \\
 \dot{\rho_{\rm r}} + 4H\rho_{\rm r} = \Gamma \dot{\phi}^2 &&
 \label{eq:reqn} 
\eea
%
with the potential given by $V(\phi) = \frac12 m^2 \phi^2$. Here, the
derivative of the potential is taken with respect to $\phi$, and $H$
is the Hubble parameter given by
%
\beq
  H^2 = \frac{1}{3M_{G}^2}(\rho_{\phi} + \rho_{\rm r}),
\eeq
%
with $\rho_{\phi}$ being the energy density of the inflaton and $M_{G}
\simeq 2.4 \times 10^{18}$ GeV denoting the reduced Planck scale.

For successful warm inflation, we require the following five
conditions:
%
\bed
  \item[(i)]  $\rho_{\rm r} \ll V$,
  \item[(ii)] ${1 \over 2} {\dot{\phi}}^2 \ll V$,
  \item[(iii)] $H \ll \Gamma$,
  \item[(iv)] $|\dot{\rho_{\rm r}}| \ll 4H\rho_{\rm r},\,\Gamma\dot{\phi}^2$,
  \item[(v)]  $|\ddot{\phi}| \ll (3H + \Gamma)\dot{\phi},\,V'(\phi)$.
\eed
%
The first two conditions are required in order to have inflation, the
third is the criterion for warm inflation as opposed to standard
inflation, without the fourth requirement it would be unreasonable to
assume that $\Gamma$ is constant, and the final criterion is the {\it
  slow rolling} condition for the inflaton dynamics. If these
conditions are satisfied, the equations (\ref{eq:eqn}) and
(\ref{eq:reqn}) reduce to
%
\bea
  \Gamma\dot{\phi} +V' \simeq 0 
    &\Longleftrightarrow&
  \dot{\phi} \simeq - \frac{V'}{\Gamma}
             \simeq - \frac{m^2 \phi}{\Gamma}, \\ 
  4H\rho_{\rm r} \simeq \Gamma\dot{\phi}^2
    &\Longleftrightarrow&
  \rho_{\rm r} \simeq \frac{\Gamma\dot{\phi}^2}{4H}
           \simeq \frac{V'^2}{4H\Gamma} 
           \simeq \sqrt{\frac38} \frac{m^3 M_{G} \phi}{\Gamma}, \\
    &\Longleftrightarrow&
  T_{\rm r} = \lmk \frac{30 \rho_{\rm r}}{g_{\ast} \pi^2} \rmk^{\frac14}
        \simeq \lmk \frac{675}{4 g_{\ast}^2 \pi^4} \rmk^{\frac18}   
               \lmk \frac{m^3 M_{G} \phi}{\Gamma} \rmk^{\frac14},
\eea
%
where $g_{\ast}$ is the number of the relativistic degrees of freedom.

Conditions (i) and (ii) imply the dominance of the vacuum energy. The
first is satisfied for
%
\beq
  \phi \gg \phi_{\rm end} \equiv \sqrt{\frac{3}{2}} \frac{m}{\Gamma} M_{G},
\eeq
%
and the second is then obeyed for all value of $\phi$ provided that
%
\beq 
m \ll \Gamma \, .
\eeq
%
Thus, warm inflation ends at $\phi = \phi_{\rm end}$. At that time,
the temperature of radiation becomes
%
\beq
  T_{\rm end} \equiv
  T_{\rm r}(\phi = \phi_{\rm end}) \sim m \sqrt{\frac{M_{G}}{\Gamma}}.
\eeq
%
The condition (iii) implies that the dominant friction term in the
inflaton equation of motion is given by the coupling to other
particles rather than by the Hubble expansion and is satisfied for
%
\beq
  \phi \ll \sqrt{6}\,\frac{\Gamma}{m} M_{G}.
\eeq
%
The condition (iv) implies the constancy of the energy density of
radiation and is valid as long as $\phi > \phi_{\rm end}$. The last
requirement is the so-called slow-roll condition and is also satisfied
if $m \ll \Gamma$. Combining these results, we conclude that warm
inflation takes place while $\phi$ is in the range given by
%
\beq \label{range}
\phi_{\rm end} \lesssim \phi \lesssim (\Gamma/m) M_{G} 
\eeq
%
provided that $m \ll \Gamma$.

The number $N(\phi)$ of e-foldings of inflation between when the
inflaton field has the value $\phi$ in the range given by
(\ref{range}) and when $\phi = \phi_{\rm end}$ can easily be estimated
and yields the following relation between $\phi_{N}$ (the initial
value of $\phi$ which gives N e-foldings) and $N$:
%
\beq
  N = \int H dt \simeq \frac{\Gamma}{\sqrt{6}\,m M_{G}}
                        (\phi_{N}-\phi_{\rm end}),
\eeq
%
with the result $\phi_{N} \simeq \sqrt{6}\,N m M_{G} / \Gamma$. Taking
the scale of the CMB anisotropies measured by the COBE satellite to
correspond to $N = 60$, then
%
\beq
\phi_{\rm COBE} = \phi_{60}
\sim 150\,m M_{G} / \Gamma \sim 150\,\phi_{\rm end} \,.
\eeq

For $T_{\rm r} > H$, which corresponds to $\phi <
(M_{G}/m\Gamma)^{1/3} M_{G}$, thermal fluctuations dominate over
quantum fluctuations \cite{thermal}.  As shown in \cite{thermal2} the
root mean square of fluctuations of the inflaton is given by
%
\beq
  \la (\delta \phi)^2 \ra \simeq \frac{1}{2\pi^{2}} 
                                    \sqrt{\Gamma H} T_{\rm r}.
\eeq
%
Based on these initial conditions for fluctuations generated during
inflation,, the final \footnote{Final means when the scale re-enters
  the Hubble radius at late times.} amplitude of the curvature
perturbation $\Phi_{A}$ (the relativistic gravitational potential in
longitudinal gauge - see \cite{MFB}) on a comoving scale whose
physical wavelength equals the Hubble radius during the period of warm
inflation at $\phi=\phi_{N}$ is given by
\cite{flucts1,flucts2,thermal2}
%
\beq
  \Phi_{A} \sim f H \frac{\sqrt{\la (\delta \phi)^2 \ra}}{\dot{\phi}} 
           \sim 0.02 \lmk \frac{\Gamma^9 \phi_{N}^3}{M_{G}^{9} m^{3}} 
                    \rmk^{\frac18}, 
\eeq
%
where $f=3/5~(2/3)$ in the matter (radiation) domination (this result
was derived using the full general relativistic theory \cite{MFB} of
linear cosmological fluctuations in \cite{flucts2}). The COBE
normalization of CMB anisotropies requires $\Phi_{A} \simeq 3\times
10^{-5}$ at $N\simeq 60$ \cite{COBE}. This leads to the requirement
%
\beq
  \Phi_{A}(N=60) \sim 0.1 \lmk \frac{\Gamma}{M_{G}} 
                      \rmk^{\frac34}
                 \sim 10^{-5}.
\eeq
%
which yields $\Gamma \sim 10^{13}$ GeV.

In addition, the spectral index $n_{s}$ can be estimated to be
\cite{thermal2,flucts2}
%
\bea
  n_{s} - 1 &=& \frac{\dot{\phi}}{H} \frac{d}{d\phi}(\ln \Phi_{A})
            \sim \frac{3\sqrt{6}}{8} \frac{m M_{G}}{\Gamma \phi_{N}}, \\ 
            &\sim& 0.006 \qquad \qquad \qquad \qquad \qquad \qquad
              {\rm for} \quad \phi = \phi_{COBE} \sim 
                  150 \lmk \frac{m M_{G}}{\Gamma} \rmk.
\eea

Even after warm inflation ends, the friction term in equation of
motion of the inflaton $\phi$ is still large so that the inflaton
continues to slow-roll instead of oscillating coherently. Hence, the
discussion of spontaneous baryogenesis in the reheating stage done in
Ref. \cite{CK} does not directly apply to the case of warm inflation.
After warm inflation ends, the dynamics of the inflaton is given by
%
\beq
 \left\{
 \begin{array}{l}
  \phi = \phi_{\rm end} \exp \lhk 
            - \frac{m^2}{\Gamma}(t - t_{\rm end})
                             \rhk, \\
  \dot\phi = - \sqrt{\frac32} \frac{m^3}{\Gamma^2} M_{G}\,\exp \lhk 
            - \frac{m^2}{\Gamma}(t - t_{\rm end})
                             \rhk, 
  \label{eq:dotphi}
 \end{array}\right.
\eeq
%
with $t_{\rm end} \simeq \Gamma/m^2$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Baryo/leptogenesis in warm inflation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\label{sec:baryogenesis}

In this subsection, we show that spontaneous baryo/leptogenesis can
easily be realized in warm inflation. As is mentioned briefly in the
Introduction, we assume that the inflaton couples derivatively to the
$B-L$ current via an interaction Lagrangian
%
\beq
  \CL_{\rm eff} = \frac{c}{M}\,\del_{\mu}\phi\,J_{B-L}^{\mu},
  \label{eq:derivative}
\eeq
%
where $c$ is a coupling constant and $M$ is the cutoff scale. Assuming
that $\phi$ is homogeneous, the above coupling becomes
%
\beq
  \CL_{\rm eff} = \frac{c}{M}\, 
                    \dot\phi\,n_{B-L}
                = \frac{c}{M}\, 
                    \dot\phi\,(n_{b-l} - n_{\bar{b-l}}) 
                = \mu(t)\,n_{B-L},
\eeq
%
with $\mu(t)$ defined as
%
\beq
  \mu(t) \equiv \frac{c}{M}\,\dot\phi.
\eeq
%

In contrast to the standard inflation models, radiation always exists
in thermal inflation. Then, if the time derivative $\dot\phi$ is
effectively nonzero, $\mu(t)$ becomes the effective time-dependent
chemical potential, which induces the $B-L$ asymmetry even in thermal
equilibrium. Such a thermal equilibrium baryo/leptogenesis scenario is
discussed in the context of quintessence \cite{LFZ,FNT,Yamaguchi}.
However, for this mechanism to work in the context of warm inflation,
the decoupling temperature of the $B-L$ violating operator would have
to be fine-tuned to be equal to the temperature of the radiation at
the end of warm inflation. Obviously, it needs to be lower or equal -
else there would be no thermal equilibrium for baryo- or leptogenesis
to occur towards the end of the period of inflation. But the
decoupling temperature cannot be lower either.
% This is because the $B-L$ asymmetry
%produced before warm inflation ends is diluted away, 
Since $\dot\phi$ decays exponentially after warm inflation ends, then
if the decoupling temperature were lower than $T_{\rm end}$, then
there would be a time interval after the end of warm inflation during
which $B-L$ violating processes were in thermal equilibrium but the
chemical potential for $B-L$ number was effectively zero, and during
which therefore the $B-L$ number density would be driven to zero.
Because of this fine tuning, we do not consider this possibility any
further.

In this paper, we consider another possibility, namely one in which
the $B-L$ asymmetry is generated dynamically like in the Affleck-Dine
baryogenesis scenario. In this case, the upper bound on the decoupling
temperature for $B-L$ violating processes no longer is present.
Taking the coupling (\ref{eq:derivative}) into account, the equation
of motion of the inflaton is changed to
%
\beq
 \ddot{\phi} + (3H + \Gamma)\dot{\phi} 
   - \frac{c}{M} (\dot{n}_{B-L} + 3 H n_{B-L}) + V'(\phi) = 0. 
 \label{eq:eom2}
\eeq
%
When the $B-L$ current is not conserved, the divergence of the current
does not disappear and is replaced by a current violating operator. We
simply assume that the $B-L$ current is not conserved and such an
operator just gives rise to an additional decay of the inflaton, which
changes the equation of motion of the inflaton to
%
\beq
 \ddot{\phi} + (3H + \Gamma + \Gamma_{\not{B-L}})\dot{\phi} 
      + V'(\phi) = 0. 
 \label{eq:eom3}
\eeq
%
Here $\Gamma_{\not{B-L}}$ is the decay width due to the additional
decay.  If it is much smaller than $\Gamma$, the dynamics of the
inflaton is not changed much.  Comparing the two equations
(\ref{eq:eom2}) and (\ref{eq:eom3}) \footnote{As for this comparison,
  a subtlety was raised in Ref. \cite{DF}. While Eq. (\ref{eq:eom2})
  is an operator equation, Eq. (\ref{eq:eom3}) is obtained after
  vacuum averaging. The authors of \cite{DF} showed that the average
  value $\la \dot{n}_{B-L} \ra$ is complicated and not given by the
  above simple comparison when the inflaton oscillates coherently.
  However, in our case, the inflaton does not oscillate coherently but
  slowly rolls so that this subtlety does not matter. In particular,
  for the values of the parameters which we use, the classical
  approximation is justified, and there is no {\it energy problem} as
  discussed in the second reference of \cite{DF}.}, it follows that
the time evolution of the $B-L$ number density is given by
%
\beq
  \dot{n}_{B-L} + 3 H n_{B-L} = 
    - \frac{M}{c} \Gamma_{\not{B-L}} \dot{\phi}.
\eeq

As given in Eq. (\ref{eq:dotphi}), $\dot\phi$ decays exponentially
after warm inflation so that, except for the dilution due to the
adiabatic expansion of the universe, $n_{B-L}$ changes significantly
only during the short period $\Delta t \simeq \Gamma/m^2$ after
inflation.  Then, the ratio between the $B-L$ number density $n_{B-L}$
and the entropy density $s = \frac{2\pi^2}{45} g_{\ast s} T^3$ can be
roughly estimated to be
%
\bea
  \frac{n_{B-L}}{s} &\simeq& 
     \frac{M}{c}\,\Gamma_{\not{B-L}} \frac{m^3}{\Gamma^2} M_{G}
     \Delta t \left/ \frac{2\pi^2}{45} g_{\ast s} T_{\rm end}^3
              \right. \non \\
                    &\simeq&
     0.02\,\frac{1}{c}\,\frac{M \Gamma_{\not{B-L}}}{m^2}
          \lmk \frac{\Gamma}{M_{G}} \rmk^{\frac12}.
\eea
%
Here we took $g_{\ast s} \sim 100$. If $T_{\rm end}$ is higher than
the temperature of the electroweak phase transition, a part of the
$B-L$ asymmetry at that time is converted into the baryon asymmetry
through the sphaleron processes \cite{sphaleron}. Then, the
baryon-to-entropy is given by
%
\bea
  \frac{n_B}{s} &\simeq&
                  \frac{8}{23} \frac{n_{B-L}}{s} \non \\
                &\simeq& 
     0.01\,\frac{1}{c}\,\frac{M \Gamma_{\not{B-L}}}{m^2}
          \lmk \frac{\Gamma}{M_{G}} \rmk^{\frac12}.
\eea

As an example, we consider two cases, where the $B-L$ violating
interaction is given by (I) a dimension five operator and (II) a
dimension six operator. In Case (I), the decay width
$\Gamma_{\not{B-L}}$ can be roughly estimated (by dimensional
analysis) to be
%
\bea
  \Gamma_{\not{B-L}} &\sim& \frac{T^3}{M^2} \non \\
                     &\sim& 
      \frac{m^3 M_{G}^{\frac32}}{M^2 \Gamma^{\frac32}}
      \qquad \qquad {\rm for} \quad T = T_{\rm end}.
\eea
% 
Then, the baryon-to-entropy is given by
%
\bea
  \frac{n_B}{s} &\sim&
     0.01\,\frac{1}{c}\,\frac{m M_{G}}{\Gamma M} \non \\
                &\sim&
     0.01 \frac{m}{\Gamma}
     \qquad \qquad {\rm for} \quad c \sim 1, M = M_{G}.
\eea
%
For $M=M_{G}$, the decoupling temperature becomes $T_{D} \sim M_{G}$,
which is much higher than $T_{\rm end}$. Taking into account $\Gamma
\sim 10^{13}$ GeV and given the constraint $\Phi_{A} \sim 10^{-5}$, we
can explain $n_B/s \sim 10^{-10}$ if $m \sim 10^{5}$ GeV, which leads
to $T_{\rm end} \sim 10^8$ GeV,

In the same way, in Case (II), the decay width $\Gamma_{\not{B-L}}$
can be roughly estimated to be
%
\bea
  \Gamma_{\not{B-L}} &\sim& \frac{T^5}{M^4} \non \\
                     &\sim& 
      \frac{m^5 M_{G}^{\frac52}}{M^4 \Gamma^{\frac52}}
      \qquad \qquad {\rm for} \quad T = T_{\rm end}.
\eea
% 
Then, the baryon-to-entropy is given by
%
\bea
  \frac{n_B}{s} &\sim&
     0.01\,\frac{1}{c}\,\frac{m^3 M_{G}^2}{\Gamma^2 M^3} \non \\
                &\sim&
     0.01 \frac{m^3}{\Gamma^2 M_{G}}
     \qquad \qquad {\rm for} \quad c \sim 1, M = M_{G}.
\eea
%
Taking into account $\Gamma \sim 10^{13}$ GeV and given the
requirement $\Phi_{A} \sim 10^{-5}$, we can explain $n_B/s \sim
10^{-10}$ if $m \sim 10^{12}$ GeV, which leads to $T_{\rm end} \sim
10^{15}$ GeV.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion and Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\label{sec:con}

In this paper, we have discussed spontaneous baryo/leptogenesis in
warm inflation. Though radiation always exists in warm inflation,
spontaneous baryogenesis in thermal equilibrium does not work in
general because for such a baryogenesis mechanism to be successful
would require a fine-tuning of the decoupling temperature of the $B-L$
violating interaction. Instead, we considered another possibility, in
which the $B-L$ asymmetry is generated dynamically. In the standard
inflation models, the inflaton oscillates coherently after inflation.
On the other hand, in warm inflation, the inflaton still slowly rolls
even after inflation ends. We have shown that spontaneous baryogenesis
can be implemented rather easily in this situation. We have shown that
during the short period just after warm inflation ends, a sufficient
$B-L$ asymmetry can be generated to explain the presently observed
baryon asymmetry.

For successful warm inflation, the inflaton must couple to a very
large number of particles in order to maintain a large decay width.
This may be viewed as a disadvantage, but it also renders it rather
likely that interaction terms such as those which we postulate exist.
Note that some models for warm inflation motivated by string theory
have been proposed.  All we assumed in this paper is the existence of
a derivative coupling of the inflaton to the $B-L$ current. It would
be of interest to explore whether in the string-motivated models for
warm inflation such a derivative coupling can be accommodated given
the couplings of $\phi$ to a large number of other fields.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{ACKNOWLEDGMENTS}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

M.Y. is partially supported by the Japanese Grant-in-Aid for
Scientific Research from the Ministry of Education, Culture, Sports,
Science, and Technology. At Brown, this work is supported in part by
the United States Department of Energy under Contract
DE-FG0291ER40688, Task A,

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\end{document}



