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\begin{document}

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\begin{titlepage}
\begin{flushright}
%\preprint{
BA-02-40\\
%March 2002\\
%}
\end{flushright}
\vskip 2cm
\begin{center}
%\title
{\Large\bf
Inflation and Leptogenesis \\
with Five Dimensional $SO(10)$
%Supersymmetric Grand Unification
}
\vskip 1cm
{\normalsize\bf
%\author{
Bumseok Kyae\footnote{bkyae@bartol.udel.edu} and
Qaisar Shafi\footnote{shafi@bxclu.bartol.udel.edu}
}
\vskip 0.5cm
{\it Bartol Research Institute, University of Delaware, \\Newark,
DE~~19716,~~USA\\[0.1truecm]}

%
%%\maketitle

\end{center}
\vskip .5cm

%\date{\today}
%\pacs{PACS: 11.25.Mj, 12.10.Dm, 98.80.Cq}


\begin{abstract}

We present a five dimensional $SO(10)$ model compactified
on $S^1/(Z_2\times Z_2')$, which yields
$SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_X$
below the compactification scale.
The gauge symmetry $SU(5)\times U(1)_X$ is preserved on one
of the fixed points, while `flipped' $SU(5)'\times U(1)'_X$ is respected
on the other fixed point.
Inflation is associated with $U(1)_X$ breaking,
and is implemented through F-term scalar potentials on the two fixed points.
A brane-localized Einstein-Hilbert term allows both branes
to have positive tensions during inflation.
The scale of $U(1)_X$ breaking is fixed from $\delta T/T$ measurements
to be around $10^{16}$ GeV,
and cosmic strings produced during $U(1)_X$ breaking are inflated away.
The scalar spectral index $n=0.96-0.99$, and there is
negligible contribution ($\lapproxeq 1\%$)
from gravitational waves to the quadrupole anisotropy.
The inflaton field decays into right-handed neutrinos whose subsequent
out of equilibrium decay yields the observed baryon asymmetry
via leptogenesis.

\end{abstract}
\end{titlepage}

\newpage

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


In a recent paper~\cite{KS},
we showed how supersymmetric (SUSY) inflation can be realized
in five dimensional (5D) models in which the fifth dimension is compactified on
the orbifold $S^1/(Z_2\times Z_2')$.
Orbifold symmetry breaking in higher dimensional grand unified theories (GUTs)
have recently attracted a great deal of attention
because of the apparent ease with which they can circumvent
two particularly pressing problems encountered
in four dimensional (4D) SUSY GUTs~\cite{orbifold}.
Namely, the doublet-triplet (DT) splitting problem
and the problem caused by dimension five nucleon decay.
The apparent reluctance of the proton to decay,
as shown by the recent lower limits on its lifetime~\cite{proton},
seems to be in broad disagreement with the predicted rates
from dimension five processes in minimal SUSY $SU(5)$ and $SO(10)$ models.
The existence of the orbifold dimension enables one to implement
DT splitting and simultaneously suppress (or even eliminate)
dimension five proton decay.

The inflationary scenario described in~\cite{KS} was inspired
by the above particle physics considerations and
has some novel features.
The primordial density (temperature) fluctuations are proportional to
$(M/M_{\rm Planck})^2$, along the lines of the 4D model
proposed in~\cite{hybrid}.
Here $M$ refers to the scale of some symmetry breaking that is
associated with inflation, and $M_{\rm Planck}=1.2\times 10^{19}$ GeV
denotes the Planck scale.
In an $SO(10)$ model, for example, the orbifold breaking can be used to yield
the subgroup $H=SU(4)_c\times SU(2)_L\times SU(2)_R$~\cite{dermisek},
so that inflation is associated with the breaking of $H$
to the MSSM gauge group~\cite{khalil,KS}.
The anisotropy measurements~\cite{cobe} can provide a determination of $M$
independently of any particle physics considerations.
$M$ turns out to be quite close (or equal) to the SUSY GUT scale
of around $10^{16}$ GeV.
Last but not least, the scalar spectral index of density fluctuations is
very close to unity ($n=0.96-0.99$)~\cite{khalil,KS}.
The gravitational wave contribution to the quadrupole anisotropy is found
to be essentially negligible ($\lapproxeq 1 \%$).
In an $SO(10)$ model with the subgroup $H$ given above,
the inflaton decays into the MSSM singlet (right-handed) neutrinos,
whose out of equilibrium decay leads to the observed baryon asymmetry via
leptogenesis~\cite{lepto,ls}.
As we will see,
this is also the case even with a different subgroup $H$ of $SO(10)$.
Baryogenesis via leptogenesis appears to be a rather generic feature
of 5D $SO(10)$ based inflationary models considered here.

Although our considerations are quite general, in this paper we focus on
an example based on $SO(10)$ with subgroup
$SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_X$ obtained from
$S^1/(Z_2\times Z_2')$ orbifolding.
The standard $SU(5)\times U(1)_X$ is preserved on one brane, while
`flipped' $SU(5)'\times U(1)'_X$ is respected on the other brane.
All massless modes from the chiral component of the 5D vector multiplet
turn out to be vector-like,
so they can pair up easily and become superheavy.
Inflation is associated with the breaking of $U(1)_X$, followed by its decay
into right-handed neutrinos, which subsequently generate
a primordial lepton asymmetry.
The gravitino constraint on the reheat temperature~\cite{gravitino}
imposes important constraints on the masses of the right-handed neutrinos
which can be folded together with the information now available
from the oscillation experiments~\cite{nuoscil}.

As emphasized in~\cite{KS}, implementation in five dimension of the
inflationary scenario considered in \cite{hybrid} requires some care.
Note that the five dimensional setup is the appropriate one because of
the proximity of the scale of inflation and the comapactification scale
(obthe are of order $M_{GUT}$).
The inflaton potential must be localized on the orbifold fixed points
(branes), since a superpotential in the bulk is not allowed.
For a vanishing bulk cosmological constant, a three space inflationary
solution triggered by non-zero brane tensions (or vacuum energies)
exists~\cite{KS}.
However, 5D Einstein equations often require that the signs of the brane
tensions on the two branes are opposite, which  is undesirable.
As shown in~\cite{KS}, this problem can be circumvented
by introducing a brane-localized
Einstein-Hilbert term in the action.
The two brane tensions are both positive during inflation,
and they vanish when it ends.
The radion's brane-localized mass term is sufficiently large
to keep the size of the extra dimension stable.

The plan of our paper is as follows. In section 2, we review cosmology
in a five dimensional setting, and discuss in particular the transition from
an inflationary to the radiation dominated epoch.
In section 3, we discuss the orbifold breaking of $SO(10)$
to $SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_X$.
Section 4 summarizes the salient features of the inflationary scenario and
subsequent leptogenesis.
Our conclusions are summarized in section 5.


\section{5D Cosmology}

%%
%%We intend to embed the 4 dimensional F-term supersymmetric inflation
%%to 5 dimensional supersymmetric theory.
%%Since $N=2$ SUSY does not allow the superpotential,
%%we need to break $N=2$ SUSY to $N=1$ by $Z_2$ orbifolding.
%%Then on orbifold fixed points (branes), only $N=1$ SUSY is preserved,
%%and the superpotentials can be introduced on the points.
%%In this set up, therefore, inflation is expected to be implemented
%%by F-term scalar potentials localized on the branes.
%%
%Our approach to inflationary cosmology is largely motivated by recent
%attempts~\cite{} to construct realistic five dimensional (5D) SUSY GUTs
%in which the fifth dimension corresponds
%to the orbifold $S^1/(Z_2\times Z_2')$,
%so that the effective low energy theory can display only $N=1$ SUSY.
%Since $N=2$ SUSY in the bulk does not allow the superpotential,
%four dimensional (4D) F-term inflation should be realized
%by suitable scalar potentials introduced on the orbifold fixed points
%(branes) where superpotentials are allowed.
%We can introduce here the 4D $N=1$ SUSY model described
%in \cite{hybrid}, which have the attractive feature that inflation is
%directly linked to some symmetry breaking phase transition,
%such that $\delta T/T\propto (M/M_{\rm Planck})^2$, where $M$ denotes
%the corresponding symmetry breaking scale.

We consider 5D spacetime $x^M=(x^{\mu},y)$, $\mu=0,1,2,3$, where the fifth
dimension is compactified on an $S^1/Z_2$ orbifold, and
the (SUGRA) action given by
\begin{eqnarray} \label{action}
S=\int d^4x \int_{-y_c}^{y_c}dy~e\bigg[\frac{M_5^3}{2}~R_5+{\cal L}_B
+\frac{\delta(y)}{e_5^5}\bigg(\frac{M_4^2}{2}~\bar{R}_4
+{\cal L}_{I}\bigg)
+\frac{\delta(y-y_c)}{e_5^5}~
%\bigg(\frac{M_{II}^2}{2}~\bar{R}_4+
{\cal L}_{II}
%\bigg)
\bigg] ~,
\end{eqnarray}
where $R_5$ ($\bar{R}_4$) stands for the 5 dimensional (4 dimensional)
Einstein-Hilbert term, and ${\cal L}_B$ (${\cal L}_I$, ${\cal L}_{II}$)
denotes some unspecified bulk (brane)
contributions to the full Lagrangian.
The brane scalar curvature term $\bar{R}_4(\bar{g}_{\mu\nu})$
is defined through the induced metric,
$\bar{g}_{\mu\nu}(x)\equiv g_{\mu\nu}(x,y=0)$ ($\mu,\nu=0,1,2,3$).
The brane-localized Einstein-Hilbert terms in Eq.~(\ref{action})
are allowed even in SUGRA, but should, of course, be accompanied
by a brane gravitino kinetic term as well as other terms,
as spelled out in the off-shell SUGRA formalism \cite{kyae}.
Here we assume that the bulk cosmological constant is zero.

For the cosmological solution let us take the following metric ansatz,
\begin{eqnarray} \label{metric}
ds^2=\beta^2(t,y)\bigg(-dt^2+a^2(t)~d\vec{x}^2\bigg)+dy^2 ~,
\end{eqnarray}
which shows that the three dimensional space is homogeneous and isotropic.
The non-vanishing components of the 5D Einstein tensor derived
from Eq.~(\ref{action}) are
\begin{eqnarray}
G^0_0&=&3\bigg[\bigg(\frac{\beta''}{\beta}\bigg)
+\bigg(\frac{\beta'}{\beta}\bigg)^2~\bigg]
-\frac{3}{\beta^2}\bigg[\bigg(\frac{\dot{\beta}}{\beta}
+\frac{\dot{a}}{a}\bigg)^2~\bigg]
-\delta(y)\frac{M_4^2}{M_5^3}\frac{3}{\beta^2}\bigg[
\bigg(\frac{\dot{\beta}}{\beta}
+\frac{\dot{a}}{a}\bigg)^2~\bigg]
~, \label{eins00} \\
G^i_i&=&3\bigg[\bigg(\frac{\beta''}{\beta}\bigg)
+\bigg(\frac{\beta'}{\beta}\bigg)^2~\bigg]
-\frac{1}{\beta^2}\bigg[2\frac{\ddot{\beta}}{\beta}+2\frac{\ddot{a}}{a}
+4\frac{\dot{\beta}}{\beta}\frac{\dot{a}}{a}
-\bigg(\frac{\dot{\beta}}{\beta}\bigg)^2+\bigg(\frac{\dot{a}}{a}\bigg)^2
~\bigg] \nonumber \\
&&-\delta(y)\frac{M_4^2}{M_5^3}\frac{1}{\beta^2}\bigg[
2\frac{\ddot{\beta}}{\beta}+2\frac{\ddot{a}}{a}
+4\frac{\dot{\beta}}{\beta}\frac{\dot{a}}{a}
-\bigg(\frac{\dot{\beta}}{\beta}\bigg)^2+\bigg(\frac{\dot{a}}{a}\bigg)^2
~\bigg]
~, \label{einsii} \\
G^5_5&=&6\bigg[\bigg(\frac{\beta'}{\beta}\bigg)^2~\bigg]-\frac{3}{\beta^2}
\bigg[\frac{\ddot{\beta}}{\beta}+\frac{\ddot{a}}{a}
+3\frac{\dot{\beta}}{\beta}\frac{\dot{a}}{a}+\bigg(\frac{\dot{a}}{a}\bigg)^2
~\bigg]
%&&-\delta(y)\frac{M_4^2}{M_5^3}\frac{3}{\beta^2}
%\bigg[\frac{\ddot{\beta}}{\beta}+\frac{\ddot{a}}{a}
%+3\frac{\dot{\beta}}{\beta}\frac{\dot{a}}{a}+\bigg(\frac{\dot{a}}{a}\bigg)^2
%~\bigg]
~, \label{eins55} \\
G_{05}&=&-3\bigg[\bigg(\frac{\beta'}{\beta}\bigg)^{\cdot}~ \bigg] ~,
\label{eins05}
\end{eqnarray}
where primes and dots respectively denote derivatives with respect to
$y$ and $t$, and the terms accompanied by delta functions
arise from the brane-localized Einstein-Hilbert term.

Let us first discuss inflation under this set up.
Since 5D $N=1$ SUSY does not allow a superpotential
(and the corresponding F-term scalar potential) in the bulk,
we introduce inflaton scalar potentials on the two branes
where only 4D $N=1$ SUSY is preserved~\cite{KS,king}.
The energy-momentum tensor during inflation is given by
\begin{eqnarray}
T^0_0~=~T^i_i&=&-\delta(y)\frac{V_{1}(t)}{M_5^3}
-\delta(y-y_c)\frac{V_{2}(t)}{M_5^3} ~,   \\
T^5_5&=&0 ~,
\end{eqnarray}
where $i=1,2,3$, and $V_{1}$($V_2$) is the scalar potential on B1(B2).
During inflation, the inflaton `rolls down' sufficiently slowly,
which provides a large enough number of e-foldings to resolve the horizon and
flatness problems.
The end of inflation is marked by the breaking of the `slow roll' conditions,
and the inflaton rolls quickly to the true vacuum with flat 4D spacetime.
%
%But once the inflaton reaches
%at the point on the potential where the flatness conditions could not be
%satisfied any more, it falls down
%
%
We make the approximation that the vacuum energies are essentially
instantaneously converted into radiation, such that
%
%they could be roughly
%supposed to have the form of the step functions with respect to time,
%
\begin{eqnarray} \label{localpot}
V_{1(2)}(t)=\Lambda_{1(2)}\times \theta\bigg(-(t-t_0)/t_*\bigg) ~,
\end{eqnarray}
where $t_0$ denotes the time when inflation terminates,
and $t_*$ is a parameter with length (or time) dimension, and
$\Lambda_{1(2)}$ ($>0$) is a cosmological constant
during inflation at B1(B2).
Clearly, $V_{1(2)}=\Lambda_{1(2)}$ for $t<t_0$, and vanishes for $t>t_0$.
Note that we assume that the inflaton is a bulk field
so that the positive vacuum energies on the two branes reduce to
zero simultaneously.

The exact solution is
\begin{eqnarray}
&&\beta(t,y)=H_0\Theta(t)~|y|+c+\Delta^2(t) ~, \label{beta} \\
&&a(t)={\rm exp}\bigg[H_0\Theta(t)~t\bigg] ~, \label{a}
\end{eqnarray}
where $H_0$ ($>0$) is the Hubble constant during inflation, and
$\Theta(t)$ and $\Delta(t)$ are given by
\begin{eqnarray}
\Theta(t)&\equiv&\theta\bigg(-(t-t_0)/t_*\bigg) ~,  \label{solB}\\
\Delta(t)&\equiv&\delta\bigg((t-t_0)/t_*\bigg) ~.   \label{solA}
\end{eqnarray}
The integration constant $c$ in Eq.~(\ref{beta}) can be taken to be positive
which avoids a naked singularity within the interval $-y_c<y<y_c$
in the solution for $t<t_0$.
It was already shown in~\cite{KS} that
the solution for $t<t_0$ is given by Eqs.~(\ref{beta}) and (\ref{a}).
The solution for $t>t_0$ is trivial
since both sides of Einstein equation vanish.
The divergences expected at $t=t_0$ from the time derivatives,
$\dot{\beta}$, $\ddot{\beta}$, $\dot{a}$, $\ddot{a}$, etc.,
appearing in Eqs.~(\ref{eins00})--(\ref{eins05}) do not materialize
because of the larger powers of $\Delta^2(t)$ contained in $\beta(t,y)$s
in the denominators \cite{kkl2}.

In reality, though, the brane vacuum energies in Eq.~(\ref{localpot})
would have large but finite inclination around $t=t_0$, and
the delta functions discussed above would be replaced by a finite function.
If the transition from $V_{1(2)}=\Lambda_{1(2)}$ to $V_{1(2)}=0$ occurs for
$t_0-\varepsilon t_*\lapproxeq t\lapproxeq t_0+\varepsilon t_*$
and $\Delta(t)\sim 1/\varepsilon$ during the transition,
deviations from the exact solution of order $H_0^2\varepsilon^2$
arise in Eqs.~(\ref{eins00})--(\ref{eins05}).
These small quantities could be compensated
by small fluctuations of graviton, matter and vacuum energies
of order $O(\varepsilon^2)$
that are present only for $|t-t_0|/t_*\lapproxeq \varepsilon$.
We could treat these perturbatively around the exact background solution,
but will not pursue it here.

The introduction of the brane-localized Einstein-Hilbert term $\bar{R}_4$
does not affect the bulk solutions Eqs.~(\ref{beta}) and (\ref{a}),
but it does modify the boundary conditions.
For $t<t_0$ the solution should satisfy the boundary conditions
at $y=0$ and $y=y_c$,
\begin{eqnarray} \label{bdy1}
&&\frac{H_0}{c}-\frac{1}{2}\frac{M_4^2}{M_5^3}~\frac{H_0^2}{c^2}
=-\frac{\Lambda_{1}}{6M_5^3}~, \\
&&\frac{H_0}{c+H_0y_c}
=\frac{\Lambda_{2}}{6M_5^3}~
\label{bdy2}
\end{eqnarray}
Hence $H_0/c$ and $y_c$ are determined by $\Lambda_1$ and $\Lambda_2$.
Note that the brane cosmological constants $\Lambda_1$
and $\Lambda_2$ are related to the Hubble constant $H_0$.
While the non-zero brane cosmological constants are responsible for
inflating the 3-space, their subsequent vanishing
guarantees that the 4D spacetime is flat.
Since $\Lambda_2$ must be fine-tuned to zero
when $\Lambda_1=0$ \cite{selftun},
it is natural that the scalar field controlling the end of inflation is
introduced in the bulk.

From Eqs.~(\ref{bdy1})--(\ref{bdy2}) we note that the brane cosmological
constants (tensions) $\Lambda_1$ and $\Lambda_2$ should have opposite signs
in the absence of the brane-localized Einstein-Hibert term at $y=0$.
However, a suitably large value of $M_4/M_5$ \cite{braneR}
can flip the sign of $\Lambda_1$~\cite{KS}, so that both $\Lambda_1$ and
$\Lambda_2$ are positive.
In fact we could introduce another brane-localized Einstein-Hilbert term
at $y=y_c$. But its relatively small coefficient $M_4'\lapproxeq M_5$
does not change the result.
Thus, a brane-localized Einstein-Hibert term at $y=0$ seems essential
for successful F-term inflation in the 5D SUSY framework.
Its introduction does not conflict with any symmetry,
and in~\cite{KS} a simple model for realizing a large ratio $M_4/M_5$
was proposed.

The 4D reduced Planck mass ($\equiv (M_{\rm Planck}/8\pi)^{1/2}$)
is given by
%
%the coefficient of the effective 4D Einstein-Hilbert term
\begin{eqnarray}
M_{P}^2&=& M_5^3\int_{-y_c}^{y_c}dy\beta^2+M_4^2c^2 ~, \nonumber \\
&=&M_5^3y_c\bigg[\frac{2}{3}H_0^2\Theta^2(t)y_c^2+2cH_0\Theta(t)y_c+
2(c+\Delta^2(t))^2~\bigg]+M_4^2c^2 ~,
\end{eqnarray}
while the 4D effective cosmological constant is calculated to be
\begin{eqnarray}
\Lambda_{\rm eff}&=&\int_{-y_c}^{y_c}dy\beta^4\bigg[M_5^3\bigg(
4\bigg(\frac{\beta''}{\beta}\bigg)+6\bigg(\frac{\beta'}{\beta}\bigg)^2\bigg)
+\delta(y)\Lambda_1+\delta(y-y_c)\Lambda_2\bigg]  \nonumber \\
&=&3H_0^2\Theta^2(t) M_5^3y_c\bigg[\frac{2}{3}H_0^2\Theta^2(t)y_c^2
+2cH_0\Theta(t)y_c+2(c+\Delta^2(t))^2~\bigg] +M_4^2c^2 \nonumber \\
&=&3H_0^2\Theta^2(t)M_P^2 ~,
\end{eqnarray}
which vanishes when $\Lambda_1=\Lambda_2=0$.

%
%Next, let us discuss the late cosmology after inflation in this set up.
%Inflation was implemented only by brane-localized vacuum energies,
%because the F term scalar potentials are allowed only on the branes.
%On the other hand,
%
After inflation, the inflaton decays into brane
and (subsequently) bulk fields,
and they reheat the whole 5 dimensional universe.
To quantify the inflaton and radiation (or matter) dominated epochs,
we use the perfect fluid approximation,
\begin{eqnarray} \label{fluid}
T^M_M &=&\frac{1}{2y_cM_5^3}
~{\rm diag.}\bigg(-\rho_B(t),~p_B(t),~p_B(t),~p_B(t),~P_B(t)\bigg)
\nonumber \\
&&+~\delta(y)\frac{1}{M_5^3}~{\rm diag.}\bigg(
-\rho_b(t),~p_b(t),~p_b(t),~p_b(t),~0\bigg) ~,
\end{eqnarray}
where we normalize the bulk part with the circumference of
the extra dimension, so its components have the same mass dimension
as their brane counterparts.
For $t>t_0$ $\beta=c$, and $a(t)$ in Eqs.~(\ref{eins00})--(\ref{eins05})
satisfies the 4D Friedmann equation~\cite{kim-kyae}
\begin{eqnarray} \label{fried1}
&&\bigg(\frac{\dot{a}}{a}\bigg)^2=\frac{c^2}{3M_4^2}~\rho_b
=\frac{c^2}{6M_5^3y_c}~\rho_B ~, \\
&&~~\frac{\ddot{a}}{a}=\frac{-c^2}{6M_4^2}\bigg(\rho_b+3p_b\bigg)
=\frac{-c^2}{12M_5^3y_c}\bigg(\rho_B+3p_B\bigg) ~. \label{fried2}
\end{eqnarray}
Eqs.~(\ref{fried1}) and (\ref{fried2}) show that $\rho_B$ and $p_B$ are
suppressed compared to $\rho_b$ and $p_b$ by the factor $(2M_5^3y_c/M_4^2)$,
since $M_4/M_5$ is large.
Hence, if brane matter is not abundant enough to satisfy the relations,
sufficient bulk matter should accumulate near the brane.

The energy densities and pressures are related by the equations of state:
$p_b=\rho_b/3$, $p_B=\rho_B/3$ for the radiation dominated era, and
$p_b=p_B=0$ for the matter dominated era.
The pressure along the fifth direction $P_B$ is given by~\cite{kim-kyae},
\begin{eqnarray} \label{fried3}
P_B=\frac{-1}{2}\bigg(\rho_B-3p_B\bigg)\equiv \omega_5\times\rho_B ~.
\end{eqnarray}
Thus, $\omega_5$ is $0$ ($-1/2$) for the radiation (matter) epoch.
Eqs.~(\ref{fried1})--(\ref{fried3}) satisfy the 5 dimensional continuity
equation $\nabla_MT^M_N=0$.

We have tacitly assumed that the interval separating the two branes
(orbifold fixed points) remains fixed during inflation.
The dynamics of the orbifold fixed points,
unlike the D-brane case~\cite{Dbrane},
is governed only by the $g_{55}(x,y)$ component of the metric tensor.
The real fields $e_5^5$, $B_5$, and the chiral fermion $\psi^2_{5R}$
in the 5D gravity multiplet are assigned even parity under $Z_2$ \cite{kyae},
and they compose an $N=1$ chiral multiplet on the branes.
The associated superfield can acquire a superheavy mass and
its scalar component can develop a vacuum expectation value (VEV)
on the brane.
%
%for example via the term $ME(M-E)$ in the superpotential,
%in which $M$ ($>1/y_c$) is a dimensionful parameter.
With superheavy brane-localized mass terms,
their low-lying KK mass spectrum is shifted so that even the lightest mode
obtain the compactification scale mass~\cite{localmass}.
Its mass is much greater than $H_0$,
so that the interval distance is stabilized.

So far we have discussed only $S^1/Z_2$ orbifold compactification.
The results can be directly extended to $S^1/(Z_2\times Z_2')$ case.
Within the framework discussed in this section,
we can accomodate any promising 4D SUSY inflationary model.
We discuss one particular model below which comes from compactifying
$SO(10)$ on $S^1/(Z_2\times Z_2')$.

% when two sol. gives flat sol.?
%
%In 4 dimensional supersymmetric theories, inflation is driven
%when the inflaton is set at a non-minimum point of the scalar potential.
%When the inflaton rolls down to a minimum point of the scalar potential,
%which is a supersymmetric vacuum state, however,
%inflation ends and flat space-time is respored.
%
%$\Lambda_B<0$ and, non-noscale sugra case discussions
%

\section{${\bf SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_X}$
Model}

%
%In the previous section, we noted that in the inflationary model
%with $G_{PS}$ gauge symmetry, a nonrenormalizable inflaton superpotential
%is necessary to avoid monopole problem.
%However, even in that case, the parameters in the inflaton superpotentials
%should be carefully chosen so that the mechanism works \cite{khalil}.

We consider $N=2$ (in 4D sense) SUSY $SO(10)$ model in 5D spacetime,
where the 5th dimension is compactified on an $S^1/(Z_2\times Z_2')$.
$Z_2$ reflects $y\rightarrow -y$, and $Z_2'$ reflects
$y'\rightarrow -y'$, with $y'=y+y_c/2$.
There are two independent orbifold fixed points (branes)
at $y=0$ and $y=y_c/2$.
The $S^1/(Z_2\times Z_2')$ orbifold compactification is exploited to yield
$N=1$ SUSY as well as break $SO(10)$ to some suitable subgroup.

Under $SU(5)\times U(1)_X$, the $SO(10)$ generators are split into
\footnote{The $SO(2n)$ generators are represented as
$\left(\begin{array}{cc}
A+C&B+S\\
B-S&A-C
\end{array}\right)$, where $A$,$B$, $C$ are $n\times n$ antisymmetric matrices
and $B$ is an $n\times n$ symmetric matrix \cite{zee}.
By an unitary transformation,
the generators are given by
$\left(\begin{array}{cc}
A-iS&C+iB\\
C-iB&A+iS
\end{array}\right)$, where $A$ and $S$ denote $U(n)$ generators, and
$C\pm iB$ transform as $n(n-1)/2$ and $\overline{n(n-1)/2}$
under $SU(n)$.}
\begin{eqnarray} \label{so10}
T_{SO(10)}=\left[\begin{array}{c|c}
{\bf 24}_0+{\bf 1}_0& {\bf 10}_{-4}\\
\hline
{\bf \overline{10}}_{4}&{\bf \overline{24}}_0-{\bf 1}_0
\end{array}\right]_{10\times 10} ~,
\end{eqnarray}
where the subscripts labelling the $SU(5)$ representations indicate
$U(1)_X$ charges, and the subscript ``$10\times 10$'' denotes the matrix
dimension.  Also, ${\bf 24}$ ($={\bf \overline{24}}$) corresponds
to $SU(5)$ generators while
${\rm diag}~({\bf 1}_{5\times 5},-{\bf 1}_{5\times 5})$
is $U(1)_X$ generator.
The $5\times 5$ matrices ${\bf 24}_0$ and ${\bf 10}_{-4}$ are further
decomposed under $SU(3)_c\times SU(2)_L\times U(1)_Y$ as
\begin{eqnarray} \label{24}
&&{\bf 24}_{0}=\left(\begin{array}{cc}
{\bf (8,1)}_{0}+{\bf (1,1)}_0 & {\bf (3,\overline{2})}_{-5/6}\\
{\bf (\overline{3},2)}_{5/6} & {\bf (1,3)}_{0}-{\bf (1,1)}_0
\end{array}\right)_{0}~,  \\
&&{\bf 10}_{-4}=\left(\begin{array}{cc}
{\bf (\overline{3},1)}_{-2/3} & {\bf (3,2)}_{1/6}\\
{\bf (3,2)}_{1/6} & {\bf (1,1)}_{1}
\end{array}\right)_{-4} ~,
\end{eqnarray}
%
%where the subscripts labelling the $SU(3)_c\times SU(2)_L$ representations
%denote $U(1)_Y$ hypercharge, with the normalized generator
%$Y_{\rm GG}\equiv {\rm diag.}(-1/3,-1/3,-1/3,+1/2,+1/2)$.
%
Thus, each representaion carries two independent $U(1)$ charges.
Note that the two ${\bf (3,2)}_{1/6}$s in ${\bf 10}_{-4}$ are identified.
%
%The ${\bf 10}_{-4}$ and ${\bf \overline{10}}_4$ are antisymmetric matrices.
%${\bf \overline{24}}_0$ ($={\bf 24}_0$) and
%${\bf \overline{10}}_{4}$ are also decomposed in similar manner.
%

Consider the two independent $Z_2$ and $Z_2'$ group elements,
\begin{eqnarray} \label{p1}
P&\equiv&{\rm diag.}\bigg(I_{3\times 3},~I_{2\times 2},
-I_{3\times 3},-I_{2\times 2}\bigg)~, \\
P'&\equiv&{\rm diag.}\bigg(-I_{3\times 3},~I_{2\times 2},
~I_{3\times 3},-I_{2\times 2}\bigg)~,  \label{p2}
\end{eqnarray}
which satisfy $P^2=P'^2={\bf 1}_{5\times 5}$.
Under the operations $PT_{SO(10)}P^{-1}$ and $P'T_{SO(10)}P^{'-1}$,
the matrix elements of $T_{SO(10)}$ transform as
\begin{eqnarray} \label{so10/z2z2}
\left[\begin{array}{cc|cc}
{\bf (8,1)}_{0}^{++} &
~{\bf (3,\overline{2})}_{-5/6}^{+-} &
{\bf (\overline{3},1)}_{-2/3}^{--} & {\bf (3,2)}_{1/6}^{-+} \\
%\hline
{\bf (\overline{3},2)}_{5/6}^{+-} &
{\bf (1,3)}_{0}^{++} &
{\bf (3,2)}_{1/6}^{-+} & {\bf (1,1)}_{1}^{--} \\
\hline
~{\bf (3,1)}_{2/3}^{--} & ~{\bf (\overline{3},\overline{2})}_{-1/6}^{-+} &
{\bf (8,1)}_{0}^{++}
& {\bf (\overline{3},2)}_{5/6}^{+-} \\
%\hline
~{\bf (\overline{3},\overline{2})}_{-1/6}^{-+} & ~{\bf (1,1)}_{-1}^{--} &
~{\bf (3,\overline{2})}_{-5/6}^{+-} &
{\bf (1,3)}_{0}^{++}
\end{array}\right]_{10\times 10} ~,
\end{eqnarray}
where the superscripts of the matrix elements indicate the eigenvalues of the
$P$ and $P'$ operations.
Here we omitted the two $U(1)$ generators (${\bf (1,1)_0^{++}}$s)
to avoid too much clutter.
As shown in Eqs.~(\ref{so10}) and (\ref{24}), they appear
in the diagonal part of the matrix (\ref{so10/z2z2}).
The eigenvlaues of $P$ and $P'$ are the imposed parities
(or boundary conditions) of fields in the adjoint representations.
%
%under the reflectons $y\rightarrow -y$ and $y'\rightarrow -y'$
%
The wave function of a field with parity $(+-)$, for instance,
must vanish on the brane at $y=y_c/2$, and only fields assigned $(++)$
parities contain massless modes in their KK spectrum.
With $Z_2\times Z_2'$, the $SO(10)$ gauge group is broken to
$SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_X$.

An $N=2$ gauge multiplet conisists of an $N=1$ vector multiplet
($V^a=(A^{a}_\mu,\lambda^{1a})$) and an $N=1$ chiral
multiplet ($\Sigma^a=((\Phi^a+iA_5^a)/\sqrt{2},\lambda^{2a})$).
In order to break $N=2$ SUSY to $N=1$, opposite parities
must be assigned to the vector and the chiral multiplets
in the same representation.
The parities of $N=1$ vector multiplets coincide with the parities of
the corresponding $SO(10)$ generators in Eq.~(\ref{so10/z2z2}).
From the assigned eigenvalues in Eq.~(\ref{so10/z2z2}),
only the gauge multiplets associated with
${\bf (8,1)}_0^{++}$, ${\bf (1,3)}_0^{++}$, and two ${\bf (1,1)}_0^{++}$,
which correspond to the $SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_X$
generators, contain massless modes.
Therefore, at low energy the theory is effectively described by
a $4D$ $N=1$ supersymmetric theory
with $SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_X$.

As seen in Eq.~(\ref{so10/z2z2}), all of the gauge multiplets associated with
the diagonal components ${\bf 24}_0$, ${\bf 1}_0$
in Eq.~(\ref{so10}) survive at B1.
Thus, on B1 $SU(5)\times U(1)_X$ is preserved \cite{hebecker}.
On the other hand, in Eq.~(\ref{so10/z2z2}),
the elements with $(++)$ and $(-+)$ parities can compose a second (distinct)
set of $SU(5)'\times U(1)_X'$ gauge multiplets.
In the $SU(5)$ generator at B1,
${\bf 24}_0$ ($={\bf (8,1)}_0^{++}+{\bf (1,3)}_0^{++}+{\bf (1,1)}_0^{++}
+{\bf (3,\overline{2})}_{-5/6}^{+-}+{\bf (\overline{3},2)}_{5/6}^{+-}$), the
${\bf (3,\overline{2})}_{-5/6}^{+-}$ and ${\bf (\overline{3},2)}_{5/6}^{+-}$
are replaced by ${\bf (3,2)}_{1/6}^{-+}$ and
${\bf (\overline{3},\overline{2}})_{-1/6}^{-+}$ at B2
which belong in ${\bf 10}_{-4}$ and ${\bf \overline{10}}_{4}$
respectively of $SU(5)\times U(1)_X$,
\begin{eqnarray}
{\bf 24}_0'={\bf (8,1)}_0^{++}+{\bf (1,3)}_0^{++}+{\bf (1,1)}_0^{++}
+{\bf (3,2)}_{1/6}^{-+}+{\bf (\overline{3},\overline{2}})_{-1/6}^{-+}~~~
{\rm at~B2}~,
\end{eqnarray}
where the assigned hypercharges coincide
with those given from `flipped' $SU(5)'\times U(1)_X'$~\cite{flipped}.
The $U(1)_X'$ genertor at B2 is defined as
\begin{eqnarray}
{\rm diag}({\bf 1}_{3\times 3}, -{\bf 1}_{2\times 2},
-{\bf 1}_{3\times 3}, {\bf 1}_{2\times 2}) ~.
\end{eqnarray}
%
%instead of ${\rm diag}({\bf 1}_{5\times 5}, -{\bf 1}_{5\times 5})$.
Thus, the $U(1)_X'$ charges of the surviving elements at B2
turn out to be zero, while the other components are
assigned $-4$ or $4$.
The $U(1)_X'$ generator and the matrix elements with $(++)$, $(-+)$ parities
in Eq.~(\ref{so10/z2z2})
can be block-diagonalized to the form in Eq.~(\ref{so10})
through the unitary transformation
\begin{eqnarray}
U=\left(\begin{array}{c|ccc}
I_{3\times 3} & 0 & 0 & 0 \\ \hline
0 & 0 & 0 & I_{2\times 2} \\
0 & 0 & I_{3\times 3} & 0 \\
0 & I_{2\times 2} & 0 & 0
\end{array}\right)_{10\times 10} ~.
\end{eqnarray}
We conclude that the gauge multiplets surviving at B2
are associated with a second (flipped) $SU(5)'\times U(1)_X'$
embedded in $SO(10)$~\cite{flipped}.
%
%Indeed, if the $SU(5)$ in $SU(5)\times U(1)_X$
%respected at B1 is the Georgi-Glashow $SU(5)$,
%the $SU(5)'\times U(1)_X'$ preserved at B2 should be the `flipped' one
%$SU(5)'\times U(1)_X'$, in which the MSSM hypercharge
%is defined as $Y\equiv -(Y_{\rm GG}-Y_{X}')/5$ \cite{flipped}.
%Thus, in this paper we take $SU(5)\times U(1)_X$ at B1 and
%$SU(5)'\times U(1)_X'$ at B2.

With opposite parities assigned to the chiral multiplets,
two vector-like pairs
$\Sigma_{{\bf (\overline{3},1)}_{-2/3}^{++}}$,
$\Sigma_{{\bf (3,1)}_{2/3}^{++}}$ and $\Sigma_{{\bf (1,1)}_{1}^{++}}$,
$\Sigma_{{\bf (1,1)}_{-1}^{++}}$ contain massless modes.
Although the non-vanishing chiral multiplets at B1 are
$\Sigma_{{\bf 10}_{-4}}$
($=\Sigma_{{\bf (\overline{3},1)}_{-2/3}^{++}}
+\Sigma_{{\bf (3,2)}_{1/6}^{+-}}
+\Sigma_{{\bf (1,1)}_{1}^{++}}$) and
$\Sigma_{{\bf \overline{10}}_{4}}$
($=\Sigma_{{\bf (3,1)}_{2/3}^{++}}
+\Sigma_{{\bf (\overline{3},\overline{2})}_{-1/6}^{+-}}
+\Sigma_{{\bf (1,1)}_{-1}^{++}}$),
$\Sigma_{{\bf (3,2)}_{1/6}^{+-}}$ and
$\Sigma_{{\bf (\overline{3},\overline{2})}_{-1/6}^{+-}}$ are replaced by
$\Sigma_{{\bf (3,\overline{2})}_{-5/6}^{-+}}$ and
$\Sigma_{{\bf (\overline{3},2)}_{5/6}^{-+}}$
at B2 that are contained in $\Sigma_{{\bf 24}_0}$ and
$\Sigma_{{\bf \overline{24}}_0}$ at B1.
Together with the vector-like pairs contaning massless modes,
they compose ${\bf 10}_{-4}'$ and ${\bf \overline{10}_4}'$-plets
of $SU(5)'\times U(1)_X'$,
\begin{eqnarray}
&&\Sigma_{{\bf 10}_{-4}'}
=\Sigma_{{\bf (\overline{3},1)}_{-2/3}^{++}}+
\Sigma_{{\bf (3,\overline{2})}_{-5/6}^{-+}}
+\Sigma_{{\bf (1,1)}_{-1}^{++}}~,~~{\rm  and}  \\
&&\Sigma_{{\bf \overline{10}}_{4}'}
=\Sigma_{{\bf (3,1)}_{2/3}^{++}}+\Sigma_{{\bf (\overline{3},2)}_{5/6}^{-+}}
+\Sigma_{{\bf (1,1)}_{1}^{++}} ~~~~~~~~~{\rm at ~ B2} ~.
\end{eqnarray}

Now let us discuss the $N=2$ (bulk) hypermultiplet $H$ $(=(\phi,\psi))$,
$H^c$ ($=(\phi^c,\psi^c)$) in the vector representations ${\bf 10}$,
${\bf 10^c}$ ($={\bf 10}$) of $SO(10)$, where
$H$ and $H^c$ are $N=1$ chiral multiplets.
Under $SU(5)\times U(1)_X$ and $SU(3)_c\times SU(2)_L\times U(1)_Y$,
${\bf 10}$ and ${\bf 10^c}$ are
\begin{eqnarray}
{\bf 10}=
\left(\begin{array}{c}
{\bf 5}_{-2} \\
\hline
{\bf \overline{5}}_{2}
\end{array}\right)=
\left(\begin{array}{c}
~{\bf (3,1)}_{-1/3}^{+-}  \\
{\bf (1,2)}_{1/2}^{++}  \\
\hline
{\bf (\overline{3},1)}_{1/3}^{-+}  \\
~{\bf (1,\overline{2})}_{-1/2}^{--}
\end{array}\right)~, ~~
{\bf 10^c}=
\left(\begin{array}{c}
{\bf 5^c}_{2} \\
\hline
{\bf \overline{5^c}}_{-2}
\end{array}\right)=
\left(\begin{array}{c}
{\bf (3^c,1)}_{1/3}^{-+}  \\
~{\bf (1,2^c)}_{-1/2}^{--}  \\
\hline
~{\bf (\overline{3}^c,1)}_{-1/3}^{+-}  \\
{\bf (1,\overline{2}^c)}_{1/2}^{++}
\end{array}\right) ~,
\end{eqnarray}
where the subscripts $\pm 2$ are $U(1)_X$ charges and the remaining subscripts
indicate the hypercharge $Y$.
The superscripts on the matrix elements denote the eigenvalues of
the $P$ and $P'$ operations.
As in the $N=2$ vector multiplet,
opposite parities must be assigned for $H$ and $H^c$
to break $N=2$ SUSY to $N=1$.   The massless modes are contained
in the two doublets ${\bf (1,2)}_{1/2}^{++}$ and
${\bf (1,\overline{2^c})}_{1/2}^{++}$ ($={\bf (1,2)}_{1/2}^{++}$).
While the surviving representations at B1,
${\bf (3,1)}_{-1/3}^{+-}$ and ${\bf (1,2)}_{1/2}^{++}$
(also ${\bf (\overline{3}^c,1)}_{-1/3}^{+-}$ and
${\bf (1,\overline{2}^c)}_{1/2}^{++}$) compose two ${\bf 5}_{-2}$
(or ${\bf \overline{5}^c}_{-2}$) of $SU(5)\times U(1)_X$,
at B2  the non-vanishing representations are
two ${\bf \overline{5}}_{2}'$ of $SU(5)'\times U(1)_X'$,
\begin{eqnarray}
{\bf \overline{5}}_{2}' ={\bf (\overline{3},1)}_{1/3}^{-+}
+{\bf (1,2)}_{1/2}^{++} ~~\bigg({\rm or}~~{\bf (3^c,1)}_{1/3}^{-+}
+{\bf (1,\overline{2}^c)}_{1/2}^{++}\bigg) ~~{\rm at~B2}~.
\end{eqnarray}

In this model, the $SU(2)$ $R$-symmetry which generally exists in $N=2$
supersymmetric theories is explicitly broken to $U(1)_R$.
Since $N=1$ SUSY is present on both branes,
$U(1)_R$ symmetry should be respected.
We note that different $U(1)_R$ charges can be assigned
to $H_{{\bf 10}_{-4}}$ and $H^c_{{\bf 10^c}_{4}}$
as shown in Table I \cite{nomura}.
%
%The results of Table I are consistent with our choice of
%the $U(1)_R$ charges $1/2$ ($-1/2$) and $-1/2$ ($1/2$) for the SUSY parameters
%$\theta^1$ ($d\theta^1$) and $\theta^2$ ($d\theta^2$), respectively.
%
\vskip 0.6cm
\begin{center}
\begin{tabular}{|c||ccc|c|} \hline
$U(1)_R$&  & $V$,~ $\Sigma$ & & $H$, $H^c$
\\
\hline \hline
$1$ & & & & $\phi^c$
\\
$1/2$ & & $\lambda^1$& & $\psi^c$
\\
$0$ & ~~$A_\mu$& & $\Phi$, $A_5$~~ & $\phi$
\\
$-1/2$ & & $\lambda^2$& & $\psi$
\\
\hline
\end{tabular}
\vskip 0.4cm
{\bf Table I.~}$U(1)_R$ charges of the vector and hypermultiplets.
\end{center}

To construct a realistic model, which includes inflation, based on
$SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_X$,
we introduce a $U(1)_{PQ}$ axion symmetry and
`matter' parity $Z_2^m$~\cite{khalil}.
For simplicity, let us assume that the MSSM matter superfields as well as
the right-handed neutrinos are brane fields residing at B1.\footnote{If
the first two quark and lepton families reside on B2 where
$SU(5)'\times U(1)_X'$ is preserved, undesirable mass relations between
the down-type quarks and the charged leptons do not arise.
Mixings between the first two and the third families
can be generated by introducing bulk superheavy hypermultiplets
in the spinor representations of $SO(10)$~\cite{KS2}.  }
They belong in ${\bf 10}_i$, ${\bf \overline{5}}_i$,
and ${\bf 1}_i$ of $SU(5)$, where $i$ is the family idex.
Their assigned $U(1)_X$, $U(1)_R$ and $U(1)_{PQ}$ charges and matter parities
appear in Table II.
\vskip 0.6cm
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|} \hline
Fields & $S$& $~N_H~$& $~\overline{N}_H~$ &$~{\bf 10}_B^{(')}~$&
$~{\bf \overline{10}}_B^{(')}~$ & ${\bf 1}_i$  &
$~~{\bf \overline{5}}_i~~$ & $~{\bf 10}_i~$
\\ \hline \hline
$X^{(')}$ & $0$& $5$& $-5$& $-4$& $4$& $5$& $-3$& $1$
\\
$R$ & $1$ & $0$& $0$& $0$ & $0$ & $1/2$ & $1/2$ & $1/2$
\\
$PQ$ & $0$& $0$& $0$& $0$ & $0$ & $0$ & $-1$ & $-1/2$
\\
$Z_2^m$ & $+$& $+$& $+$& $-$& $-$& $-$& $-$& $-$
\\
\hline \hline
Fields & ~$h_1^{(')}$~ & $h_2^{(')}$ & $~\overline{h}_1^{(')}~$ &
$~\overline{h}_2^{(')}~$ & $\Sigma_1$ & $\Sigma_2$ & $\overline{\Sigma}_1$ &
$\overline{\Sigma}_2$
\\
\hline \hline
$X^{(')}$& $-2$& $-2$& $2$ & $2$ & $0$ & $0$& $0$ & $0$
\\
$R$ & $0$& $1$& $0$ & $1$& $1/2$ & $1/2$ & $0$ & $0$
\\
$PQ$  & $1$& $1$& $3/2$ & $3/2$& $-1$ & $-3/2$ & $1$ & $3/2$
\\
$Z_2^m$& $+$& $-$& $+$& $-$& $+$& $+$& $+$& $+$
\\
\hline
\end{tabular}
\vskip 0.4cm
{\bf Table II.~}$U(1)_X^{(')}$, $U(1)_R$, $U(1)_{PQ}$ charges
and matter partities of the superfields.
\end{center}

We introduce two pairs of hypermultiplets
$(H_{\bf 10},H^c_{\bf 10^c})$ and
$(H_{\bf \overline{10}},H^c_{\bf \overline{10}^c})$
($=(H_{\bf 10},H^c_{\bf 10^c})$) in the bulk.
The two $SU(5)$ Higgs multiplets $h_1$ and $\overline{h}_1$ (${\bf 5}$ and
${\bf \overline{5}}$) arise from $H_{\bf 10}$ and $H_{\bf \overline{10}}$,
and their $U(1)_R$ charges are chosen to be zero.
As discussed above,
the $N=2$ superpartners $H_{\bf 10}^c$ and $H_{\bf \overline{10}}^c$
also provide superfields $h_2$ and $\overline{h}_2$
with ${\bf 5}^{(')}$ and ${\bf \overline{5}}^{(')}$ representations at B1 (B2).
However, their $U(1)_R$ charges are unity unlike $h_1$ and $\overline{h}_1$.
To make them superheavy
%
%mass terms for $h_2$ and $\overline{h}_2$,
we can introduce another pair of ${\bf 5}$ and ${\bf \overline{5}}$
with zero $U(1)_R$ charges and `$-$' matter parities on the brane.
The multiplets $\Sigma_{{\bf 10}_{-4}}$ ($\equiv {\bf 10}_{B}$) and
$\Sigma_{{\bf \overline{10}}_{4}}$ ($\equiv {\bf \overline{10}}_B$) also
contain massless modes.  We can also make them superheavy by pairing them
with two (vector-like) chiral multiplets ${\bf 10}_b$ and
${\bf \overline{10}}_b$ carrying $U(1)_R$ charges of unity on the brane.
The superpotential at B1, neglecting the superheavy particles' contributions
except for the inflatons, is given by
\begin{eqnarray} \label{3211}
W&=&\kappa S\bigg(N_H\overline{N}_H
%+\lambda_1{\bf 10}_B{\bf \overline{10}}_B
%
%+\frac{\rho_1}{M_P}N_Hh_1\overline{h}_b
%+\frac{\rho_2}{M_P}\overline{N}_Hh_b\overline{h}_1
-M^2\bigg)
%\nonumber \\
%&&+\eta_1N_Hh_2\overline{h}_b+\eta_2\overline{N}_Hh_b\overline{h}_2
+\frac{\sigma_1}{M_P}\Sigma_1\Sigma_2h_1\overline{h}_1
+\frac{\sigma_2}{M_P}\Sigma_1\Sigma_2\overline{\Sigma}_1\overline{\Sigma}_2
 \\
&&+y^{(d)}_{ij}{\bf 10}_i{\bf 10}_jh_1+
y^{(ul)}_{ij}{\bf 10}_i{\bf \overline{5}}_j\overline{h}_1
+y^{(n)}_{ij}{\bf 1}_i{\bf \overline{5}}_jh_1
+\frac{y^{(m)}_{ij}}{M_P}{\bf 1}_i{\bf 1}_j\overline{N}_H\overline{N}_H ~,
 \nonumber
\end{eqnarray}
where $S$, $N_H$, $\overline{N}_H$, $\Sigma_{1,2}$ and
$\overline{\Sigma}_{1,2}$ are singlet fields.
Their assigned quantum numbers appear in Table II.
While $S$ and $h_1$, $\overline{h}_1$ are bulk fields, the rest are
brane fields residing on B1.
From Eq.~(\ref{3211}),
it is straightforward to show that the
SUSY vacuum corresponds to $\langle S\rangle=0$, while
$N_H$ and $\overline{N}_H$ develop VEVs of order $M$.
(After SUSY breaking {\it $\grave{a}$ la} $N=1$ supergravity
(SUGRA), $\langle S\rangle$ acquires a VEV
of order $m_{3/2}$ (gravitino mass)).
This breaks $SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_X$
to the MSSM gauge group.
From the last term in Eq.~(\ref{3211}), the VEV of $\overline{N}_H$
also provides masses to the right-handed Majorana neutrinos.
%
%Then the `$\tau$' neutrino mass turns out to be in eV range
%via the seesaw mechanism.
%

Because of the presence of the soft terms,
$\Sigma_{1,2}$ and $\overline{\Sigma}_{1,2}$, which carry $U(1)_{PQ}$ charges,
can obtain intermediate scale VEVs of order $\sqrt{m_{3/2}M_P}$.
This leads to a $\mu$ term of order $m_{3/2}$ in MSSM
as desired \cite{mu, mu2}.
Of course, the presence of $U(1)_{PQ}$ also resolves
the strong CP problem~\cite{axion}.
%
%As a result of the $U(1)_{PQ}$ symmetry breaking at the intermediate scale,
%there exists a very light axion solving the strong CP problem
%in the model \cite{axion}.
%

The Higgs fields $h_1$ and $\overline{h}_1$ contain color triplets
as well as weak doublets.
Since the triplets in $h_1$ and $\overline{h}_1$ are just superheavy KK modes,
a small coefficient ($\mu\sim$ TeV) accompanying $h_1\overline{h}_1$
more than adequately suppresses dimension 5 operators that induce proton decay.
Proton decay can still proceed via superheavy gauge bosons with masses
$\approx \pi/y_c$ and are adequately suppressed ($\tau_p\sim 10^{34-36}$ yrs).

%Soft SUSY breaking effect and instanton effect
%break $U(1)_R$  explicitly to $Z_2$.
%Then odd parities under the subgroup $Z_2$ are assigned to the quark and
%lepton sector fields and $\Sigma_{1,2}$, $\overline{\Sigma}_{1,2}$.
%Since $\Sigma$ fields get VEV, this discrete
%symmetry is spontaneously broken and domain walls are created.
%Therefore, in this model, we must assume that
%the $U(1)_{PQ}$ breaking takes place before or during inflation
%so that the induced domain walls are washed out.
%At low energy, remaining symmetry is SM gauge group$\times Z_2^{mp}$.

\section{Inflation and Leptogenesis}

The first two terms in the superpotential (\ref{3211}) are ideally suited
for realizing an inflationary scenario along the lines
described in \cite{KS,hybrid,khalil}.
We will not provide any details except to note that the breaking of $U(1)_X$
takes place near the end of inflation which can lead to the appearance of
cosmic strings. Since the symmetry breaking scale of $U(1)_X$ is
determined from inflation to be close to $10^{16}$ GeV \cite{hybrid},
the cosmic strings are superheavy and therefore not so desirable
(because of potential conflict with the recent $\delta T/T$ measurements.)
They can be simply avoided by following the strategy outlined
in \cite{khalil}, in which suitable non-renormalizable terms are added
to Eq.~(\ref{3211}),
such that $U(1)_X$ is broken along the inflationary trajectory and the strings
are inflated away.  Remarkably, the salient features of the inflationary
scenario are not affected by the addition of such higher order terms.
However, there is some impact on the prediction regarding the scalar spectral
index $n$ of density fluctuations.
Depending on the value of the superpotential coupling parameter
$\kappa$ ($\sim 10^{-3}-{\rm few}\times 10^{-4}$),
$n$ can now vary over a somewhat wider range, with the central value
being very close to 0.98.   For $\kappa$ somewhat smaller (larger) than
$10^{-3}$, $n$ varies between 0.96 and 0.99.

The inflationary epoch ends with the decay of the oscillating fields
$S$, $N_H$, $\overline{N}_H$ into the MSSM singlet (right-handed) neutrinos,
whose subsequent out of equilibrium decays yield the observed
baryon asymmetry via leptogenesis~\cite{lepto,ls}.
The production of the right-handed neutrinos and sneutrinos proceeds via
the superpotential couplings $\nu^c_i\nu^c_j\overline{N}_H\overline{N}_H$
and $SN_H\overline{N}_H$ on B1, where $\nu^c_{i,j}$ ($i, j=1,2,3$)
denote the $SU(5)$ singlet (right-handed neutrinos)
carrying non-zero $U(1)_X$ charges.
Taking account of the atmospheric and solar neutrino oscillation
data~\cite{nuoscil},
and assuming a hierarchical pattern of neutrino masses
(both left- and right-handed ones), it turns out that the inflaton decays
into the lightest (the first family right handed neutrino)~\cite{KS,pati}.
The discussion proceeds along the lines discussed in~\cite{lasymm},
where it was shown that a baryon asymmetry of the
desired magnitude is readily obtained.

The 5D inflationary solution requires positive vacuum energies on both branes
B1 and B2~\cite{KS}.  While inflation could be driven by the first two term
in Eq.~(\ref{3211}) at B1,
an appropriate scalar potential on B2 is also necessary.
Since the boundary conditions in Eq.~(\ref{bdy1}) and (\ref{bdy2})
require $\Lambda_1$ and $\Lambda_2$ to simultaneously vanish,
it is natural for $S$ to be a bulk field.
The VEVs of $S$ on the two branes can be adjusted
such that the boundary conditions are satisfied.
As an example, we can introduce the following superpotential at B2,
\begin{eqnarray} \label{b1superpot}
W_{B2}=\kappa_2S(Z\overline{Z}-M_2^2) ~,
\end{eqnarray}
where $Z$ and $\overline{Z}$ are $SO(10)$ singlet superfields
with opposite $U(1)_R$ charges.
Thus, only the gauge symmetry $U(1)_X$ on B1 is broken during inflation.

\section{Conclusion}

Inspired by recent attempts to construct realistic 5D SUSY GUT models,
we have presented a realistic inflationary scenario in this setting,
along the lines proposed in~\cite{KS}.
Inflation is implemented through F-term scalar potentials on the two branes,
which is allowed by 5D Einstein gravity.
We have discussed the transition from the inflationary
to the radiation dominated phase, and
provided a realistic 5D SUSY $SO(10)$ model in which compactification on an
$S^1/(Z_2\times Z_2')$ orbifold leads to the gauge symmetry
$SU(3)_c\times SU(2)_L\times U(1)_Y\times U(1)_X$.
Inflation is associated with the breaking of the gauge symmetry $U(1)_X$,
at a scale very close to $M_{\rm GUT}$.
Baryogenesis via leptogenesis is very natural in this approach,
and the scalar spectral index $n=0.96-0.99$.
The gravitational contribution to the quadrupole
anisotropy is found to be very tiny (less than or of order $1\%$).

\vskip 0.3cm
\noindent {\bf Acknowledgments}

\noindent
We acknowledge helpful discussions with S. M. Barr and Tianjun Li.
The work is partially supported
by DOE under contract number DE-FG02-91ER40626.

\begin{thebibliography}{99}

\def\apj#1#2#3{Astrophys.\ J.\ {\bf #1}, #2 (#3)}
\def\ijmp#1#2#3{Int.\ J.\ Mod.\ Phys.\ {\bf #1}, #2 (#3)}
\def\mpl#1#2#3{Mod.\ Phys.\ Lett.\ {\bf #1}, #2 (#3)}
\def\nat#1#2#3{Nature\ {\bf #1}, #2 (#3)}
\def\npb#1#2#3{Nucl.\ Phys.\ {\bf B#1}, #2 (#3)}
\def\plb#1#2#3{Phys.\ Lett.\ {\bf B#1}, #2 (#3)}
\def\prd#1#2#3{Phys.\ Rev.\ {\bf D#1}, #2 (#3)}
\def\prl#1#2#3{Phys.\ Rev.\ Lett.\ {\bf #1}, #2 (#3)}
\def\prt#1#2#3{Phys.\ Rep.\ {\bf #1}, #2 (#3)}
\def\sjnp#1#2#3{Sov.\ J.\ Nucl.\ Phys.\ {\bf #1}, #2 (#3)}
\def\zp#1#2#3{Z.\ Phys.\ {\bf #1}, #2 (#3)}


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