\section{Introduction}
The current accelerated expansion of the universe is usually explained by invoking a Dark Energy (DE)  component\footnote{For alternative approaches which try to avoid dark energy by invoking large scale inhomogeneities see, for instance, \cite{void1,void2,reviews}.}  which today comprises more than $70\%$ of the total energy in the universe (for reviews see \cite{DEreview,Caldwell:2009ix,Silvestri:2009hh}).  The case of a pure cosmological constant, with $w_{\Lambda}\equiv p_{\Lambda}/\rho_{\Lambda}=-1$ marks the divide to the `phantom' realm.  Phantom dark energy models are described  by systems with
\begin{equation}
w_{p} = \frac{p_p}{\rho_p} <-1
\end{equation}
 and  have the intriguing feature that the energy density in the universe increases with expansion,
 \begin{equation}
 \rho_p\sim a^{-3(1+w_p)} .
\end{equation}
Hence a universe with  low $\sim mev$ scale accelerated expansion can eventually reach energy scales close to the GUT scale, for instance.  For some examples of cosmological scenarios using phantom energy see \cite{phantomcosmology,bfk}. The question that we want to ask is whether it is possible to exploit this feature of phantom cosmology and turn the dark energy driven acceleration into a GUT scale inflationary phase\footnote{For a brief list of papers that propose various mechanism of connecting the current accelerated expansion to inflation see \cite{acc_infl}}. The idea then would be to construct a cyclic model where dark-energy/inflationary phases are interspersed with decelerating radiation/matter phases.

Several problems immediately appear. Firstly, unless the equation of state for the phantom phase, $w_p$, is extremely close to $-1$, the phantom acceleration will be much faster than the deSitter expansion, and cannot be reconciled with data. Density perturbations produced
during a phantom phase will give rise to a blue spectrum, and consistency with the current WMAP 5-yr data at the $2 \sigma$ level with tensor modes included~\cite{WMAP5}    requires  $-1>w_{p}>- 1.01$. Secondly, it is well known that phantom cosmology typically ends in a Big-rip singularity, rather than the standard radiation phase which follows inflation. Remarkably, we find that both these problems can be addressed when we include interactions between the ``phantom fluid'' and some hidden sector  matter. Such interactions  ameliorate the phantom acceleration phase to an asymptotic de Sitter type expansion, once the phantom energy density reaches a critical value. It is easy to arrange this transition to occur around the GUT scale, which is appropriate for inflationary cosmology. This also automatically avoids the big rip singularity as the space time now approaches a deSitter universe. The density perturbations
can have a variety of possibilities, allowing for agreement with observations  \cite{WMAP5}.  Moreover,
in our scenario the universe transitions to an asymptotic deSitter
phase independent of the value of $\omega_p$, and thus avoids having to  fine-tune $w_p$ very close to $-1$. As an additional advantage over the usual slow-roll inflationary scenario, in our phantom-driven inflationary model one does not have to tune the flatness of the potential usually necessary to obtain the large number  of efoldings and near scale-invariant spectrum. In addition, the hierarchy
between  the $\sim meV$ dark energy scale and the GUT inflation scale can re-expressed in terms of four parameters that take values of $\mathcal{O}(1)$ to $\mathcal{O}(10^2)$. We cannot however address the ``coincidence'' problem in our picture.
Finally,  there is the question of how to construct a theoretically self-consistent model of phantom energy. We will comment on this problem
shortly.

 Before delving into the details of our specific realization of the ``phantom cyclic model'', let us
 outline the  basic picture  by considering just a simple  two fluid model, phantom matter ($\rho_p$) + radiation ($\rho_r$). The cosmology we want to realize is the following:
although the scale factor always increases monotonically with time, the energy density ``cycles'', at least approximately. Each cycle is divided into two different phases: (a) Radiation dominated phase, which starts at an energy density $\rho_r=\lambda_{\mt{max}}^4$. As the universe expands radiation gets diluted, the Hubble parameter decreases and reaches a minimum when $\rho_r=\rho_p=\lambda_{\mt{min}}^4$. From here on we enter (b) the  phantom energy dominated phase. In realistic cosmology the radiation phase should give way to matter domination at energy densities $\sim (10\ eV)^4$, before giving way to phantom domination, but for simplicity we are going to ignore this slight complication. Thus for a typical scenario which would be consistent with dark energy and inflationary paradigm, $\lambda_{\mt{max}}\sim 10^{15}\ GeV\sim 10^{-3}M_p$ or the GUT scale, and $\lambda_{\mt{min}}\sim meV\sim 10^{-30}M_p$ corresponding to the scale of current energy density.  Now, in the phantom phase, as the universe expands the energy density increases, and so does the Hubble rate. Initially, depending upon how negative the phantom equation of state parameter, $\omega_p$, is this increase in energy density can be quite fast. However, in our model we will see that once the energy density reaches close to a critical scale $\lambda_{\mt{max}}$, which is determined by the interactions between the hidden and ordinary matter sector, the energy density and the Hubble parameter asymptote to a constant giving rise to a near exponential expansion. This inflationary phase can end via the reheating mechanism described in section \ref{sec:reheating},~\ref{sec:transition} after which we enter the radiation dominated era of the  next cycle.

A similar idea to the one in this paper has previously been presented by \cite{etexpcyclic}
who dubbed this model "the eternally expanding cyclic universe."  They too (in their Section 4.3)
suggested an alternating increasing/decreasing energy density. However the theory behind
their model is quite different from ours, and consequently their predictions for resultant
density perturbations are different as well \footnote{They always
predict a negative, even if extremely small, tilt of the spectral
index while our model will allow a variety of possibilities (as discussed in later sections of this paper).}.
One major problem of all phantom type models is the vacuum stability due to the null energy condition violation.  \cite{etexpcyclic} examine a consistent way to solve this problem, based on a deformation of the ghost condensate model of Arkani-Hamed et all \cite{ghostcondensate}.

Although the interactions between phantom fluid and ordinary matter can lead to an inflationary
spacetime, we are still left with a graceful exit problem, or how to ensure that the universe enters the
standard radiation dominated era. Depending upon the specific model different ``reheating mechanisms'' may be
able to trigger such a transition. We  focus on a model where the phantom fluid consists of a ghost like
scalar field coupled to some hidden matter sector. Such a fluid closely resembles the interacting DE-DM
models~\cite{Amendo,coupled,adinstab,trodden} except that the scalar field instead of being an ordinary
quintessence field, has negative kinetic energy like a ghost. Although field theory with ghosts is plagued
with problems of unitarity/instability \cite{ghostinstab},  recent developments attempting
to address these problems
include progress in  non-local~\cite{non-local} and Lee-Wick~ \cite{lee-wick}  higher derivative  models;
see also \cite{etexpcyclic} and \cite{ghost}.
As we will see, in our model  the transition from phantom to  radiation phase and vice-versa is achieved partly by suitably choosing the interaction strength between the scalar field and the hidden matter sector, and partly due to the presence of interactions between the hidden and ordinary matter sector. (There is no direct coupling between the ghost field and  ordinary matter.)

As emphasized before, the main reason why the cosmology described above can replace the standard inflationary paradigm is because in the phantom phase the energy density increases even though the universe continues to expand. Thus after the usual dilution in a radiation dominated phase, the phantom phase followed by reheating ensures that the universe again  becomes hot and therefore can reproduce the successes of the Big Bang Model, such as Big Bang Nucleosynthesis and Cosmic Microwave Background Radiation. There is another essential similarity between our scenario and the inflationary paradigm.  The essential reason why inflation solves  the standard cosmological puzzles is because our observable universe (of radius $\sim H_0^{-1}$) can originate from a very tiny region at the beginning of inflation. Something very similar happens in our model as well, the universe expands by a huge factor in every cycle. In our model the number of e-foldings in the radiation and the phantom phase is given by
\begin{equation}
{\cal N}_{\mathtt{rad}}\sim \ln\left( \frac{\lambda_{\mt{max}}}{\lambda_{\mt{min}}}\right)\mbox{ and }{\cal N}_{\mathtt{phan}}\sim \frac{-4}{3(1+\omega_p)}\ln\left( \frac{\lambda_{\mt{max}}}{\lambda_{\mt{min}}}\right).
\end{equation}
 In order to have a successful GUT scale inflationary paradigm we need ${\cal N}_{\mathtt{inf}}\gtrsim60$. Thus one gets:
\begin{eqnarray}
{\cal N}_{\mathtt{tot}}&=&{\cal N}_{\mathtt{phan}}+{\cal N}_{\mathtt{rad}}+{\cal N}_{\mathtt{inf}}\\\nonumber
&\sim&\LF1 -\frac{4}{3(1+\omega_p)}\right)\ln\left( \frac{\lambda_{\mt{max}}}{\lambda_{\mt{min}}}\right)+{\cal N}_{\mathtt{inf}}
\end{eqnarray}
Just to get an idea, if we take $\lambda_{\mt{min}}\sim mev$, $\lambda_{\mt{max}}\sim 10^{15}Gev$, $\omega_p\sim -1.3$, and ${\cal N}_{\mathtt{inf}}\sim 60$, we get  ${\cal N}_{\mathtt{tot}}\sim 400$.  What this means is that only a very very tiny portion (for the chosen example, $e^{-400}$ th) of our observable universe will ultimately grow to become the observable universe in the next cycle at the same energy density scale. Another essential similarity between the standard inflationary scenario and our phantom based model, is the production of a huge amount of entropy in every cycle. Even in a ``cyclic'' scenario if one wants to address the usual cosmological puzzles, such as flatness, homogeneity, etc, as well as produce cosmological perturbations with the correct amplitude required for galaxy formation, entropy production seems to be an inevitable requirement~\cite{entropy}. Like in inflation, in our scenario a huge amount of entropy is produced during reheating when most of the phantom energy is converted into radiation. This is essential to ensuring the cyclicity of energy density even though the scale factor of the
universe is monotonically increasing~\footnote{It may be possible to embed the scenario in phantom cyclic models~\cite{phantom-cyclic,bfk} with actual phases of contraction, but we are not going to explore this possibility here.}.

 It is worth mentioning that in most cyclic models proposed  the scale factor $a(t)$ has a contracting-expanding behaviour, where both a bounce, near the Big Bang, and a turnaround, when the scale factor becomes large are needed. A brief and incomplete compilation of some of the papers that have proposed cyclic cosmologies can be found in \cite{cyclicmodels,bfk}. Such cosmologies provide  natural solutions to the flatness and horizon problems of standard Big Bang scenario. Some variants  also  avoid the issue of initial conditions, provided entropy produced during one cycle is not transferred to the next. In this case  the cycles will not grow, i.e. become larger (Tolman's argument\cite{tolman}) from one to the next, so we can no longer define a beginning of the universe.  For instance the authors of \cite{cyclic} developed  an ekpyrotic inspired cyclical model as an alternative to inflation, where a phase of slow contraction before the bang is responsible for generating a nearly scale invariant spectrum of perturbations that seed the large scale structure formation. In contrast, the standard inflationary scenario assumes a short  phase that occurs after the big bang when the universe is rapidly expanding and nearly scale invariant perturbations are generated.

The paper is organized as follows. In section \ref{sec:phantom} we present our model of phantom fluid and discuss the cosmology relevant for dark energy. In section \ref{sec:reheating}, we discuss how including interactions can lead to an inflationary space-time along with partially reheating the universe. In section \ref{sec:transition} we provide a specific example where transitions from the phantom-inflation phase to radiation and vice-versa can be orchestrated giving us a cyclic model of the universe. In section \ref{sec:numbers} we discuss the different observational constraints coming from inflation, Big Bang Nucleosynthesis (BBN), and dark energy experiments. Finally, we conclude with a summary of the scenario presented and issues that needs to be addressed further.

\section{Phantom Dark Energy}\label{sec:phantom}
The purpose of this section is to implement a model for the phantom component that would drive the super-accelarated expansion. For now we will not include regular matter nor radiation, but rather focus solely on the components necessary to obtain a phantom phase. One can possibly implement the cosmology sketched before in many different ways. Here we are going  to realize the above picture using a ghost-like scalar field (with negative kinetic energy) $\phi$ coupled to a hidden matter sector which we denote by index ``$h$''. There are two equivalent approaches to describe this type of coupling. One starting from an action, where we allow for a direct coupling term between the hidden sector and the scalar field. The second approach is to consider two fluids that can exchange energy while maintaining the conservation of the total stress-energy tensor, as required by diffeomorphism invariance, although the individual stress-energy tensors are not conserved.

Our system will be described by the following action:
\begin{equation}\label{actionPDE}
S=\,\int\,\text{d}^4x\sqrt{-g}\left[M_p^2 \frac{R}{2}+\frac{(\nabla\phi)^2}{2}+C(\phi)\mathcal{L}_{h}\right]\ ,
\end{equation}
where $\mathcal{L}_h$ does not depend on $\phi$, being the action describing a perfect barotropic fluid. Here we work with a spatially flat FRW metric with signature $(-,+++)$.  Notice that the kinetic term for the scalar field comes with the `wrong' sign, as appropriate for a ghost.
In the phantom dominated phase, described here,  the Hubble equation derived from the action above looks like
\begin{equation}
H^2= \frac{1}{3M_p^2}(\rho_{\phi}+\rho_{h})=\frac{1}{3M_p^2}\left[-\frac{\dot{\phi}^2}{2}+C(\phi)\tilde{\rho}_{h}\right]\ ,
\label{hubble}
\end{equation}
where we have assumed the field $\phi$ to be homogenous. Here a dot represents the derivative with respect to cosmic time, $t$ and  $\tilde{\rho}_h$ denotes the bare energy density of the hidden sector, which is $\phi$ independent. We will assume that it behaves like a perfect barotropic fluid, satisfying the continuity equation:
\begin{equation}\label{contbare}
\dot{\tilde{\rho}}_h+3H(\tilde{\rho}_h+\tilde{p}_h)=0
\end{equation}
with an equation of state
\begin{equation}\label{defomega}
\tilde{p}_h=\omega\tilde{\rho}_h
\end{equation}
One can also include a potential for the scalar field, and its effects are discussed briefly in the appendix, but for the purpose of illustration we are going to set it to zero.

The interaction that we are going to consider between the hidden matter sector and $\phi$ is going to be very similar to the interactions considered in  coupled quintessence (or interacting DE-DM) models~\cite{coupled, adinstab,trodden}. From the action in equation (\ref{actionPDE}) we get two additional equations of motion.
\begin{eqnarray}
\ddot{\phi}+3H\dot{\phi}&=&2\rho_h\mu(\phi)/M_p\label{KG}\\
\dot{\rho}_h+3H(1+\omega)\rho_h&=&2\rho_h\dot{\phi}\mu(\phi)/M_p\, ,
\label{cont-h}
\end{eqnarray}
where we have defined
\begin{equation}\label{Cpara}
\mu(\phi) \equiv\frac{1}{C} \frac{dC}{d\phi}\ ,
\end{equation}
and we are going to assume hence forth that $\mu(\phi)$  always remains positive.
As one can see, both the Klein-Gordon equation for $\phi$ and the continuity equation for the hidden matter sector $\rho_h$ are augmented by interaction terms in the right hand side of the equations (\ref{KG},\ref{cont-h}). Although  $\mu$  in general depends on $\phi$, to understand the phantom phase let us consider a constant $\mu$ to begin with. It is easy to check that the above interaction is consistent with the conservation of the total energy momentum tensor:
\begin{equation}
\dot{\rho}_{\mathtt{tot}}+3H(\rho_{\mathtt{tot}}+p_{\mathtt{tot}})=0
\end{equation}
where $\rho_{\mathtt{tot}}\equiv \rho_h+\rho_{\phi}$. To see this we remind the readers that the Klein-Gordon equation can be recast as
$$
\frac{d}{dt}\left[-\frac{\dot{\phi}^2}{2}\right]-3H\dot{\phi}^2=-2\rho_h\mu\dot{\phi}
$$
\begin{equation}
\Rightarrow \dot{\rho}_{\phi}+3H(\rho_{\phi}+p_{\phi})=-2\rho_{\phi}\dot{\phi}\mu(\phi)\ ,
\label{cont-phi}
\end{equation}
since the energy density and pressure for a phantom scalar field are given by
\begin{equation}
\rho_{\phi}=p_{\phi}=K=-\frac{\dot{\phi}^2}{2}
\end{equation}
Thus the source terms in the individual conservation equations (\ref{cont-h}) and (\ref{cont-phi}) cancel each other.

One can  solve the hidden matter continuity equation exactly to find
\begin{equation}
\rho_{h}=\rho_{h0}C(\phi)\left(\frac{a}{a_0}\right)^{-3(1+\omega)}
\end{equation}
where we have chosen the convention that at $a=a_0$, $C(\phi)=1$ and $\rho_{h}=\rho_{h0}$. The $a^{-3(1+\omega)}$ dependence reflects the usual dilution of the energy density of an ideal fluid with expansion.   Depending upon the scale of energy density relative to the mass of the hidden matter particles, they can either behave as non-relativistic matter $(\omega=0)$ or like a relativistic species ($\omega=1/3$)\footnote{In this context we note that we have a choice in how we interpret the augmentation of the energy density with growth of $\phi$. One can either think of this growth as simply the increase in mass of the hidden matter particles if the mass depends on $\phi$, or creation of the hidden matter particles through its interactions with $\phi$, or a combination of the two. To keep things simple we are going to assume that the mass of the hidden matter particles remains a constant, but its number density increases.}. We will for most part consider a light degree of freedom, so that approximately it behaves like radiation.

For the special case when $\mu$ is a constant the Coupling function is given by
\begin{equation}
C(\phi)=e^{2\mu(\phi-\phi_0)/M_p}
\label{exponential}
\end{equation}
Now, coming back to the evolution equations, we only need to solve the Hubble equation (\ref{hubble}) and the Klein-Gordon equation, the latter  simplifying to
\begin{equation}
\label{KG2}
\ddot{\phi}+3H\dot{\phi}=\frac{2\mu\rho_{h0}}{M_p}e^{2\mu(\phi-\phi_0)/M_p}\left(\frac{a}{a_0}\right)^{-3(1+\omega)}\equiv V'_{\mathtt{eff}}(\phi)
\end{equation}
It is as if the phantom scalar field is evolving under the influence of an ``effective potential'' given by
\begin{equation}
V_{\mathtt{eff}}(\phi)=\rho_{h}(\phi)=\rho_{h0}e^{2\mu(\phi-\phi_0)/M_p}\left(\frac{a}{a_0}\right)^{-3(1+\omega)}\label{effpotential}
\end{equation}
An important thing to note  is the +ve sign appearing in front of $V'_{\mathtt{eff}}$ in equation (\ref{KG2}) because $\phi$ is a ghost field with  negative kinetic energy. It is clear now that because of the peculiar properties of the phantom field, $\phi$ actually rolls up the effective potential $e^{2\mu\phi/M_p}$.

We have the following late time attractor power-law solutions:
\begin{eqnarray}
a(t)&=&a_0 \left(\frac{t}{t_0}\right)^n\mbox{ \& }\phi= \phi_0+pM_p\ln \left(\frac{t}{t_0}\right)\\ \nonumber
&\Longleftrightarrow&e^{\frac{\phi}{M_p}}= e^{\frac{\phi_0}{M_p}}\left(\frac{t}{t_0}\right)^p
\label{ansatz}
\end{eqnarray}
with
\begin{equation}
n=-\frac{1-{\omega}}{4\mu^2-3/2(1-\omega^2)}\mbox{ and }p=-\frac{4\mu}{4\mu^2-3/2(1-\omega^2)}
\label{power}
\end{equation}
We have verified (see appendix \ref{sec:stability}) that these late time attractors are indeed stable\footnote{We have not investigated whether these solutions suffer from any hydrodynamic instabilities of the nature found in some interacting quintessence models \cite{adinstab}, and we leave this for a future exercise.}.

In this phase the scalar field and the hidden matter are tightly coupled and evolve as a single fluid with an effective equation of state parameter
\begin{equation}\label{eqstatemat}
\omega_{p}\equiv {p_{\phi}+\frac{p_h}{\rho_\phi+\rho_h}}\to-\frac{8}{3}\frac{\mu^2}{(1-\omega)}+\omega
\end{equation}
The asymptotic value is attained during the late time attractor phase. In the phantom phase, expansion of the scale factor is controlled by $\omega_p$, since $a(t)\sim t^{\frac{2}{3(1+\omega_p)}}$. These are analogues to the coupled quintessence solutions~\cite{coupled,adinstab,trodden}. A detailed derivation of all results presented in this section can be found in the appendix \ref{sec:stability}. The crucial thing to note is that  as long as
\begin{equation}\label{muphantom}
\mu^2>\frac{3}{8}(1-\omega^2)
\end{equation}
we have a phantom phase, i.e. $\omega_p<-1$. In particular for $\omega=1/3$ the last condition gives a constraint on $\mu$,
\begin{equation}
\mu>\frac{1}{\sqrt{3}}
\label{mu-cond}
\end{equation}
In passing we also note that in this phase most of the energy density is actually stored in the hidden sector; one can check that the tracking ratio between scalar field and the hidden matter density is given by
\begin{equation}
-\frac{\rho_{\phi}}{\rho_{h}}=-\frac{K}{\rho_{h}}=\frac{8\mu^2}{3(1-\omega)^2+8\mu^2}<1
\label{ratio-ph}
\end{equation}

One can also calculate the amount by which $\phi$ evolves during this phase. A straight forward calculation gives us
\begin{equation}
\Delta\phi_{p}=\frac{2}{\mu}\LT1-\frac{4}{3(1+\omega)}\right]\ln\left(\frac{\lambda_{\mt{max}}}{\lambda_{\mt{min}}}\right)
\label{phantom-phi}
\end{equation}
Here $\lambda_{\mt{min}}$ and $\lambda_{\mt{max}}$ represent the energy scale at which the phantom phase begins and  ends respectively.

In summary, in this section we have found under what conditions a phantom phase could be described by a stable late time attractor for ghost-like fields coupled exponentially  to a perfect fluid.
The purely phantom sector we have studied in this section is problematic as an inflationary model.
For example, for a constant (purely) phantom equation of state $\omega_p<-1$, typically one would obtain a  blue spectrum (see equation (\ref{pps})) which would be inconsistent with the WMAP data. This conclusion is of course valid with the assumption that primordial perturbations are generated mostly during the phantom phase.  Thus unless $\mu$ is fine tuned to be very close to the critical value (which gives rise to $\omega_p=-1$), we will not be able to reproduce the inflationary near scale-invariant spectrum.  Thus far we have considered the phantom sector alone; in the next section we include interactions with the standard model which ameliorate some of the problems of a phantom sector alone.

\section{``Partial Reheating'' and Late time deSitter Phase}\label{sec:reheating}
In the previous section we realized the phantom phase through an interacting phantom scalar field and hidden matter sector. In the absence of any new physics this phase is going to last till the Big Rip singularity, as is well known in phantom cosmology. In order for the next  cycle to begin we need to first find a ``reheating'' mechanism which converts most of the phantom energy density to radiation.

What we find, quite remarkably, is that once we include interaction between the hidden matter sector and Standard Model particles (namely we allow the hidden sector particles to be converted to light degrees of freedom of the standard model), which generically exist, it naturally ameliorates the phantom like acceleration to a near exponential inflationary expansion.

 As we will see, such interactions begin the process of reheating the universe by  producing a radiation bath. Unfortunately the interactions don't provide us with a graceful exit from the inflationary phase, but we will discuss how this issue can also be addressed in the next section.

To understand how interactions effect the cosmological evolution we will use  Boltzmann equations in the following form:
\begin{eqnarray}
\dot{\rho}_h+3H(1+\omega)\rho_h&=&-\rho_h\Gamma+2\rho_h\dot{\phi}\frac{\mu}{M_p}\label{hidensector}\\
\ddot{\phi}+3H\dot{\phi}&=&2\rho_h\frac{\mu}{M_p}\label{KGannih}\\
\dot{\rho}_{\gamma}+4H\rho_{\gamma}&=&\rho_h\Gamma\label{radiation}
\end{eqnarray}
along with the Hubble equation
\begin{equation}
H^2=\frac{1}{3M_p^2}\left(\rho_h-\frac{\dot{\phi}^2}{2}+\rho_{\gamma}\right)
\end{equation}
Here by $\rho_{\gamma}$ we include all the light degrees of freedom which do not couple to the phantom field, and $\Gamma$ is the annihilation rate of the hidden matter particles into all these other light degrees of freedom. We have  ignored  the inverse process of creation of the hidden matter particles from the rest of the matter under the assumption that the equilibrium density of the hidden matter sector is small compared to normal radiation. Now, the annihilation rate, per hidden sector particle, is given in general by
\begin{equation}
\Gamma_{h\bar{h}\rightarrow \gamma\ga}=n_h<\sigma|v|>_{h\bar{h}\rightarrow \gamma\ga}
\end{equation}
where  $<\sigma|v|>$ is the average over all initial and final states of the differential cross-section times the relative velocities of the annihilating particles. Usually one is used to consider the opposite process  while trying to determine when a given species freezes out. In the latter case since the photons are in thermal equilibrium, one can use thermal distribution functions to compute the ``thermally averaged''  $<\sigma|v|>$. However, in our case in order to compute $<\sigma|v|>$ we would need information regarding the velocity distribution of the produced hidden particles from $\phi$. In the absence of any micro-physical theory of such an interaction, for the purpose of illustration, here we are simply going to assume that $<\sigma|v|>$ is a constant set by the details of the interaction, so that the interaction rate per hidden sector particle is given by
\begin{equation}
\Gamma=\frac{\rho_h}{m^3}
\end{equation}
Here $m$ is an energy scale we introduce as a free parameter. More generally one expects $\Gamma$ to go as some power law, $\Gamma\sim \rho_h^{\lambda}$, the power being determined by the micro-physics, but most of our  results and conclusions should hold qualitatively as long as $\lambda>1$.

It is easy to check that the above set of equations (\ref{hidensector}-\ref{radiation}) have an asymptotic de Sitter late time attractor solution where all the energy densities and the Hubble parameter tend to a constant. Defining the following dimensionless variables,
\begin{equation}\nonumber
 \xi\equiv \frac{\rho_h}{M_p^2H^2} \quad\text{ and}\quad \eta\equiv\frac{\Gamma}{H}=\frac{\rho_h}{m^3H}
\end{equation}
 we have:
\begin{eqnarray}
\xi&\rightarrow& \frac{27\omega-9+3\sqrt{9(3\omega-1)^2+192\mu^2}}{ 8\mu^2}\label{dSx}\\
\eta&\rightarrow& \frac{3\omega-9}{2}+\frac{1}{2}\sqrt{9(3\omega-1)^2+192\mu^2}\label{dSy}\\
\dot{\phi}&=&\frac{2\rho_h\mu}{3H}\rightarrow \frac{2\mu m^3\eta} {3M_p}\label{dSphi}\\
\rho_{\gamma}&=&\frac{\rho_h^2}{4H m^3}\rightarrow \frac{m^6\eta^3}{4M^2_p\xi}\label{dSrad}
\end{eqnarray}
As a  consistency check we have evolved these equations  numerically, see Fig.1, and have verified  that the asymptotic values are exactly the ones predicted by the above set of equations.
Let us make a few observations. Since we are specializing to $\omega=1/3$, let us look at the asymptotic value of the energy densities in this case:
\begin{eqnarray}
\rho_{\gamma}&=&\frac{16\,\sqrt{3}\mu\,{m}^{6}}{9M_p^2} \left( \,\sqrt {3}\mu -1\right)^3\nonumber\\
\rho_{h}&=&\frac{16\,\sqrt{3}\mu\,{m}^{6}} {9M_p^2} \left( \,\sqrt {3}\mu -1\right)^2\label{asymptotic}\\
\rho_{\phi}&=&-\frac{32\,\mu^2\,{m}^{6}}{9M_p^2} \left( \,\sqrt {3}\mu -1\right)^2\nonumber
\end{eqnarray}
Note that the solutions above are consistent only if $\mu> \frac{1}{\sqrt{3}}$, but we know this is also a requirement  for the existence of  a phantom phase\footnote{see equation (\ref{mu-cond})} which will eventually settle to this deSitter attractor. In particular, we find that for $\mu\gg 1$, which is the case we will eventually be focusing on, radiation dominates over the hidden sector:
\begin{equation}
{\cal R}\equiv \frac{\rho_h}{\rho_{\gamma}}\approx  \frac{1}{4\sqrt{3}\mu}\ll1\mbox{ for }\mu\gg 1
\label{ratio}
\end{equation}
In other words, as $\mu$ increases, the conversion from hidden matter to radiation becomes more efficient, and in particular radiation can easily dominate the total energy density. However, this does not mean that we can enter into the usual radiation dominated epoch because the equation of state for all the energy densities essentially approach $\omega_p=-1$, as all the energy densities approach the constant asymptotic values in equation (\ref{asymptotic}). Physically, the energy density that the phantom field was pumping into the hidden matter sector is now transferred to radiation through its interaction with the hidden sector, in such a way that we approach a deSitter universe. This phase however is good for inflationary cosmology as fluctuations produced during this de Sitter phase are expected to give rise to a near scale-invariant spectrum (allowed, although not favored, by  the WMAP data). In order for the amplitude of the fluctuations to be consistent with observations we require
\begin{equation}
\rho_{\gamma}\approx 10^{-12}M_p^4\Rightarrow m\sim 10^{-2}M_p\ .
\end{equation}
We will return to the constraint coming from the number of efoldings required for a successful inflationary paradigm later.

In this section we have found that the big rip singularity could be avoided in this model if the hidden sector particles are converted to light degrees of freedom of the Standard Model. As the energy densities approach their asymptotic values the universe will enter in a deSitter phase. Of the three components, radiation is dominant, yet the universe is inflating. This is due to the interplay between the coupling of the hidden sector to the phantom field and light degrees of freedom of the Standard Model which leads to a state where all the energy densities approach a constant value.
\section{Transition to Radiation and Cyclicity}\label{sec:transition}
In the above section we saw that interactions between hidden matter and radiation can ameliorate the phantom like acceleration to exponential inflation but cannot provide a graceful exit from the deSitter inflationary phase. So far we have assumed that $\mu$ remains a constant leading to an exponential coupling between the hidden sector and the phantom field  as in equation (\ref{exponential}). If $\mu(\phi)$ is not a constant a graceful exit becomes possible. Here we explore  the case when $\mu(\phi)$ is periodic. For simplicity we will just assume a step function for $\mu(\phi)$, where
\begin{eqnarray}
\mu&=&\mu_p\gg 1 \mbox{ for }0<\phi<\phi_R\\
\label{mur}
\mu&=&\mu_r\ll 1 \mbox{ for }\phi_R<\phi<\phi_0
\end{eqnarray}
and then the pattern repeats itself. The first phase, when $\mu\gg1$, reproduces the phantom phase discussed  in the previous section, leading eventually to the late time de Sitter like attractor evolution. However, now this inflationary phase ends once $\phi$ reaches the transition value $\phi_R$. We are also going to assume that $\omega=1/3$ in this scenario. We remind the readers that $\omega$ describes the hidden sector, as in equation (\ref{defomega}). As we will see shortly, the periodicity in $\mu$  will ensure that we enter the standard radiation dominated era which lasts till $\phi$ rolls to $\phi_0$.

The evolution of the universe during one cycle can be described in our model using three phases, as indicated in Fig. 1. Phase I
provides the  ``Reheating'' from inflationary expansion. Phase II describes a standard radiation dominated era. Then a short phantom phase IIIA ensues, during which the energy is driven from the
$meV$ to the GUT scale.   This is followed by a deSitter inflationary phase IIIB.  After one such cycle
is complete, the next one begins, again cycling through Phases I, II, IIIA, and IIIB successively.

\subsection{Phase I: Reheating}
\label{reheating}
The reheating process is most easily understood when $\mu_r=0$ (defined in equation (\ref{mur})), so let
us first focus on this simple case. As we discussed in the previous section, as long as $\mu$ is large,
although radiation is the dominant energy density, the deSitter phase continues. However, if and when $
\mu$ sharply falls to zero, the phantom phase indeed ends. Two things happen. Firstly, since  initially $
\Gamma$ is comparable to $H$, as it can be seen from equation  (\ref{dSy}), there is rapid conversion of the
hidden matter to radiation, but the hidden matter sector now no longer gets replenished by the scalar
 field.  Secondly, the driving term in the right hand side of the Klein-Gordon equation for the phantom scalar field (\ref{KG}) is now absent and as a result $\phi$ slows due to Hubble damping,
 $\rho_{\phi}\sim a^{-6}$ and eventually comes to a halt.  At the beginning of the reheating phase  the
 energy densities of
radiation, hidden matter and $\phi$ are approximately given by the asymptotic values of  the late time de
Sitter phase (\ref{asymptotic}). As we had pointed out before, for $\mu\gg1$  radiation is the dominant
energy density component in this asymptotic phase. The ``reheating'' phase further ensures that
radiation continues to dominate the energy density.  If one tracks the ratio of the energy densities
between hidden matter and radiation, $\mathscr{R}$, then it starts with the asymptotic value given by equation
(\ref{ratio}), decreases rapidly during conversion, and then approaches a constant, $\mathscr{R}_{\mathtt{min}}$,
once the conversion ends. Note, since $\phi$ slows down, $\rho_h$ redshifts almost as radiation and
therefore  maintains an approximately constant tracking ratio approaching $\mathscr{R}_{\mathtt{min}}$. In appendix
\ref{sec:ratiomin} we calculated this asymptotic ratio to be
\begin{equation}
\rs_{\mathtt{min}}\approx \frac{1}{6}\frac{\sqrt{3}}{\mu_p^2}\label{rfinalrad0}\mbox{ for }\mu_r=0\ ,\mu_p\gg1
\end{equation}

Qualitatively, it turns out that one can distinguish two different regimes in the reheating phase depicted in Fig.1 as phases I.$\mathrm{A}$ and I.$\mathrm{B}$. Numerically we found that even after $\mu\rightarrow 0$, it takes a while for the radiation energy density to start decreasing substantially. The reason is somewhat technical and the reader is referred to the appendix \ref{sec:ratiomin} for details. Intuitively, the main reason is that initially the Hubble damping of radiation is compensated by the hidden matter decays into  radiation,  $4H\rho_{\gamma}\sim \rho_h\Gamma$. Since radiation is the dominant component of the energy density, this in turn leads to $H$ being approximately constant, as can be  seen in phase I.$\mathrm{A}$ of Fig.1.
Once $\rho_h$ decreases appreciably so that $\rho_h\Gamma\ll 4H\rho_{\gamma}$, the radiation energy density starts to decrease appreciably and therefore so does the Hubble rate. This is depicted in phase I.$\mathrm{B}$ of Fig.1. At some point, the conversion from hidden matter to radiation effectively stops,
marking the end of the reheating phase.

Although the above discussion has been for $\mu_r=0$,
we note that for a non-zero but small $\mu_r$, the reheating phases (I.$\mathrm{A,B}$)
follow basically the same pattern as in the $\mu_r=0$ case.
The asymptotic tracking ratio~\footnote{Unlike in the $\mu_r=0$ case where the tracking ratio keeps decreasing and asymptotically approaches $\rs_{\mathtt{min}}$, when $\mu_r\neq 0$, $\rs_{\mathtt{min}}$ is actually a minimum of the tracking ratio that is attained. Since $\phi$ never comes to a halt, the ratio does increase  from its minimum value  of $\rs_{\mathtt{min}}$, but this increase is very slight.} between hidden matter and radiation receives a slight correction:
\begin{equation}
\rs_{\mathtt{min}}\sim \frac{1}{18}\frac{\sqrt{3}(3+8\mu_r)}{\mu_p^2}\label{rfinalrad}\mbox{ for }\mu_r\ll1\ ,\mu_p\gg1
\end{equation}
We have checked this numerically, and some of the more technical details  are  discussed in Appendix \ref{sec:ratiomin}.

\subsection{Phase II: ``Standard'' Radiation Domination}
After the reheating phase, since hidden matter is no longer converted into radiation, the latter  starts to evolve as $a(t)^{-4}$, and consequently $H\sim 1/2t$ as in the standard radiation dominated era. In the meantime $\phi$ continues to be Hubble damped. Once the scalar field effectively stops evolving, the hidden matter starts redshifting as radiation and thus settles down to its constant tracking ratio given in equation (\ref{rfinalrad0}). In particular, one can see from (\ref{rfinalrad}) that for sufficiently large values of $\mu_p$, this ratio can be quite small and easily satisfy constraints coming from BBN and CMB. BBN/CMB only constrains the abundance of dark radiation component to be less than around 10\%~\cite{bbn}. In Fig.1 we refer to the phase when $\phi$ is being Hubble damped as phase II.$\mathrm{A}$, and the subsequent radiative phase as phase II.$\mathrm{B}$.

If $\mu_r$ is precisely zero, then the radiation  dominated  phase IIA can continue forever because
$\phi$ will effectively come to a halt, and unless the value of $\phi_0$ is fine-tuned, $\phi$ will never make it to the next large $\mu$ phase. As a result the next phantom phase will not begin and the  cyclic picture cannot be sustained. This is why a small but non-zero value of $\mu_r$ is essential to maintaining cyclicity without having to resort to unnatural fine-tuning.

 For a non-zero but small $\mu_r$, the  reheating phases (I.$\mathrm{A,B}$) and the Hubble damping phase (II.$\mathrm{A}$) follow very much the same pattern as in the $\mu_r=0$ case, as discussed
 at the end of the previous section \ref{reheating}.
The main difference when $\mu_r\neq 0$, as compared to the $\mu_r=0$ case, appears in the radiative phase II.$\mathrm{B}$. This is because the non-zero driving term in the Klein-Gordon equation now ensures that instead of coming to a halt, $\phi$ now tracks radiation.  After the initial phase of Hubble damping, the driving term on the right hand side of the KG equation catches up with the Hubble damping term. At this point the scalar field enters a  phase where its energy density approximately tracks that of radiation. This can be seen in Fig.1 where we have divided the radiation phase into two parts. II.$\mathrm{A}$ refers to the regime when the scalar energy density is still being Hubble damped, while  phase II.$\mathrm{B}$ refers to the tracking phase where both the hidden matter  and the scalar energy densities are tracking that of radiation.

This tracking behavior can be approximately obtained as follows: From the KG equation we have
\begin{equation}
3H\dot{\phi}=\frac{2\rho_h\mu_r}{M_p}\approx \frac{2\mathscr{R}_{\mathtt{min}}\rho_{\gamma}\mu_r}{M_p}\approx  6\mathscr{R}_{\mathtt{min}}M_p\mu_r H^2
\label{slow-roll}
\end{equation}

In the above we have ignored the $e^{2\mu_r\phi}$ dependence of $\rho_h$, as $\phi$ is rolling very slowly, and $\mu_r\ll1$. We will perform a consistency check later. Also, we have ignored the contributions to the energy density coming from the hidden matter and the scalar field as compared to normal radiation. Again, this is justified as ${\cal R}_{\mathtt{min}}\ll1$ and $\phi$ is rolling slowly. Choosing the ansatz
\begin{equation}
\phi=M_pp_r\ln\left(\frac{t}{t_R}\right)\Rightarrow \dot{\phi}=\frac{M_pp_r}{t}
\end{equation}
we find that (\ref{slow-roll}) can indeed be satisfied provided
\begin{equation}
p_r=\mathscr{R}_{\mathtt{min}}\mu_r
\end{equation}
In the above analysis we have used the fact that in a radiation dominated universe $H\sim 1/2t$. In particular our analysis tells us that the tracking ratio between the kinetic energy of $\phi$ and radiation is indeed very small
\begin{equation}
\frac{K_{\phi}}{\rho_{\gamma}}=-\frac{2}{3}(\mu_r \mathscr{R}_{\mathtt{min}})^2
\end{equation}
justifying our earlier assumption. In the next subsection we will also see that during this phase $\phi$ evolves rather slowly, so that $C(\phi)$ changes only  by an ${\cal O}(1)$ factor ensuring that the hidden matter indeed  behaves as radiation to a very good approximation.

This  radiation dominated tracking era lasts till $\phi$ reaches $\phi_0$ and rolls over to the large $\mu$ region. The next  phase of phantom domination, phase III.$\mathrm{A}$, can now begin.
\subsection{Cyclicity}
\begin{figure}
\includegraphics[width=.5\textwidth, height=6cm]{Figure.eps}
\caption{Numerical solutions for the energy densities as we complete a cycle from a deStiter phase back to it. Here we have chosen: $\omega=\frac{1}{3}$, $\mu_p=5$, $\mu_r=.1$, $m=10^{-2}M_p$ and we have set $M_p$ to one.  Note the six  distinct phases: I.A and I.B corresponding to reheating; II.A and II.B corresponding to a radiation dominated universe; III.A and III.B corresponding to the phantom and dS phase respectively. In order to make all phases clearly distinct we chose the transition set the minimum energy density at around $10^{-45}$ instead of the realistic $meV^{4}$ }
\end{figure}
To better understand the transition from one cycle to the next let us discuss the various phases we observe in Fig. 1, where we plot a numerical solution for the energy densities of the three components from one deSitter inflationary phase to the next. The plot does not correspond to realistic values for
$\lambda_{\mathtt{max}}$ or $\lambda_{\mathtt{min}}$, but captures all the essential features of the different phases. The plot starts (extreme left) at $t=0$  and $\phi=\phi_R$, corresponding to the beginning of the reheating phase I.$\mathrm{A}$.  We have estimated in Appendix  \ref{sec:ratiomin}, equation  (\ref{ttrans}) how long ($t_{1A}$) the I.$\mathrm{A}$ sub-phase lasts
\begin{equation}
t_{1A}=\left(\frac{3M_p^4\xi}{m^6 \eta^3}\right)^\frac{1}{2}
\end{equation}
During this phase the field $\phi$ evolves approximately a distance of:
\begin{equation}\label{delta1}
\Delta\phi_{1A}=\dot{\phi}_{dS}\,\int_0^{t_{1A}}\,\text{d} t\,e^{-3H_{dS}t}=\frac{\dot{\phi}_{dS}}{3 H_{dS}}\left[1-e^{-3H_{dS}t_{1A}}\right]
\end{equation}
Above we have used:
\begin{equation}
\dot{\phi}(t_{1A})=\dot{\phi}_{dS}e^{-3H_{dS}t_{1A}}
\end{equation}
where $\dot{\phi}_{dS}$ represents the asymptotic value in the deSitter phase, given by (\ref{dSphi}) and $H_{dS}$ is the Hubble rate during the inflationary phase and can be solved for from the definitions of $\xi$ and $\eta$.

Phase  I.$\mathrm{A}$ is followed by phase I.$\mathrm{B}$, where although the conversion from hidden matter to radiation takes place, the Hubble rate starts to decrease appreciably as well. As soon as the conversion is no longer efficient  we enter a regime, phase II.$\mathrm{A}$, where radiation starts to redshift as $a^{-4}$ marking the beginning of a Standard radiation dominated era. In Phase  II.$\mathrm{A}$ radiation and the hidden sector energies approximately track each other while the field $\phi$ is still being Hubble damped. This phase ends when the scalar field is no longer Hubble damped and starts to track radiation as well, a phase we refer to as II.$\mathrm{B}$. Next we estimate the time $t_{2A}$, when this transition from the Hubble damping  phase II.$\mathrm{A}$ to tracking phase II.$\mathrm{B}$ occurs. It can be defined as the time when the Hubble damping term in the l.h.s. of equation (\ref{KGannih}) is equal to the coupling term on the r.h.s. Under the assumption that we are in a radiation dominated phase and that the hidden sector energy density tracks the radiation energy density with the ratio $\rs_{\mathtt{min}}$ we get:
\begin{eqnarray}\label{td}
\frac{3\dot{\phi}(t_{2A})} {2 t_{2A}}&=&\frac{2\mu_r}{M_p}\rs_{\mathtt{min}}\rho_{\gamma}(t_{2A})\nonumber\\
\Rightarrow t_{2A}&=&\frac{t_{1A}^3\dot{\phi}^2(t_{1A})}{M_p^2\mu_r^2\rs_{\mathtt{min}}^2}
\end{eqnarray}
This last equation can be rewritten in terms of our parameters in the following form:
\begin{equation}\label{ratiotttd}
\frac{t_{1A}}{t_{2A}}=\frac{1}{144}\frac{\mu_r^2(3+8\mu_r)^2\eta}{\xi\mu_p^6}e^{6H_{dS}t_{1A}}
\end{equation}
With  $\omega=1/3$ and in the limit $\mu_p\gg 1$  the exponent $6 H_{dS}t_{1A}$ becomes $\sqrt{3}$, so we see, as expected, that $t_{2A}\gg t_{1A}$.

We can compute the distance the field $\phi$ evolves during the phases I.$\mathrm{B}$ and II.$\mathrm{A}$ since the Hubble friction term is dominant during this time.
\begin{eqnarray}\label{delta2}
\Delta\phi_{1B+2A}&=&\int_{t_{1A}}^{t_{2A}}\,\,\text{d} t\dot{\phi}=2\dot{\phi}(t_{1A})t_{1A}\left(1-\sqrt{\frac{t_{1A}}{t_{2A}}}\right)\nonumber\\
&\approx&\frac{4\mu_p}{\sqrt{3}}\sqrt{\frac{\xi}{\eta}}M_p e^{-\frac{\sqrt{3}}{2}}
\end{eqnarray}

In the next phase,  II.$\mathrm{B}$, as one can see from the plot,  all the energy components are tracking each other. Since this phase will lasts until almost 'today' and it began at $t_{2A}$ we have:
\begin{equation}\label{phi-radnonexp}
\frac{\Delta\phi_{2B}}{M_p}\approx 2\mathscr{R}_{\mathtt{min}}\mu_r\ln\left(\frac{(\rho_{\gamma}(t_{2A}))^\frac{1}{4}} {meV}\right) \approx 120 \mathscr{R}_{\mathtt{min}}\mu_r
\end{equation}
where  $\Delta\phi_{2B}$ is the distance the scalar field evolves during phase II.$\mathrm{B}$. In particular we note that this means
\begin{equation}
e^\frac{\mu_r\Delta\phi_{2B}}{M_p}\approx e^{120 \mathscr{R}_{\mathtt{min}}\mu_r^2}
\label{phi-rad}
\end{equation}
Since $\mathscr{R}_{\mathtt{min}}$ is a small number, for sufficiently small values of $\mu_r$ it is easy to see that the exponential will only contribute to an ${\cal O}(1)$ factor to the energy density of the hidden matter sector. In other words unlike the $\mu_r=0$ case, although  the ratio $\mathscr{R}$ does not monotonically decrease during the radiation phase, but rather starts to increase as $\phi$ evolves, this increase is very slow. This justifies our earlier assumption of hidden matter approximately behaving as radiation.

The advantage of having a non-zero $\mu_r$ is that it keeps the scalar field rolling, albeit slowly, ensuring passage to the next phantom phase when $\phi$ reaches $\phi_0$. Therefore, no fine-tuning is involved in  restarting  the phantom era. To see this observe that
\begin{equation}
\Delta\phi_{2B}=(\phi_0-\phi_R)-(\Delta\phi_{1A}+\Delta\phi_{1B+2A})
\end{equation}
and crucially for realistic values of the parameters the three different $\Delta\phi$'s are of the same order of magnitude. If for instance, it turned out that $\Delta\phi_{2B}\ll\Delta\phi_{1A}+\Delta\phi_{1B+2A}$, that would have meant fine tuning the range $\phi_0-\phi_R$ to cancel $\Delta\phi_{1A}+\Delta\phi_{1B+2A}$ to very high precision. In fact, this is what one has to do as $\mu_r\rightarrow 0$.

A related nice feature of the model is that the exponential hierarchy between the scales of inflation and dark energy is rather easy to arrange. To see  this let us try to obtain $\lambda_{\mathtt{min}}$ in terms of $\phi_0-\phi_R$. By re-arranging equation (\ref{phi-radnonexp}) one finds that the energy density at the beginning of the phantom phase is given by~\footnote{Above we have used a  value for $\rho_{\gamma}^{1/4}(t_{2A})$ of $10^{-3}M_P$, slightly overestimating it. The actual value is typically lower than this, because of the additional Hubble dilution of the energy density of radiation from $t_{1A}$ to $t_{2A}$. For instance, with the parameters used in the numerical solution for Fig. 1 we have $\rho_{\gamma}^{1/4}(t_{2A})\sim 10^{-5}M_p$.}
$$
\lambda_{\mathtt{min}}\sim \lambda_{\mathtt{max}}\exp\left(-\frac{\Delta\phi_{2B}}{2\mathscr{R}_{\mathtt{min}}\mu_r M_p}\right)$$
\begin{equation}\sim 10^{-3}M_p\exp\left(-\frac{\phi_0-\phi_R-\Delta\phi_{1A}-\Delta\phi_{1B+2A}}{2\mathscr{R}_{\mathtt{min}}\mu_r M_p}\right)\label{hierarchy}
\end{equation}
Since $\mathscr{R}_{\mathtt{min}},\mu_r$ are small numbers as compared to $\Delta\phi/M_p$'s, it is easy to arrange the exponential suppression of $\rho_0$ as compared to the GUT scale. Conversely, we can reformulate the ``smallness'' problem associated with dark energy in terms of four parameters: $\mu_p, \mu_r,\phi_R, \phi_0$, which as we shall see shortly have values that range from $\mathcal{O}(1)$ to $\mathcal{O}(10^2)$.  In  the next section (below equation (\ref{move}))
we will provide specific numerical examples which make this more evident.

Finally, we come to the phantom phases III.$\mathrm{A,B}$. The transition to the phantom phase, which is supposed to be happening during the present cosmological epoch, occurs once $\phi$ reaches $\phi_0$ and $\mu$ transitions suddenly to its high value, $\mu_p$. The hidden matter + scalar field energy densities catch up with radiation, start evolving as a phantom fluid with equation of state given by (\ref{eqstatemat}), and come to dominate the universe. The radiation keeps getting diluted as $a^{-4}$ during this phase which we refer to as phase III.$\mathrm{A}$. Since the increase in energy density in the phantom phase occurs at a very short time scale, all the features of this phase are not very discernable in the log-log plot in Fig.1, but we have checked them numerically. As we have seen before, the annihilation term will ultimately cause this behavior to transition to a de Sitter phase, when $\Gamma\sim H$ or $\rho_h\sim m^6/M_p^2$. At this point we enter, what  we  call phase III.$\mathrm{B}$, when all the energy densities become comparable and then tend towards constant values, leading to an asymptotic de Sitter space-time. It is during this phase we expect to generate a scale invariant spectrum of perturbations. This phase will last until the field $\phi$ has evolved a distance of $\phi_0+\phi_R$, when we re-start the cycle.

\section{Numbers and Constraints}\label{sec:numbers}
So far we have provided general constraints coming from different observations on the couplings and the scales. For the purpose of illustration let us provide some typical values which conform to these constraints and in the process we will also be able to understand the different phases of evolution better. Let us start with the inflationary phase. As already discussed, to obtain the correct amplitude of inflationary fluctuations we will take the reheating to occur at approximately GUT scale energy densities which implies
\begin{equation}
m\sim 10^{-2}M_p
\end{equation}

During the course of 60 e-foldings\footnote{We choose 60 as a generic typical example for GUT scale
inflation; the actual number of e-foldings might be lower or larger than this.},
we find, using equations (\ref{dSy},\ref{dSphi}), that the field  $\phi$ evolves a distance of:
\begin{equation}\label{deltainf}
\Delta\phi_{3B}=40 M_p\mu_p\xi
\end{equation}
As a prototype example, for $\mu_p=4$ this gives us $\Delta \phi_{3B}\sim 200$. This only gives us a constraint on the range when $\mu$ is large. For instance,  for $\mu_p=4$ the number of efoldings during the phantom phase turns out to be (using equation (\ref{phantom-phi})) $\Delta\phi_{3A}\sim 30$.  To be consistent with inflation we must have $\phi_R\gtrsim 200+30=230$.  Given the
fact that these field values are transPlanckian, it is possible that higher order terms in the
Lagrangian should not be neglected; we proceed here with the assumption that our starting point
is sensible nonetheless.

Depending on the number of e-foldings during the de Sitter phase we find two distinct cases for the spectrum of fluctuations.
Case I.  If the range of $\phi$ is such that one gets more than 60 e-foldings of de Sitter,
then the CMB fluctuations are scale invariant.
As we noted in the introduction, $n_s\approx 1$  is still consistent with observations \cite{WMAP5} once one allows the possibility of running of the tilt and/or tensor modes.
If the range in $\phi$ is such that we have only around 60 e-foldings then the CMB fluctuations  at large scales can show a transition from a blue (when the phantom phase will be operating and the reheating mechanism hasn't kicked in completely) to a scale invariant spectrum. This could be an interesting and rather unique signature of the model.
 Case II. If the number of efoldings in the dS phase is shorter than 60 e-foldings, the
 fluctuations that we are observing in the sky must have been generated in the phantom phase. This gives a rather stringent constraint on how far below $-1$ the phantom equation of state $\omega_p$ can be. For a ``super-inflationary'' space-time sourced by phantom fluid, the spectral tilt is expected to be given by
\begin{equation}\label{pps}
\eta_s-1=\frac{6(1+\omega_p)}{ 1+3\omega_p}>0
\end{equation}
implying  a blue spectrum. Therefore to be consistent with the observations $\omega_p$ has to be very close to $-1$. For instance, if we include tensor modes, according to \cite{WMAP5} at the $2\sigma$ level we find
\begin{equation}
\eta_s<1.01\Rightarrow \omega_p>-1.01
\end{equation}
implying $\mu_p<.581$. This bound is very restrictive, remembering that in order to have a phantom phase we need $\mu_p>1/\sqrt{3}\approx 0.577$. In the rest of the section we are not going to discuss this possibility any further and concentrate on Case I with $\mu_p\gg1$ which seems more attractive.

Let's next look at the constraint coming from BBN and WMAP on the amount of dark radiation~\cite{bbn}. To  be consistent with the data the amount of dark radiation has to be limited to within 10\% of ordinary radiation. This essentially imposes a constraint on $\mu_p$, but a rather weak one
\begin{equation}
\rs_{\mathtt{min}}<0.1\Rightarrow \mu_p\gtrsim 0.17
\label{bbn}
\end{equation}
which is easily satisfied.

What about the range of $\Delta\phi$ during radiation domination? This of course depends on the value of $\mu_p,\mu_r$. Just to get an idea, we find for $\mu_p=4,\mu_r=0.2$, we need
\begin{eqnarray}
\label{move}
\Delta\phi&=&\Delta\phi_{1A}+\Delta\phi_{1B+2A}+\Delta\phi_{2B}=\phi_0-\phi_R\sim\nonumber\\
&\sim& 0.70+2.16+0.66
\end{eqnarray}
 As one can see, all the values in the above equation
 are of the same order of magnitude  and therefore no fine tuning seems to be involved.

Thus the picture that emerges from the above estimates is that for the scenario to work we need a relatively longer phases in $\phi$ when $\mu$ is large (\ref{deltainf}), followed by
shorter phases when $\mu$ is small.  However, even the discrepancy in $\Delta\phi$ between
equations (\ref{deltainf}) and (\ref{move}) is only a few orders of magnitude.  Similarly, $\mu_r$
and $\mu_p$ (in the above example) differ again by only a few orders of magnitude.
It is these numbers  (the  values of $\Delta\phi$ and the two values of $\mu$)
that determine the heirarchy between today's $meV$ scale
and the GUT scale of inflation via equation (\ref{hierarchy}).  So we see that, indeed,
little fine-tuning is required to explain these disparate mass scales.

Finally, let us come to constraints from dark energy. The main constraint comes from the equation of state parameter. A combined $2 \sigma$ bound from CMB+BAO+SN  is given by $-0.88>\omega_p>-1.14$~\cite{WMAP5}. This provides a constraint on $\mu_p$
\begin{equation}
\mu_p<0.61
\label{mu-constraint}
\end{equation}
This may seem too small, but we realize that currently we are undergoing the phase transition from small $\mu$ to a large $\mu$ region, and therefore it is easy to arrange that the ``current'' value of $\mu$ satisfies inequality (\ref{mu-constraint}) and has not yet reached the constant maximum value $\mu_p$. Equivalently, the equation of state parameter today has not yet reached its late time phantom phase value given by equation (\ref{eqstatemat}).

\section{Conclusions}
In this paper we have studied a cyclical model of the universe where the energy density cycles between a minimum value, typically of the order of $meV^{4}$ and a maximum value roughly set by the GUT scale. This effectively provides a connection between the current accelerated expansion we observe today and the inflationary era in the past. The scale factor continues to grow from one cycle to the next
(there is no "turnaround").
In order to achieve this model we postulated the existence of some `hidden' sector matter  coupled
  to a ghost like scalar field. This mechanism is responsible for a super-accelerated phantom  expansion. Allowing for hidden sector particles to be converted to light degrees of freedom of the standard model ameliorates the phantom behavior, effectively transitioning to a deSitter like expansion, therefore avoiding the Big-Rip singularity. Although dominated by radiation, in this phase all the energy densities remain constant. Therefore, if most of the cosmological perturbations are generated during this exponential inflationary era the spectrum is expected to be scale invariant. Even if not favored by the data, this is still a possibility. Another possibility would be to  have some   of the fluctuations generated during the phantom phase which will show up as a blue tilt in the  spectrum. In that case  we will see a running of the tilt which could be a unique  possible signature of the model.

In order to achieve the cyclic behavior we have postulated that the coupling between the ghost scalar field and the hidden sector is constant piecewise. This procedure might seem ad-hoc, but it is just the simplest possibility. We found that no fine tuning seems to be involved when requiring to have a long enough radiation/matter dominated phase. It is also worth mentioning that the ``smallnesss'' problem associated with dark energy  is circumvented. The only parameter we need in order to describe the current acceleration is the coupling between the hidden sector and the ghost field, and it has a value not much greater than one.

\begin{acknowledgments}
T B would like to acknowledge the hospitality of the physics department at University  of Minnesota at Minneapolis. KF and CI are supported by the US Department of Energy and MCTP via the Univ. of Michigan.
\end{acknowledgments}


\section{Introduction}

\indent

  Since the original discovery of black hole radiation by Hawking
\cite{1}, the studies on this topic have not terminated. There are
many different methods for the derivation of Hawking radiation
\cite{2,3,4,5,6,7,8,9}. In \cite{10,11}, a semiclassical method for
the derivation of Hawking radiation was formulated by Parikh and
Wilczek based on the quantum tunneling picture. In such a method,
the radiated particles of a black hole are treated as
$s$-waves.\footnote{~ This is reasonable because for an observer at
infinity, the radiation of a black hole is spherically symmetric, no
matter whether the black hole is rotating or not.} When a particle
is radiated from the black hole horizon, it tunnels through a
barrier that is made by the tunneling particle itself due to the
horizon's contraction \cite{10,11}. To use the WKB approximation,
the tunneling rate of an $s$-wave from inside to outside the black
hole horizon is given by
$$
  \Gamma=\Gamma_{0}\exp(-2\mbox{Im}{\cal I}) ~.
  \eqno{(1)}  $$
Here, ${\cal I}$ is the action of the tunneling particle,
$\Gamma_{0}$ is a normalization factor. On the other hand, a black
hole's radiation satisfies the law of Boltzmann distribution
classically, thus the emission rate of a particle of energy $E$ from
a black hole horizon can be expressed by
$$
  \Gamma=\Gamma_{0}\exp(-\beta E) ~,
  \eqno{(2)}  $$
where $\beta=2\pi/\kappa$, $\kappa$ is the surface gravity of the
horizon. To compare (2) with (1), the Hawking temperature of a black
hole can be derived.

  After the original work of Parikh and Wilczek, many developments
on this topic have been carried out \cite{12,13,14,15}, and many
applications of this method for the derivation of Hawking radiation
of different types of black holes have been done
\cite{16,17,18,19,20,21,22,23,24,25,26,27}. In this paper, we study
the Hawking radiation of general four-dimensional rotating black
holes from the tunneling approach. We use the null-geodesic method
of Parikh and Wilczek \cite{10,11} to calculate the action of a
tunneling particle. In \cite{21}, Hawking temperature of Kerr and
Kerr-Newman black holes have been derived from tunneling approach
using dragging coordinate systems. In such a kind of coordinate
system, the spacetime of a four-dimensional rotating black hole has
been contracted to a three-dimensional slice. Thus the topology of
the spacetime of a rotating black hole has been changed to use the
method of \cite{21}. In order to keep the spacetime topology of a
rotating black hole, we choose a reference system that is
co-rotating with the event horizon to eliminate the motion of $\phi$
degree of freedom of a tunneling particle. We obtain that for a
general four-dimensional rotating black hole, its thermal radiation
temperature derived from the tunneling approach is in accordance
with its Hawking temperature derived from black hole thermodynamics.
These contents are given in Section 2. In Section 3, we give the
explicit result of the Hawking temperature of the Kerr-Newman-AdS
black hole from the tunneling approach. In Section 4, we discuss
some of the problems.

\section{Hawking temperature of four-dimensional rotating
         black holes from tunneling}

\indent

  The metric of a four-dimensional spherically symmetric black hole
can be expressed as
$$
  ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+r^{2}d\Omega^{2}
  \eqno{(3)}  $$
generally. Following \cite{10,11}, the imaginary part of the action
of a tunneling particle in terms of an $s$-wave can be calculated
from
$$
  \mbox{Im}{\cal I}=\mbox{Im}\int_{r_{h}(M)}^{r_{h}(M-E)}p_{r} ~ dr
    =\mbox{Im}\int_{r_{h}(M)}^{r_{h}(M-E)}\int_{0}^{p_{r}}
     dp_{r}^{\prime} ~ dr ~,
  \eqno{(4)}  $$
where $r_{h}$ is the radius of the outer horizon, $M$ is the total
mass of the black hole, $M-E$ is the total mass of the black hole
after the particle is emitted, $E$ is the energy of the tunneling
particle. To make use of the Hamilton's equation
$$
  \dot{r}=\frac{dH}{dp_{r}}=\frac{d(M-\omega)}{dp_{r}} ~,
  \eqno{(5)}  $$
we can write
$$
  dp_{r}=\frac{d(M-\omega)}{\dot{r}} ~.
  \eqno{(6)}  $$
To substitute (6) into (4), we have
$$
  \mbox{Im}{\cal I}=\mbox{Im}\int_{r_{h}(M)}^{r_{h}(M-E)}
    \int_{0}^{E}\frac{d(M-\omega)}{\dot{r}} ~ dr =
    \mbox{Im}\int_{0}^{E}\int^{r_{h}(M)}_{r_{h}(M-E)}
    \frac{dr}{\dot{r}} ~ d\omega ~.
  \eqno{(7)}  $$
Usually, the Hawking temperature of a black hole is very small,
zero-mass particles will possess the main part of the whole
radiation spectrum. For a tunneling particle of zero-mass in terms
of an $s$-wave, it moves in a radial null geodesic. To transform the
metric (3) to the Painlev{\'e} form, $\dot{r}$ can be obtained from
$ds^{2}=0$ \cite{10,11}.

  The metric of a four-dimensional rotating black hole can be cast
in the form
$$
  ds^{2}=-g_{tt}(r,\theta)dt^{2}+g_{rr}(r,\theta)dr^{2}+
         g_{\theta\theta}(r,\theta)d\theta^{2}+
         g_{\phi\phi}(r,\theta)d\phi^{2}-2g_{t\phi}(r,\theta)dtd\phi
  \eqno{(8)}  $$
generally. For the tunneling of a rotating black hole, we can still
use the $s$-wave approximation, this is because for an observer at
infinity, the radiation of a rotating black hole is still
spherically symmetric. However, when a particle is tunneling through
the horizon of a rotating black hole, it will be dragged by the
rotation of the black hole. Thus, a tunneling particle will have
motion in the $\phi$ degree of freedom, i.e. $d\phi\neq 0$, which
means that in the calculation of the action of a tunneling particle
in formula (4), we need also to consider the contribution to the
action that comes from the motion on the $\phi$ degree of freedom,
as we can see in \cite{21}. Meanwhile, in the equation of the null
geodesic, we cannot set $d\phi=0$, thus, $\dot{r}$ cannot be
obtained from $ds^{2}=0$ conveniently.

  In order to eliminate the motion of $\phi$ degree of freedom
of a tunneling particle, we can choose a reference system that is
co-rotating with the black hole horizon. This can be realized
through the rotating coordinate transformation
$$
  \phi^{\prime}=\phi-\Omega_{h}t    ~~~~~~~~ \mbox{or} ~~~~~~~~
  \phi=\phi^{\prime}+\Omega_{h}t ~,
  \eqno{(9)}  $$
where $\Omega_{h}$ is the angular velocity of the event horizon of a
rotating black hole, which is a constant and is defined by
$$
  \Omega_{h}=\frac{g_{t\phi}}{g_{\phi\phi}}\bigg\vert_{r=r_{h}} ~.
  \eqno{(10)}  $$
In (10) and in the following, we use $r_{h}$ to represent the radius
of the event horizon of a rotating black hole. In such a co-rotating
reference system, the observers located at the horizon cannot
observe the rotation of the black hole, they will find that the
angular velocity $\Omega_{h}^{\prime}$ of the black hole is zero.
Because the tunneling of a particle takes place at the horizon, it
will not be dragged by the rotation of the black hole to observe
from such a co-rotating reference system. Therefore we have
$d\phi^{\prime}=0$ for a tunneling particle, i.e., a tunneling
particle has no motion in the $\phi^{\prime}$ degree of freedom.
This makes us be able to use equation (4) to calculate the action.
Meanwhile, in obtaining the expression of $\dot{r}$ from the
null-geodesic method, we can set $d\phi^{\prime}=0$.

  Under the coordinate transformation (9), the metric (8) turns to
the form
$$
  ds^{2}=-G_{tt}(r,\theta)dt^{2}+g_{rr}(r,\theta)dr^{2}+
         g_{\theta\theta}(r,\theta)d\theta^{2}+
         g_{\phi\phi}(r,\theta)d\phi^{\prime ~ 2}-
         2g_{t\phi}^{\prime}(r,\theta)dtd\phi^{\prime} ~,
  \eqno{(11)}  $$
where
\setcounter{equation}{11}
\begin{eqnarray}
    G_{tt} & = & g_{tt}+2g_{t\phi}\Omega_{h}-g_{\phi\phi}
      \Omega_{h}^{2} ~,           \\
  g_{t\phi}^{\prime} & = & g_{t\phi}-\Omega_{h}g_{\phi\phi} ~.
\end{eqnarray}
Because of (10), we have
$$
  g_{t\phi}^{\prime}\vert_{r=r_{h}}=0 ~.
  \eqno{(14)}  $$
This also indicates $\Omega_{h}^{\prime}=
g_{t\phi}^{\prime}/g_{\phi\phi}\vert_{r=r_{h}}=0$. On the other
hand, according to (A.5), we have
$$
  G_{tt}\big\vert_{r=r_{h}}=0 ~.
  \eqno{(15)}$$
The horizon's radius of the metric (11) is determined by
$g^{rr}\vert_{r=r_{h}}=g_{rr}^{-1}\vert_{r=r_{h}}=0$, which is the
same equation of the horizon's radius of the metric (8), thus, the
horizon's radius of a rotating black hole will not be changed under
the coordinate transformation (9). On the other hand, because of
(15), the horizon's radius for the metric (11) is also determined by
(15).

  Because in metric (11), $g_{rr}$ is singular on the horizon, in
order to calculate the action of a tunneling particle, we need to
eliminate such a coordinate singularity first. This can be realized
through the Painlev{\'e} coordinate transformation \cite{10,11}. We
use $T$ to represent the Painlev{\'e} time coordinate and make a
coordinate transformation
$$
  dt=dT-\sqrt{\frac{g_{rr}(r,\theta_{0})-1}
     {G_{tt}(r,\theta_{0})}}dr
  \eqno{(16)}$$
to the metric (11). In (16), like that in \cite{15} in studying the
tunneling from Kerr-Newman black hole, we have set $\theta$ to be a
constant in order to make the coordinate transformation integrable.
Such a manipulation is reasonable because for a tunneling particle
in terms of an $s$-wave, it satisfies $d\theta=0$, therefore we can
consider the tunneling of a particle at a constant angle
$\theta_{0}$. At last we can obtain that the physical result does
not depend on the angle $\theta_{0}$. However, the explicit integral
of (16) is not needed to be given here. Under the above coordinate
transformation, for the metric (11), we have
\setcounter{equation}{16}
\begin{eqnarray}
  ds^{2} & = & -G_{tt}(r,\theta_{0})dT^{2}+2\sqrt{G_{tt}
     (r,\theta_{0})}\sqrt{g_{rr}(r,\theta_{0})-1} ~ drdT+dr^{2}
      +g_{\phi\phi}(r,\theta_{0})d\phi^{\prime ~ 2}   \nonumber  \\
     & ~ &  -2g_{t\phi}^{\prime}(r,\theta_{0})d\phi^{\prime}
      \bigg(dT-\sqrt{\frac{g_{rr}(r,\theta_{0})-1}
      {G_{tt}(r,\theta_{0})}}dr\bigg) ~.
\end{eqnarray}
The horizon's radius for the metric (17) is determined by
$G_{tt}\big\vert_{r=r_{h}}=0$, thus, the horizon's radius for the
metric (11) is not changed after the coordinate transformation (16).
As mentioned above, for a tunneling particle in the co-rotating
reference system, it satisfies $d\phi^{\prime}=0$. Thus we have
$$
  ds^{2}=-G_{tt}(r,\theta_{0})dT^{2}+2\sqrt{G_{tt}(r,\theta_{0})}
    \sqrt{g_{rr}(r,\theta_{0})-1} ~ drdT+dr^{2} ~.
  \eqno{(18)}  $$
To suppose that the mass of the tunneling particle is zero, then its
motion is determined by the null-geodesic equation $ds^{2}=0$. To
solve this equation, we obtain
$$
  \dot{r}=\sqrt{G_{tt}(r,\theta_{0})\cdot g_{rr}(r,\theta_{0})}
    \bigg(\pm 1-\sqrt{1-\frac{1}{g_{rr}(r,\theta_{0})}} ~ \bigg) ~.
  \eqno{(19)}  $$
Because $G_{tt}\big\vert_{r=r_{h}}=0$,
$g_{rr}^{-1}\vert_{r=r_{h}}=0$, $r_{h}$ is a simple zero point of
$G_{tt}$ and $g_{rr}^{-1}$, $G_{tt}\cdot g_{rr}$ should be regular
at the horizon. The plus and minus signs in (19) correspond to
outgoing and ingoing radial null geodesics respectively. For an
outgoing tunneling particle, $\dot{r}$ is positive, we have
$$
  \dot{r}=\sqrt{G_{tt}(r,\theta_{0})\cdot g_{rr}(r,\theta_{0})}
    \bigg(1-\sqrt{1-\frac{1}{g_{rr}(r,\theta_{0})}} ~ \bigg) ~.
  \eqno{(20)}  $$

  To substitute (20) into (7), we obtain
$$
  \mbox{Im}{\cal I}=\mbox{Im}\int_{0}^{E}
      \int^{r_{h}(M)}_{r_{h}(M-E)}\frac{dr}
     {\sqrt{G_{tt}(r,\theta_{0})\cdot g_{rr}(r,\theta_{0})}
      \Big(1-\sqrt{1-\frac{1}{g_{rr}(r,\theta_{0})}}
       ~ \Big)} ~ d\omega ~.
  \eqno{(21)}  $$
To multiply $1+\sqrt{1-\frac{1}{g_{rr}(r,\theta_{0})}}$ in the
numerator and denominator of the integrand at the same time, we
obtain
$$
  \mbox{Im}{\cal I}=\mbox{Im}\int_{0}^{E}
      \int^{r_{h}(M)}_{r_{h}(M-E)}\frac
      {1+\sqrt{1-\frac{1}{g_{rr}(r,\theta_{0})}}}
      {\sqrt{G_{tt}(r,\theta_{0})\cdot g_{rr}(r,\theta_{0})} ~
      \frac{1}{g_{rr}(r,\theta_{0})}} ~ drd\omega ~.
  \eqno{(22)}  $$
For the metric of a four-dimensional rotating black hole, because
$g_{rr}$ is singular on the horizon, generally, we can write
$g_{rr}$ in the form
$$
  g_{rr}(r,\theta)=\frac{C(r,\theta)}{r-r_{h}} ~,
  \eqno{(23)}  $$
where $C(r,\theta)$ is a function regular on the horizon. To
substitute (23) into (22), we have
$$
  \mbox{Im}{\cal I}=\mbox{Im}\int_{0}^{E}
      \int^{r_{h}(M)}_{r_{h}(M-E)}\frac
      {1+\sqrt{1-\frac{r-r_{h}}{C(r,\theta_{0})}}}
      {\sqrt{G_{tt}(r,\theta_{0})\cdot g_{rr}(r,\theta_{0})} ~
      \frac{r-r_{h}}{C(r,\theta_{0})}} ~ drd\omega ~.
  \eqno{(24)}  $$
In (24), $r_{h}$ is a simple pole of the integrand. To add a small
imaginary part to the variable $r$, and to let the integral path
round the pole in a semicircle, the integral of $dr$ can be
evaluated which results
$$
  \mbox{Im}{\cal I}=2\pi\int_{0}^{E}\frac{C(r_{h},\theta_{0})}
    {\sqrt{G_{tt}(r_{h},\theta_{0})\cdot g_{rr}(r_{h},\theta_{0})}}
     ~ d\omega ~.
  \eqno{(25)}  $$
It is reasonable to suppose that the energy $E$ of the tunneling
particle is far less than the total mass $M$ of the black hole, i.e.
$E<<M$, thus, in (25), the integrand can be treated as a constant.
Therefore we obtain
$$
  \mbox{Im}{\cal I}=2\pi E\frac{C(r_{h},\theta_{0})
    \sqrt{g^{rr}(r_{h},\theta_{0})}}
    {\sqrt{G_{tt}(r_{h},\theta_{0})}} ~.
  \eqno{(26)}  $$

  Because $G_{tt}(r_{h},\theta)=0$, $g^{rr}(r_{h},\theta)=0$, near
the horizon, we can expand $G_{tt}(r,\theta_{0})$ and
$g^{rr}(r,\theta_{0})$ in the form
$$
  G_{tt}(r,\theta_{0})=G_{tt}^{\prime}(r_{h},\theta_{0})
    (r-r_{h})+\ldots ~,
  \eqno{(27)}  $$
$$
  g^{rr}(r,\theta_{0})=g^{rr\prime}(r_{h},\theta_{0})
    (r-r_{h})+\ldots ~,
  \eqno{(28)}  $$
where in (27) and (28), $\ldots$ represents high order terms of
$(r-r_{h})$. From (23), we have
$$
  g^{rr\prime}(r_{h},\theta_{0})=\frac{1}{C(r_{h},\theta_{0})} ~.
  \eqno{(29)}  $$
To substitute (27)--(29) into (26), we obtain
$$
  \mbox{Im}{\cal I}=\frac{2\pi E}{\sqrt{G_{tt}^{\prime}
     (r_{h},\theta_{0})g^{rr\prime}(r_{h},\theta_{0})}} ~.
  \eqno{(30)}  $$
To substitute (30) into (1), we can see that the tunneling rate can
be cast in the form of (2), which is the Boltzmann distribution, and
we obtain
$$
  \mbox{Im}{\cal I}=\frac{\pi E}{\kappa(r_{h})} ~.
  \eqno{(31)}  $$
To compare (31) with (30), we obtain
$$
  \kappa(r_{h})=\frac{\sqrt{G_{tt}^{\prime}(r_{h},\theta_{0})
     g^{rr\prime}(r_{h},\theta_{0})}}{2} ~.
  \eqno{(32)}  $$
Thus, we obtain the thermal temperature of a four-dimensional
rotating black hole
$$
  T_{H}=\frac{\sqrt{G_{tt}^{\prime}(r_{h},\theta_{0})
     g^{rr\prime}(r_{h},\theta_{0})}}{4\pi} ~.
  \eqno{(33)}  $$
Equation (33) is derived from the tunneling approach. On the other
hand, in Appendix A, we have derived a formula (A.12) for the
surface gravity of a four-dimensional rotating black hole from black
hole thermodynamics which is given by
$$
  \kappa(r_{h})=\lim_{r\rightarrow r_{h}}
     \frac{\partial_{r}\sqrt{G_{tt}}}{\sqrt{g_{rr}}}=
     \lim_{r\rightarrow r_{h}}
     \frac{\partial_{r}G_{tt}}{2\sqrt{G_{tt}\cdot g_{rr}}} ~.
  \eqno{(34)}$$
From black hole thermodynamics \cite{28,29}, we know that on the
horizon, $\kappa(r_{h})$ is a constant, therefore we can evaluate it
at an arbitrary angle $\theta_{0}$. To substitute (27) and (28) into
(34), we obtain
$$
  \kappa(r_{h})=\frac{\sqrt{G_{tt}^{\prime}(r_{h},\theta_{0})
     g^{rr\prime}(r_{h},\theta_{0})}}{2} ~.
  \eqno{(35)}  $$
To compare (32) with (35), we can see that they are equivalent.
Because $\kappa(r_{h})$ is a constant on the horizon, the explicit
result for the surface gravity of a rotating black hole obtained
from (35) will not depend on the parameter $\theta_{0}$. This means
that in (32) and (33), the explicit results for the surface gravity
and Hawking temperature of a rotating black hole will not depend on
the parameter $\theta_{0}$ either.

\section{Hawking temperature of Kerr-Newman-AdS black hole}

\indent

  In this section, we derive the Hawking temperature of the
Kerr-Newman-AdS black hole using (33) of Section 2. In the
Boyer-Lindquist coordinates, the metric of the Kerr-Newman-AdS is
given by \cite{30}
\setcounter{equation}{35}
\begin{eqnarray}
  ds^{2}= & - & \!\! \frac{1}{\Sigma}[\Delta_{r}-
     \Delta_{\theta}a^{2}\sin^{2}\theta]dt^{2}+
     \frac{\Sigma}{\Delta_{r}}dr^{2}+
     \frac{\Sigma}{\Delta_{\theta}}d\theta^{2}+
     \frac{1}{\Sigma\Xi^{2}}[\Delta_{\theta}(r^{2}+a^{2})^{2}
     \nonumber       \\
    & ~ & -\Delta_{r}a^{2}\sin^{2}\theta]\sin^{2}\theta d\phi^{2}
    -\frac{2a}{\Sigma\Xi}[\Delta_{\theta}(r^{2}+a^{2})-
     \Delta_{r}]\sin^{2}\theta dtd\phi ~,
\end{eqnarray}
where
$$
  \Sigma=r^{2}+a^{2}\cos^{2}\theta ~,   ~~~~~~
  \Xi=1+\frac{1}{3}\Lambda a^{2} ~,
  \eqno{(37)}  $$
$$
  \Delta_{\theta}=1+\frac{1}{3}\Lambda a^{2}\cos^{2}\theta ~,
   ~~~~~~ \Delta_{r}=(r^{2}+a^{2})
  \big(1-\frac{1}{3}\Lambda r^{2}\big)-2Mr+Q^{2} ~,
  \eqno{(38)}  $$
$\Lambda$ is the cosmological constant, $\Lambda<0$. The horizons of
the metric (36) are determined by
\setcounter{equation}{38}
\begin{eqnarray}
  \Delta_{r} & = & (r^{2}+a^{2})\big(1-\frac{1}{3}
     \Lambda r^{2}\big)-2Mr+Q^{2}     \nonumber     \\
     & = & -\frac{1}{3}\Lambda\bigg[r^{4}-\Big(\frac{3}{\Lambda}
       -a^{2}\Big)r^{2}+\frac{6M}{\Lambda}r-\frac{3}{\Lambda}
       (a^{2}+Q^{2})\bigg]     \nonumber     \\
     & = & -\frac{1}{3}\Lambda(r-r_{++})(r-r_{--})
     (r-r_{+})(r-r_{-})=0 ~.
\end{eqnarray}
The equation $\Delta_{r}=0$ has four roots \cite{31}, where $r_{++}$
and $r_{--}$ are a pair of complex conjugate roots, $r_{+}$ and
$r_{-}$ are two real positive roots, and we suppose $r_{+}>r_{-}$.
Thus, $r=r_{+}$ is the event horizon. We first calculate the Hawking
temperature at a special value of $\theta_{0}$ and we choose
$\theta_{0}=0$. To expand $g^{rr}(r,\theta_{0}=0)$ near the event
horizon $r_{h}=r_{+}$, we obtain
$$
  g^{rr}(r,\theta_{0}=0)=\frac{-\frac{1}{3}\Lambda(r_{+}-r_{++})
    (r_{+}-r_{--})(r_{+}-r_{-})}{r_{+}^{2}+a^{2}}(r-r_{+})+\ldots ~,
  \eqno{(40)}  $$
where $\ldots$ are high order terms of $(r-r_{+})$. $G_{tt}$ is
defined by (12). We can rewrite it in the form
$$
  G_{tt}=g_{tt}+g_{t\phi}\Omega_{h}+
    (g_{t\phi}-\Omega_{h}g_{\phi\phi})\Omega_{h}
    =g_{tt}+g_{t\phi}\Omega_{h}+g_{t\phi}^{\prime}\Omega_{h}
  \eqno{(41)}  $$
generally. According to (14), $g_{t\phi}^{\prime}$ is zero on the
horizon, thus the last term of (41) does not need to be considered
when we expand $G_{tt}(r,\theta_{0})$ near the horizon. The angular
velocity of the Kerr-Newman-AdS black hole defined by (10) is
$\Omega_{h}=\frac{a\Xi}{r_{+}^{2}+a^{2}}$. At $\theta_{0}=0$,
$G_{tt}(r,\theta_{0}=0)$ can be expanded as
$$
  G_{tt}(r,\theta_{0}=0)=\frac{-\frac{1}{3}\Lambda(r_{+}-r_{++})
    (r_{+}-r_{--})(r_{+}-r_{-})}{r_{+}^{2}+a^{2}}(r-r_{+})+\ldots ~,
  \eqno{(42)}  $$
where $\ldots$ are high order terms of $(r-r_{+})$. To compare (42)
and (40) with (27) and (28), we can obtain
$G_{tt}^{\prime}(r_{+},\theta_{0}=0)$ and
$g^{rr\prime}(r_{+},\theta_{0}=0)$. To substitute
$G_{tt}^{\prime}(r_{+},\theta_{0}=0)$ and
$g^{rr\prime}(r_{+},\theta_{0}=0)$ into (33), we obtain, for the
Kerr-Newman-AdS black hole,
$$
  T_{H}=-\frac{\Lambda}{12\pi(r_{+}^{2}+a^{2})}(r_{+}-r_{++})
      (r_{+}-r_{--})(r_{+}-r_{-}) ~.
  \eqno{(43)}  $$
Because $\Lambda<0$, $r_{+}$ and $r_{-}$ are positive,
$r_{+}>r_{-}$, $r_{++}$ and $r_{--}$ are complex conjugate, these
make sure that $T_{H}$ is positive. At an arbitrary value of
$\theta_{0}$, through explicit calculation, $g^{rr}(r,\theta_{0})$
and $G_{tt}(r,\theta_{0})$ can be expanded as
\setcounter{equation}{43}
\begin{eqnarray}
  g^{rr}(r,\theta_{0}) & = & \frac{-\frac{1}{3}\Lambda
    (r_{+}-r_{++})(r_{+}-r_{--})(r_{+}-r_{-})}{r_{+}^{2}+a^{2}
      \cos^{2}\theta_{0}}(r-r_{+})+\ldots ~,                \\
  G_{tt}(r,\theta_{0}) & = & \frac{-\frac{1}{3}
    \Lambda(r_{+}-r_{++})(r_{+}-r_{--})
    (r_{+}-r_{-})(r_{+}^{2}+a^{2}\cos^{2}\theta_{0})}
    {(r_{+}^{2}+a^{2})^{2}}(r-r_{+})+\ldots ~.
\end{eqnarray}
To compare (45) and (44) with (27) and (28), we can
obtain $G_{tt}^{\prime}(r_{+},\theta_{0})$ and
$g^{rr\prime}(r_{+},\theta_{0})$. To substitute
$G_{tt}^{\prime}(r_{+},\theta_{0})$ and
$g^{rr\prime}(r_{+},\theta_{0})$ into (33), we obtain again
$$
  T_{H}=-\frac{\Lambda}{12\pi(r_{+}^{2}+a^{2})}(r_{+}-r_{++})
      (r_{+}-r_{--})(r_{+}-r_{-}) ~.
  \eqno{(46)}  $$
In \cite{32}, another expression for the Hawking temperature of the
Kerr-Newman-AdS black hole has been obtained which is given by
$$
  T_{H}=\frac{3r_{+}^{4}+(a^{2}+l^{2})r_{+}^{2}-l^{2}(a^{2}+Q^{2})}
       {4\pi l^{2}r_{+}(r_{+}^{2}+a^{2})} ~,
  \eqno{(47)}  $$
where $\Lambda=-3/l^{2}$. It is not difficult to verify that these
two expressions of $T_{H}$ for the Kerr-Newman-AdS black hole are
equivalent. The result of (46) is also equal to that obtained from
(A.13) and (A.14). From this example, we can also see that the
explicit result of the Hawking temperature given by (33) does not
depend on the parameter $\theta_{0}$.

  In the case $\Lambda=0$, the metric (36) degenerates to the metric
of four-dimensional Kerr-Newman black hole. If the charge is zero,
the metric will be the Kerr black hole. Following the same approach
as above, we can also obtain their Hawking temperature from
tunneling.

\section{Discussion}

\indent

  In this paper, we have studied the Hawking radiation of general
four-dimensional rotating black holes using the tunneling method of
Parikh and Wilczek \cite{10,11}. We obtain that the tunneling rate
of a zero-mass particle is given by
$$
  \Gamma=\Gamma_{0}\exp(-\beta E)=\Gamma_{0}\exp(-E/T_{H}) ~,
  \eqno{(48)}  $$
which is just the Boltzmann distribution. The thermal temperature
$T_{H}$ of a four-dimensional rotating black hole is given by (33),
which is in accordance with the Hawking temperature derived from
black hole thermodynamics. And we have given the explicit result for
the Hawking temperature of the Kerr-Newman-AdS black hole from the
tunneling approach. In order to eliminate the motion of $\phi$
degree of freedom of a tunneling particle from a rotating black
hole, we choose a reference system that is co-rotating with the
black hole horizon. In such a co-rotating reference system, we
avoided the dimension degeneration in the method of dragging
coordinate system adopted in \cite{21} for the tunneling of a
rotating black hole.

  It is necessary to point out that if we use the method of \cite{21}
to calculate the action of a tunneling particle directly for the
Kerr-Newman-AdS black hole, then we need to consider the action that
comes from the motion on the $\phi$ degree of freedom, the
calculation will be rather complicated in this case. In order to
simplify the calculation, we have made a rotating coordinate
transformation first. At the same time, the method provided in this
paper is general for a general four-dimensional rotating black hole.
And then we applied our result to the special case of the
Kerr-Newman-AdS black hole. Another point needed to point out here
is that there are some overlaps between the approach of this paper
and the manipulation of the tunneling from the Kerr-Newman black
hole in \cite{15} using the null-geodesic method. The difference
lies in that in \cite{15} the rotating coordinate transformation for
the tunneling of a rotating black hole was not proposed clearly, and
it has not been used to a general four-dimensional rotating black
hole. While in this paper, we have studied the Hawking temperature
of a general four-dimensional rotating black hole from tunneling
using the rotating coordinate transformation clearly, and then we
applied our result to the special case of the Kerr-Newman-AdS black
hole. An alternative method for the calculation of the action of a
tunneling particle was proposed in \cite{14} from the
Hamilton--Jacobi equation approach. Such a method was applied to the
tunneling of some rotating black holes in \cite{14,15}. For the
tunneling of the Kerr-Newman-AdS black hole, to use the
Hamilton--Jacobi equation method of \cite{14}, we can also make a
rotating coordinate transformation first to simplify the
calculation. The same results of (33) and (46) will be obtained at
last. However, limited by the length of this paper, we will not give
such a derivation further in this Letter.

  The tunneling rate (48) and Hawking temperature (33) for a rotating
black hole are obtained in the reference system co-rotating with the
black hole horizon. However, because the obtained tunneling rate and
Hawking temperature of a black hole are scalars, they will not
change for an observer static relatively to infinity. Thus, we can
deduce that for an observer static relatively to infinity, the
tunneling rate and Hawking temperature of a four-dimensional
rotating black hole are still given by (48) and (33). The difference
lies in that, for a tunneling particle, or an observer, the angular
velocity of a rotating black hole is zero in the co-rotating
reference system, while it is $\Omega_{h}$ in the static reference
system. To combine the first law of black hole thermodynamics, we
can generalize the tunneling rate (48) to a particle with non-zero
angular momentum and non-zero charge.

\vskip 1cm

\noindent{\bf \Large Acknowledgements}

\indent

  The author is grateful very much for the comments made by the
referee which have improved the contents of this paper.

\vskip 2cm

\noindent{\bf \Large Appendix A. Hawking temperature of
four-dimensional rotating black holes from black hole
thermodynamics}

\indent

  In this appendix, we give an expression for the Hawking temperature
of a four-dimensional rotating black hole from black hole
thermodynamics. The metric of a four-dimensional rotating black hole
is given by (8) generally. For the metric (8), there exists the
Killing field
$$
  \xi^{\mu}=\frac{\partial}{\partial t}+
       \Omega_{h}\frac{\partial}{\partial \phi} ~,
  \eqno{({\rm A}.1)}$$
where $\Omega_{h}$ is the angular velocity of the horizon which is a
constant. Here, we mean that the horizon is the outer horizon for a
rotating black hole. Because the horizon is a null surface and
$\xi^{\mu}$ is normal to the horizon, we have on the horizon
\cite{28}
$$
  \xi^{\mu}\xi_{\mu}\vert_{r=r_{h}}=0 ~.
  \eqno{({\rm A}.2)}$$
For the metric (8), we have
$$
  \xi^{\mu}\xi_{\mu}=g_{tt}+2g_{t\phi}\Omega_{h}
     -g_{\phi\phi}\Omega_{h}^{2} ~.
  \eqno{({\rm A}.3)}$$
Here, we have defined that the square of the norm of the Killing
field is positive outside the horizon, at least for the case
$\Omega_{h}=0$. As in (12), we define
$$
    G_{tt}=g_{tt}+2g_{t\phi}\Omega_{h}-g_{\phi\phi}
      \Omega_{h}^{2} ~.
  \eqno{({\rm A}.4)}$$
Thus we have
$$
    G_{tt}\big\vert_{r=r_{h}}=0 ~.
  \eqno{({\rm A}.5)}$$
Following \cite{28} we write
$$
  \xi^{\mu}\xi_{\mu}=-\lambda^{2} ~,
  \eqno{({\rm A}.6)}$$
where $\lambda$ is a scalar function, and it is a constant on the
horizon. According to (A.3), we have $\lambda^{2}=-G_{tt}$ for the
metric (8) of a four-dimensional rotating black hole. Let
$\nabla^{\mu}$ represent the covariant derivative operator, thus
$\nabla^{\mu}(\xi^{\nu}\xi_{\nu})$ is also normal to the horizon.
Then, according to \cite{28,29}, there exists a function $\kappa$
satisfying the equation
$$
  \nabla^{\mu}(-\lambda^{2})=-2\kappa\xi^{\mu} ~,
  \eqno{({\rm A}.7)}$$
where on the horizon $\kappa(r_{h})$ is a constant and is just the
horizon's surface gravity.

  Similarly, we have the lower index equation
$$
  \nabla_{\mu}(-\lambda^{2})=-2\kappa\xi_{\mu} ~.
  \eqno{({\rm A}.8)}$$
Thus, from (A.7) and (A.8) we have
$$
  \nabla^{\mu}(-\lambda^{2})\nabla_{\mu}(-\lambda^{2})
   =-4\kappa^{2}\lambda^{2} ~.
  \eqno{({\rm A}.9)}$$
Because $\lambda^{2}$ is a scalar function, $\kappa^{2}$ is also a
scalar function. Therefore the surface gravity of a black hole
horizon is invariant under general coordinate transformations,
including the rotation of (9). From (A.3), (A.4), (A.6), (A.9), and
the axial symmetry of the metric, we obtain
$$
  4\kappa^{2}G_{tt}=g^{rr}(\partial_{r}G_{tt})^{2}
         + g^{\theta\theta}(\partial_{\theta}G_{tt})^{2} ~.
  \eqno{({\rm A}.10)}$$
Because of (A.5), we have
$$
  \lim_{r\rightarrow r_{h}}\partial_{\theta}G_{tt}=0 ~.
  \eqno{({\rm A}.11)}$$
Therefore, to take the limit $r\rightarrow r_{h}$ in both sides of
(A.10) yields
$$
  \kappa(r_{h})=\lim_{r\rightarrow r_{h}}
     \frac{\partial_{r}\sqrt{G_{tt}}}{\sqrt{g_{rr}}} ~.
  \eqno{({\rm A}.12)}$$
In (A.12), because $G_{tt}$ is zero on the horizon, the partial
derivative is taken before the limit. Thus, the Hawking temperature
of the metric (8) is given by
$$
  T_{H}=\frac{\kappa(r_{h})}{2\pi}=\lim_{r\rightarrow r_{h}}
     \frac{\partial_{r}\sqrt{G_{tt}}}{2\pi\sqrt{g_{rr}}} ~.
  \eqno{({\rm A}.13)}$$
Because $\kappa(r_{h})$ is a constant on the horizon \cite{28,29},
it can be evaluated at an arbitrary $\theta$. For convenience, it
can be evaluated at $\theta=0$ usually. For the metrics of many
four-dimensional rotating black holes, we can see that usually they
satisfy $G_{tt}\vert_{\theta=0}=g_{tt}\vert_{\theta=0}$. Thus we can
write
$$
  T_{H}=\frac{\kappa(r_{h})}{2\pi}=
     \lim_{\theta=0, ~ r\rightarrow r_{h}}
     \frac{\partial_{r}\sqrt{g_{tt}}}{2\pi\sqrt{g_{rr}}} ~.
  \eqno{({\rm A}.14)}$$
On the other hand, because $\kappa(r_{h})$ is a constant on the
horizon, this means that, in formula (A.13), the dependence of
$T_{H}$ on the variable $\theta$ is only apparent.

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