#  Constructing warm inflationary model in brane-antibrane system

M. R. Setare<EMAIL_ADDRESS>A.Sepehri<EMAIL_ADDRESS>V.Kamali 3<EMAIL_ADDRESS>1 Department of Science, Campus of Bijar,
University of Kurdistan, Bijar, Iran.
2 Faculty of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman, Iran.
3 Department of Physics, Faculty of Science, Bu-Ali Sina University, Hamedan,
65178, Iran.

###### Abstract

Recently, various observational data predict a possibility that inflation may
naturally occur in a warm region. In this scenario, radiation is produced
during the inflation epoch and reheating is avoided. The main question arises
that what is the origin of warm inflation in 4D universe? We answer to this
question in brane-antibrane system. We propose a model that allows all
cosmological parameters like the scale factor a, the Hubble parameter H and
phantom energy density depend on the equation of state parameter in transverse
dimension between two branes. Thus, an enhancement in these parameters can be
a signature of some evolutions in extra dimension. In our model, the expansion
of 4D universe is controlled by the separation distance between branes and
evolves from non-phantom phase to phantom one. Consequently, phantom-dominated
era of the universe accelerates and ends up in big-rip singularity. Also, we
show that as the tachyon potential increases, the effect of interaction
between branes on the 4D universe expansion becomes systematically more
effective, because at higher energies there exists more channels for flowing
energy from extra dimension to other four dimensions. Finally, we test our
model against WMAP and Planck data and obtain the ripping time. According to
experimental data, $N\simeq 50$ case leads to $n_{s}\simeq 0.96$, where _N_
and $n_{s}$ are the number e-folds and the spectral index respectively. This
standard case may be found in $0.01<R_{Tensor-scalar}<0.22$, where $R_{Tensor-
scalar}$ is the tensor-scalar ratio. At this point, the finite time that Big
Rip singularity occurs is $t_{rip}=33(Gyr)$.

##  I Introduction

One main problem of the inflation theory is how to attach the universe to the
end of the inflation epoch. An interesting solution of this problem is the
study of inflation in the context of warm inflation scenario m1 . In this
model, radiation is produced during the inflation epoch and reheating is
avoided. Until now, the warm inflationary universe has been discussed in many
papers m2 ; m3 ; m4 ; m5 ; m6 . Specially, the effect of interaction between
branes on the warm inflation has been considered in some scenarios m7 ; m8 ;
m9 . For example, some authors showed that for D3-$\bar{D3}$ branes with
additional flavor D7-branes, inflation may naturally occur in a warm regime.
They explained that by using a Coulomb-like or quadratic hybrid potential, a
sufficient number of e-folds may be calculated for perturbative couplings and
$O(10-10^{4})$ D3-branes. This was successful realizations of warm inflation
and lead to a sufficiently long period of accelerated expansion even for the
steep potentials dominating brane-antibrane interactions in flat space.
Furthermore, their model predicted interesting observational signatures such
as negligible tensor-to-scalar ratio m7 . Some other authors discussed that
the unstable tachyonic field which arises from the motion of D3 brane in the
background of k coincident D5 branes can be the source of inflation in warm
inflationary scenario. The whole process was considered in presence of a
radiation bath. In this model, the dissipative effects had important role and
hence the whole dynamics was analyzed in terms of the dissipative parameter.
Also, the slow-roll parameters and the cosmological observables were evaluated
by considering the fact that the dissipative parameter is a function of the
tachyonic field. This field was given in terms of the radius of the ring
produced by one D3 and k number of D5 branes m8 . In another research, the
study of warm inflation model as a mechanism that gives an end for vector
inflation theory is motivated some authors to consider the warm vector
inflation model m10 . In their scenario, the general conditions for inflation
era and end of this epoch were obtained.

Recently, some investigators m9  examined the warm inflationary model with
observational data and showed that the prediction of this model is consistent
with Planck results m11 . They explained that when the temperature of the
radiation bath is bigger or equal to Hubble parameter ($T\geq H$) and non-
trivial inflaton occupation numbers are sustained during inflation, the
amplitude of scalar curvature fluctuations can be significantly enhanced,
whereas tensor perturbations are generically unaffected due to their weak
coupling to matter fields. This generically reduces the tensor-to-scalar ratio
with respect to conventional models, changes the tilt of the scalar power
spectrum and yields a modified consistency relation for warm inflation. They
discussed that as an example, the quartic chaotic potential is in very good
agreement with the Planck results for nearly-thermal inflaton fluctuations,
whereas it seems to be ruled out in a cold scenario. They also consider other
simple models that are in agreement with the Planck data m11  within a
renormalizable model of warm inflation.

Now, the question arises that what is the origin of warm inflationary model in
4D universe? We answer to this question in brane-antibrane system. In this
system, it might be reasonable to ignore the tachyon in the ultraviolet where
the flavor branes and antibranes are well separated and the tachyon is small,
it is likely to condense and acquire large values in the infrared where the
branes meet. In this condition, inflation of universe is controlled by tachyon
potential between branes and evolves from non phantom phase to phantom one.
The tachyonic phantom energy density grows without bound and becomes infinite
in finite time, dominates all other forms of energy, such that the
gravitational repulsion and finally brings our brief epoch of universe to an
end. When the universe approaches this future singularity; the Milky Way,
solar system, Earth, and ultimately the molecules, atoms, nuclei, and nucleons
of which we are composed will be ripped by the infinite phantom energy in a
Big Rip m13 . This singularity is characterized by divergences in the scale
factor a, the Hubble parameter H and its time-derivative $\dot{H}$ at the
finite future $t=t_{rip}$.

The outline of the paper is as the following. In section II, we construct warm
inflationary model of universe in brane-antibrane system and show that all
cosmological parameters depend on the separation distance between two branes.
Also, in this section, we consider Big Rip singularity and calculate the
relation between ripping time and separation between branes in extra
dimension. In section III, we test our model against the observational data
from Planck and WMAP collaborations m11 ; m12 ; m27  and obtain the ripping
time . The last section is devoted to summary and conclusion.

##  II  The warm inflationary model of universe in D3/$\bar{D3}$ system

Previously, the dynamical behavior of warm brane-antibrane inflation in the
era that the Big Rip singularity will not occur was considered m7 ; m8 . In
this section we will enter the effects of this singularity on the results of
the derivation of scale factors, Hubble parameter and other important
parameters in warm model model of cosmology. We will show that these
parameters depend on the ripping time and are influenced by separation
distance of branes in extra dimension. In our model, equation of state
parameter in 4D universe may change due to flowing energy from extra dimension
and decrease from higher values of -1 (non-phantom phase) to lower values
(phantom one).

The D3/$\bar{D3}$ model is formulated by placing $\bar{D3}$-branes into the
D3-brane background with the following metric m14 ; m15 ; m16 ; m17 ; m18 :

$\displaystyle
ds^{2}=(1+\frac{R^{4}}{y^{4}})^{-\frac{1}{2}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+(1+\frac{R^{4}}{y^{4}})^{\frac{1}{2}}(dy^{2}+y^{2}d\Omega_{5}^{2}).$
(1)

Here

$\displaystyle R_{4}=4\pi g_{s}N\acute{\alpha}^{2},$ (2)

where $\lambda=g_{s}N=g_{YM}^{2}$ is the ’t Hooft coupling, N the number of D3
branes and $\acute{\alpha}$ the inverse string tension ($\acute{\alpha}$ =
$l_{s}^{2}$, $l_{s}$ string length). Also, $\eta_{\mu\nu}$ is the standard 3+1
dimensional Minkowski metric, the $x_{\mu}$ are the coordinates on the stack
of D3 branes and the $y$ denotes the spatial coordinate perpendicular to the
brane.

To construct four dimensional universe, we consider a set of
D3-$\overline{D3}$-brane pairs in the above background which are placed at
points $y_{1}=l/2$ and $y_{2}=-l/2$ respectively so that the separation
between the brane and antibrane is l. For the simple case of a single
D3-$\overline{D3}$-brane pair with open string tachyon, the action ism19 :

$\displaystyle S=-\tau_{3}\int
d^{9}\sigma\sum_{i=1}^{2}V(T,l)e^{-\phi}(\sqrt{-detA_{i}})$
$\displaystyle(A_{i})_{ab}=(g_{MN}-\frac{T^{2}l^{2}}{Q}g_{My}g_{yN})\partial_{a}x^{M}_{i}\partial_{b}x^{M}_{i}+F^{i}_{ab}+\frac{1}{2Q}((D_{a}T)(D_{b}T)^{\ast}+(D_{a}T)^{\ast}(D_{b}T))$
$\displaystyle+il(g_{ay}+\partial_{a}y_{i}g_{yy})(T(D_{b}T)^{\ast}-T^{\ast}(D_{b}T))+il(T(D_{a}T)^{\ast}-T^{\ast}(D_{a}T))(g_{by}+\partial_{b}y_{i}g_{yy}).$
(3)

where

$\displaystyle Q=1+T^{2}l^{2}g_{yy},$ $\displaystyle
D_{a}T=\partial_{a}T-i(A_{2,a}-A_{1,a})T,V(T,l)=g_{s}V(T)\sqrt{Q},$
$\displaystyle e^{\phi}=g_{s}(1+\frac{R^{4}}{y^{4}})^{-\frac{1}{2}},$ (4)

,$\phi$, $A_{2,a}$ and $F^{i}_{ab}$ are dilaton field, the gauge fields and
field strengths on the world-volume of the non-BPS brane, T is the tachyon
field, $\tau_{3}$ is the brane tension and V (T) is the tachyon potential. The
indices a,b denote the tangent directions of D-branes, while the indices M,N
run over the background ten-dimensional space-time directions. The Dp-brane
and the anti-Dp-brane are labeled by i = 1 and 2 respectively. Then the
separation between these D-branes is defined by $y_{2}-y_{1}=l$. Also, in
writing the above we are using the convention $2\pi\acute{\alpha}=1$.

Let us consider the only t dependence of the tachyon field T for simplicity
and set the gauge fields to zero. In this case, the static gauge the action
(3) in the region that $r>R$ simplifies to

$\displaystyle S\simeq-\frac{\tau_{3}}{g_{s}}\int
d^{4}xV(T)(\sqrt{D_{1,T}}+\sqrt{D_{2,T}})$ (5)

where $D_{1,T}=D_{2,T}\equiv D_{T}$, $V_{3}=\frac{4\pi^{2}}{3}$ is the volume
of a unit $S^{3}$ and

$\displaystyle
D_{T}=1+\frac{l^{\prime}(t)^{2}}{4}+T^{\prime}(t)^{2}+T(t)^{2}l(t)^{2}$ (6)

where the prime denotes a derivative w.r.t. its argument t.

The equations of motion obtained from this action are:

$\displaystyle(\frac{1}{\sqrt{D_{T}}}T^{\prime}(t)\acute{)}=\frac{1}{\sqrt{D_{T}}}[T(t)l(t)^{2}+\frac{V^{\prime}(T)}{V(T)}(D_{T}-T^{\prime}(t)^{2})]$
(7)
$\displaystyle(\frac{1}{\sqrt{D_{T}}}l^{\prime}(t)\acute{)}=\frac{1}{\sqrt{D_{T}}}[T(t)^{2}l(t)-\frac{V^{\prime}(T)}{V(T)}l^{\prime}(t)T^{\prime}(t))]$
(8)

Note that the prime on V (T) denotes a derivative w.r.t. its argument T and
not a derivative w.r.t. time. To solve these equations, we should determine
explicit form of V (T). There are several proposals for V (T) which satisfy
these requirements m20 , although no rigorous derivation exists. In view of
this, in the following analysis we will avoid using any specific expression
for V (T), except when needed for explicit numerical calculations. It will,
however, be necessary for us to specify the asymptotic form of the potential
for large T. A potential which satisfies above equations and has been used in
most of papers is m21 ; m22 ; m23 :

$\displaystyle V(T)=\frac{\tau_{3}}{cosh\sqrt{\pi}T}$ (9)

Employing this potential, we can solve equations (7 and 8). Approximate
solutions of these equations for old universe are:

$\displaystyle
T(t)\sim\frac{T_{+}}{(t-t_{rip})}(1+\frac{1}{l_{0}(t-t_{rip})})e^{-l_{0}(t-t_{rip})}+\frac{T_{-}}{(t-t_{rip})}(1-\frac{1}{l_{0}(t-t_{rip})})e^{l_{0}(t-t_{rip})}$
$\displaystyle l(t)\sim l_{1}(t-t_{rip})+.....$ (10)

where $l_{0}$ is the maximum separation between the brane and antibrane and
$T_{+}$ and $l_{1}$ are positive constants which will be determined in future
calculations. These equations show that when two branes are very distance from
each other (t=0), tachyon field is very small, whereas moving the branes
towards each other, the value of this field increases and becomes infinity at
$t=t_{rip}$.

The effective potential for the tachyon can be obtained from the action (5) by
setting $T^{\prime}=h^{\prime}=0$. It is:

$\displaystyle V_{eff}(T)\sim
sech\sqrt{\pi}T\sqrt{1+\frac{l^{\prime}(t)^{2}}{4}+T^{\prime}(t)^{2}+T(t)^{2}l(t)^{2}}\rightarrow$
$\displaystyle V_{eff}(T)\sim
sech\sqrt{\pi}(\frac{T_{+}}{(t-t_{rip})})\sqrt{1+\frac{T_{+}}{(t-t_{rip})^{4}}}$
(11)

This potential indicates that when the branes-antibranes become close to each
other, the interaction potential increases and at higher energies there exists
more channels for string tachyon production in this system; consequently, the
effect of string tachyon on the universe inflation becomes systematically more
effective.

Let us now discuss the warm inflationary model of universe in D3/$\bar{D3}$
system. For this, we need to compute the contribution of the flavour brane-
antibrane system to the four- dimensional universe energy momentum tensor. The
energy momentum tensor is obtained from action by calculating its functional
derivative w.r.t. the background ten-dimensional metric $g_{MN}$. The precise
relation is $T^{MN}=\frac{2}{\sqrt{-detg}}\frac{\delta S}{\delta g_{MN}}$. We
get, m19 ,

$\displaystyle T^{ab}_{i}=-V(T)\sqrt{D_{T}}g^{ab},a,b=0,1,2,3$ $\displaystyle
T^{yy}_{i}=-V(T)\frac{1}{\sqrt{D_{T}}}(T^{2}l^{2}+\frac{\acute{l}^{2}}{4})$
(12)

This equation shows that with decreasing the separation between D3 and
$\overline{D3}$ branes, tachyon energy-momentum tensors increase. This is
because that when branes and antibranes are well separated, tachyon is small,
however when two branes are close to each other, it achieves to large values.

The conservation law of energy-momentum relates the tensors calculated in
brane-antibrane system with ones associated with the four dimensional universe
with the following equation:

$\displaystyle T^{\mu\nu}=\frac{2}{\sqrt{-detg}}\frac{\delta S}{\delta
g_{MN}}\frac{\delta g_{MN}}{\delta g_{\mu\nu}}=T_{MN}\frac{\delta
g_{MN}}{\delta g_{\mu\nu}},$ (13)

where $T^{\mu\nu}$ is the energy-momentum tensor of 4D universe in ten
dimensional space-time with the metric of the form:

$ds^{2}=ds^{2}_{Uni,1}+ds^{2}_{Uni,2}+b^{2}dw^{2}+\omega^{2}d\Omega_{2},$ (14)

Here

$\displaystyle
ds^{2}_{Uni1}=ds^{2}_{Uni2}=-dt^{2}+a(t)^{2}(dx^{2}+dy^{2}+dz^{2}),$ (15)

where $w$ is the extra space-like coordinates perpendicular to two universes
and scale factors $a(t)$ and $b(t)$ are assumed to be functions of time only.
In this model, we introduce two four dimensional universe that interact with
each other and form a binary system. To obtain the energy- momentum tensor in
this system, we use of the Einstein’s field equation in presence of fluid flow
that reads as:

${R_{ij}}-\frac{1}{2}{{\mathop{\rm g}\nolimits}_{ij}}R=k{T_{ij}}.$ (16)

Setting the solution of this equation with the line element of Eq. (14) in the
conservation law of energy-momentum tensor in Eq. (13) and employing Eq. (12)
yield:

$\displaystyle
kT_{1}^{1}=\frac{{5\ddot{a}}}{a}+\frac{{{3{\dot{a}}^{2}}}}{{{a^{2}}}}+\frac{{5\dot{a}\dot{b}}}{{ab}}+\frac{{\ddot{b}}}{b}=-V(T)\sqrt{D_{T}},$
$\displaystyle
kT_{2}^{2}=\frac{{5\ddot{a}}}{a}+\frac{{{3{\dot{a}}^{2}}}}{{{a^{2}}}}+\frac{{5\dot{a}\dot{b}}}{{ab}}+\frac{{\ddot{b}}}{b}=-V(T)\sqrt{D_{T}},$
$\displaystyle
kT_{3}^{3}=\frac{{5\ddot{a}}}{a}+\frac{{{3{\dot{a}}^{2}}}}{{{a^{2}}}}+\frac{{5\dot{a}\dot{b}}}{{ab}}+\frac{{\ddot{b}}}{b}=-V(T)\sqrt{D_{T}},$
$\displaystyle
kT_{4}^{4}=\frac{{6\ddot{a}}}{a}+\frac{{6{{\dot{a}}^{2}}}}{{{a^{2}}}}=-V(T)\frac{1}{\sqrt{D_{T}}}(T^{2}l^{2}+\frac{\acute{l}^{2}}{4}),$
$\displaystyle
kT_{10}^{10}=\frac{{6{{\dot{a}}^{2}}}}{{{a^{2}}}}+\frac{{6\dot{a}\dot{b}}}{{ab}}=V(T)\sqrt{D_{T}}$
(17)

The higher-dimensional stress-energy tensor will be assumed to be that of a
perfect fluid and of the form:

$T_{i}^{j}={\mathop{\rm
diag}\nolimits}\left[{-p,-p,-p,-\bar{p},-p,-p,-p,\rho}\right],$ (18)

where $\bar{p}$ is the pressure in the extra space-like dimension. In above
equation, we are allowing the pressure in the extra dimension to be different,
in general, from the pressure in the 3D space. Hence, this stress-energy
tensor describes a homogeneous, anisotropic perfect fluid in ten dimensions.
By adopting the metric ansatzs in (1) and (14), the conservation law in (13)
and (17) and the perfect fluid stress-energy tensor in (18), the field
equations are of the form:

$-p=\frac{{5\ddot{a}}}{a}+\frac{{{3{\dot{a}}^{2}}}}{{{a^{2}}}}+\frac{{5\dot{a}\dot{b}}}{{ab}}+\frac{{\ddot{b}}}{b}=-V(T)\sqrt{D_{T}},$
(19)
$-\bar{p}=\frac{{6\ddot{a}}}{a}+\frac{{6{{\dot{a}}^{2}}}}{{{a^{2}}}}=-V(T)\frac{1}{\sqrt{D_{T}}}(T^{2}l^{2}+\frac{\acute{l}^{2}}{4}),$
(20)
$\rho=\frac{{6{{\dot{a}}^{2}}}}{{{a^{2}}}}+\frac{{6\dot{a}\dot{b}}}{{ab}}=V(T)\sqrt{D_{T}},$
(21)

where we set the higher-dimensional coupling constant equal to one, k = 1. The
equations (19),(20) and (21) help us to explain all properties of the current
universe in terms of evolutions in brane-antibrane system. These equations
constraint the pressure and density in our universe to tachyon fields and
express that any increase or decrease in these parameters is due to tachyon
potential in extra dimension.

By taking derivative from the left side of Eq. (21), we get the following
equation:

$\frac{2\dot{a}}{a}(\frac{5\ddot{a}}{a}-\frac{3\dot{a}^{2}}{a^{2}}-\frac{\dot{a}\dot{b}}{ab}+\frac{\ddot{b}}{b})+\frac{\dot{b}}{b}(\frac{2\ddot{a}}{a}-\frac{2\dot{a}\dot{b}}{ab})=\frac{\dot{\rho}}{3}.$
(22)

Substituting (19) and (20) in (22), we obtain:

$\frac{2\dot{a}}{a}(-\rho-p)+\frac{\dot{b}}{b}(-\frac{\rho}{3}-\frac{\bar{p}}{3})=\frac{\dot{\rho}}{3}.$
(23)

or

$\dot{\rho}+6\frac{\dot{a}}{a}(\rho+p)+\frac{\dot{b}}{b}(\rho+\bar{p})=0.$
(24)

By defining $H=\frac{\dot{a}}{a}$ and $\bar{H}=\frac{\dot{b}}{b}$ as Hubble
parameters in 4D universe and extra dimension, we can rewrite (21) and (24) as
following:

$6H^{2}+6H\bar{H}=\rho.$ (25) $\dot{\rho}+6H(\rho+p)+\bar{H}(\rho+\bar{p})=0.$
(26)

Eq.(26) is in fact conservation equation in ten dimensional space-time. This
equation is analogous to standard 4D FRW cosmology with $\bar{H}=0$ and
$ds^{2}_{Uni2}=0$. We are now at a stage that can enter the properties of warm
inflation into calculations. The dynamic of warm inflation in spatially flat
ten dimensional space-time is presented by this equation:

$\displaystyle\dot{\rho}+6H(\rho+p)+\bar{H}(\rho+\bar{p})=-\Gamma\dot{T}^{2}.$
(27)

Also, with similar calculations we can obtain:

$\displaystyle\dot{\rho}_{\gamma}+(8H+\bar{H})\rho_{\lambda}=\Gamma\dot{T}^{2}.$
(28)

where $\rho_{\lambda}$ is energy density of the radiation and $\Gamma$ is the
dissipative coefficient. In the above equations dot h. h means derivative with
respect to cosmic time. Warm inflation is an example of a phase transition
with dissipative effect m1 . In supercool inflation model m24 , this effect
becomes important after the end of inflation (in reheating epoch,) but in warm
inflation model the interactions of inflaton with other fields are important
during the inflationary period.

Relating the 3D and higher-dimensional pressures to the density (p =
$\omega\rho$, $\bar{p}$ = $\bar{\omega}\rho$) and employing Eq.(25) in Eq.(27)
lead to following equation:

$\displaystyle 6H\dot{H}+3\dot{H}\bar{H}+3H\dot{\bar{H}}$
$\displaystyle+9(H^{3}+H^{2}\bar{H})(1+\omega)$
$\displaystyle+9(H^{2}\bar{H}+H\bar{H}^{2})(1+\bar{\omega})=-\Gamma\dot{T}^{2},$
(29)

where $\omega$ and $\bar{\omega}$ are equation of state parameters in 4D
universes and extra dimension respectively. Notice that these parameters can
in general be time-dependent. Also, the relation ($\bar{p}$ =
$\bar{\omega}\rho$) and equations (20) and (21) yield:

$\bar{H}=\frac{\dot{H}+H^{2}\bar{\omega}}{\bar{\omega}H}.$ (30)

Substituting Eq.(30) in Eq.(29) gives us the following relation:

$\displaystyle(\bar{\omega}^{2}-2\bar{\omega}-3\bar{\omega}\omega+4)H^{2}\dot{H}$
$\displaystyle+(\bar{\omega}^{2}-6\bar{\omega}\omega+4)H^{4}$
$\displaystyle+(1+\bar{\omega})\dot{H}^{2}-\bar{\omega}H\ddot{H}=-\Gamma\dot{T}^{2}.$
(31)

Solving equations (19, 20, 21, 30, 31) simultaneously and performing some
algebra for old universe, we find a consistent solution for the Hubble
parameters and scale factors in 4D universes and extra dimension and also
tachyon, separation between branes and dissipation coefficient in terms of
ripping time:

$\displaystyle H=\frac{A}{t-t_{rip}}$ $\displaystyle a(t)=(t-t_{rip})^{A}$
$\displaystyle
A=\frac{(\bar{\omega}^{2}-2\bar{\omega}-3\bar{\omega}\omega+4)\pm\sqrt{(\bar{\omega}^{2}-2\bar{\omega}-3\bar{\omega}\omega+4)^{2}-4(\bar{\omega}^{2}-6\bar{\omega}\omega+4)(1-\bar{\omega})}}{2(\bar{\omega}^{2}-6\bar{\omega}\omega+4)},$
(32) $\displaystyle\bar{H}=\frac{B}{t-t_{rip}}$ $\displaystyle
b(t)=(t-t_{rip})^{B}$ $\displaystyle B=\frac{-1+A\bar{\omega}}{\bar{\omega}},$
(33) $\displaystyle
T\sim\frac{A^{2}+AB}{(t-t_{rip})}(1+\frac{1}{l_{0}(t-t_{rip})})e^{-l_{0}(t-t_{rip})}+\frac{A^{2}-AB}{(t-t_{rip})}(1-\frac{1}{l_{0}(t-t_{rip})})e^{l_{0}(t-t_{rip})}$
$\displaystyle l\sim(A^{2}+AB)^{\frac{9}{4}}(t-t_{rip}),$ (34)
$\displaystyle\Gamma\sim\frac{A(t-t_{rip})^{4}}{(A^{2}+AB)}(1+\frac{1}{l_{0}(t-t_{rip})})^{-2}e^{2l_{0}(t-t_{rip})}.$
(35)

Equation (32) shows that if A is smaller than zero, 3D scale factor and Hubble
parameter at a certain time (t = $t_{rip}$) are infinity. So far, researchers
have believed that if 3D equation of state parameter is lower than negative
one, Dark energy is phantom energy and at Big Rip singularity, ingredients of
which universe is made will be ripped; whereas our calculations indicate that
occurrence of this singularity is related to both $\bar{\omega}$ and $\omega$.

As can be seen from (33), the value of the scale factor is determined by two
competing terms (A and $\bar{\omega}$). When $A<0$ and $\bar{\omega}>0$, both
4D universe and extra dimension encounter with a type of singularity. Also, if
$\bar{\omega},A<0$ and $\bar{\omega}>\frac{1}{A}$ or
$A>\frac{1}{\bar{\omega}}$, we are faced with similar situation; while if
$\bar{\omega}<\frac{1}{A}$ or $A<\frac{1}{\bar{\omega}}$, Big Rip singularity
won’t happen in extra dimension. In this regard, another particular condition
($\bar{\omega},A>0$ and $\bar{\omega}<\frac{1}{A}$ or
$A<\frac{1}{\bar{\omega}}$)may arise where this singularity occurs only in the
extra dimension.

The equation (34) has some interesting results which can be used to explain
the reasons for occurance of Big Rip singularity in present era of universe.
According to these calculations, when two branes are located at a large
distance from each other ($l=l_{0}$ and t=0), tachyon field is almost zero,
while approaching the two together, the value of this field increases and
tends to infinity at $t=t_{rip}$. In this situation, brane-antibrane system
disappears and consequently one singularity happens in our four dimensional
universe. Another interesting point that comes out from this equation is that
time of this singularity is proportional to the initial distance between two
branes.

The equation (35) indicates that the dissipative coefficient decreases with
time and shrink to zero at ripping time. Also, dissipative coefficient is
controlled by both of equation of state parameters in four dimensional
universes and extra dimension. This means that the warm inflation can be
affected by phenomenological events in extra dimension and other universe.

Inserting the solutions in (32, 33, 35) back into Eq.(28), we obtain the
energy density of radiation in terms of ripping time:

$\rho_{\lambda}\sim\frac{1}{(t-t_{rip})^{8A+B}}$ (36)

As is obvious from Eq.(36), the energy density of radiation, increases with
time and tends to infinity at Big Rip singularity. This is because that with
moving two branes towards each other, tachyon inflaton radiation grows and
acquires the large values near the colliding point. Another interesting result
that can be deduce from this equation is that the radiating energy density
depends on the equation of state parameters and any enhancement or decrease in
this density can be a signature of some interactions between two universes in
extra dimension.

##  III Considering the signature of interaction between branes in
observational data

In previous section, we propose a model that allows to consider the warm
inflation of universe in brane-antibrane system. Here, we examine the
correctness of our model with observation data and obtain some important
parameters like ripping time. Until now, eight possible asymptotic solutions
for cosmological dynamics have been proposed m25 . Three of these solutions
have non-inflationary scale factor and another three ones of solutions give de
Sitter, intermediate and power-low inflationary expansions. Finally, two cases
of these solutions have asymptotic expansion with scale factor
($a=a_{0}exp(A[lnt]^{\lambda})$. This version of inflation is named
logamediate inflation m26 . Our model lets us to study Warm-tachyon
inflationary model in the scenarios of logamediate inflation with
${\lambda}=1$. In this condition, the model is converted to power-law
inflation. Using (32), the number of e-folds may be found:

$N=\int_{t_{0}}^{t}Hdt=|A|[ln\frac{t_{rip}-t_{0}}{t_{rip}-t}]$ (37)

where $t_{0}$ denotes the beginning time of inflation epoch.

In Fig.1, we present the number of e-folds _N_ for warm scenario as a function
of the t where t is the age of universe. In this plot, we choose A=-100,
$t_{0}=0$ and $t_{rip}=33(Gyr)$. We find that _N_ =50 leads to
$t_{universe}=13.5(Gyr)$. This result is compatible with with both Planck and
WMAP9 data m11 ; m12 ; m27 . It is clear that the number of e-folds _N_ is
much larger for older universe. This is because, as the age of universe
increases, the distance between branes becomes smaller and the tachyon field
acquires bigger values.

Figure 1: The number of e-folds _N_ in warm inflation scenario as a function
of $t$ for A=-100, $t_{0}=0$ and $t_{rip}=33(Gyr)$.

Another parameters that help us to test our model with experimental data are
the power-spectrum of scalar and tensor perturbations which presented by m10 ;
m28 :

$\displaystyle\Delta_{R}^{2}=-(\frac{\Gamma^{3}T^{2}}{36(4\pi)^{3}})^{\frac{1}{2}}\frac{H^{\frac{3}{2}}}{\dot{H}}$
$\displaystyle\Delta_{T}^{2}=\frac{2H^{2}}{\pi^{2}},$ (38)

where $T=[-\frac{r\dot{H}}{2C(1+\frac{r}{3}})]^{\frac{1}{4}}$,
$r=\frac{\Gamma}{3H}$ and $C=\frac{\pi^{2}g^{\ast}}{30}$ ($g^{\ast}$ is the
number of relativistic degree of freedom.). Using these parameters, we can
define the tensor-scalar ratio as m10 ; m28 :

$\displaystyle
R=-\frac{\Delta_{T}^{2}}{\Delta_{R}^{2}}=-(\frac{144(4\pi)^{3}}{\Gamma^{3}\pi^{4}T^{2}})^{\frac{1}{2}}\dot{H}H^{\frac{1}{2}},$
(39)

In Fig.2, we show the tensor-scalar ratio _R_ for warm scenario as a function
of the age of universe. In this plot, we choose A=-100, $t_{0}=0$, c=.01 and
$t_{rip}=33(Gyr)$. We find that _R_ =0.0147 leads to $t_{universe}=13.5(Gyr)$.
This result is compatible with with both Planck and WMAP9 data m11 ; m12 ; m27
. Obviously, the tensor-scalar ratio _R_ is much larger for older universe.
The reason for this is that when two old brane universes approach together,
the value of tachyon inflaton increases and tends to infinity at singularity.

Figure 2: The tensor-scalar ratio _R_ in warm inflation scenario as a function
of $t$ for A=-100, $t_{0}=0$, C=.01 and $t_{rip}=33(Gyr)$.

Finally, we compare our model with the Scalar spectral index which is defined
bym10 ; m28 :

$n_{s}-1=-\frac{dln\Delta_{R}^{2}}{dlnk}=\frac{3}{2}\varepsilon-(\frac{r^{\prime}}{4r})\eta$
(40)

Here k is co-moving wavenumber and $\varepsilon$ and $\eta$ are slow-roll
parameters of the warm inflation which are given by:

$\displaystyle\varepsilon=-\frac{1}{H}\frac{dlnH}{dt}$
$\displaystyle\eta=-\frac{\ddot{H}}{H\dot{H}},$ (41)

In Fig.3, we show the Scalar spectral index $\emph{n}_{s}$ for warm scenario
as a function of the age of universe. In this plot, we choose A=-100 and
$t_{0}=0$. Comparing this figure with figures(2,3), we find that $N\simeq 50$
case leads to $n_{s}\simeq 0.96$. This standard case is found in
$0.01<R_{Tensor-scalar}<0.22$, which is consistent with both Planck and WMAP9
data m11 ; m12 . At this point, the finite time that Big Rip singularity
occurs is $t_{rip}=33(Gyr)$.

Figure 3: The Scalar spectral index $\emph{n}_{s}$ in warm inflation scenario
as a function of $t$ for A=-100, $t_{0}=0$ and $t_{rip}=33(Gyr)$.

##  IV Summary and Discussion

In this research, we construct warm inflation in brane-antibrane system and
show that the energy density, slow-roll, Number of e-fold and perturbation
parameters can be given in terms of the time that the two branes approached
together in extra dimensions. According to our results, when the distance
between branes increases, the Number of e-fold, the spectral index and the
tensor-scalar ratio increases and tends to infinity at Big Rip singularity .
This is because, as the separation distance between branes decreases, the
interaction potential increases and at higher energies there exists more
channels for inflaton production in brane-antibrane system; consequently, the
effect of inflaton radiation from extra dimension on the universe inflation
becomes systematically more effective. We find that $N\simeq 50$ case leads to
$n_{s}\simeq 0.96$. This standard case may be found in $0.01<R_{Tensor-
scalar}<0.22$, which agrees with observational data m11 ; m12 ; m27 . At this
point, the finite time that Big Rip singularity occurs is $t_{rip}=33(Gyr)$.

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