Slinky Inflation
We present a new approach to quintessential inflation, in which both dark energy and inflation are explained by the evolution of a single scalar field. We start from a simple scalar potential with both oscillatory and exponential behavior.We employ the conventional reheating mechanism of new inflation, in which the scalar decays to light fermions with a decay width that is proportional to the scalar mass. Because our scalar mass is proportional to the Hubble rate, this gives adequate reheating at early times while shutting off at late times to preserve quintessence and satisfy nucleosynthesis constraints. We discuss a simple model which solves the horizon, flatness, and ``why now'' problems. Without any additional tuning of parameters, this model satisfies all constraints from CMB, large scale structure, and nucleosynthesis. The predictions for the inflationary spectral indices are $n_S = n_T = 1$. In this model we are currently beginning the third cosmic epoch of accelerated expansion.
slinkyf.tex
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\begin{document}

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\title{Slinky Inflation}

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\author{Gabriela Barenboim}
%\homepage[]{Your web page}
%\thanks{}
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\affiliation{Departament de F\'{\i}sica Te\`{o}rica,
 Universitat de Val\`{e}ncia, 
Carrer Dr.~Moliner 50, E-46100 Burjassot (Val\`{e}ncia),  ��Spain�}

\author{Joseph Lykken}
\affiliation{Theoretical Physics Department, Fermi National 
Accelerator Laboratory,\\ P.O.\ Box 500, Batavia, Illinois 60510 USA}

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\begin{abstract}
We present a new approach to quintessential inflation, in
which both dark energy and inflation are explained by
the evolution of a single scalar field. We
start from a simple scalar potential with both oscillatory
and exponential behavior.We employ the conventional reheating 
mechanism of new inflation, in which the
scalar decays to light fermions with a decay width that is
proportional to the scalar mass. Because our scalar mass is
proportional to the Hubble rate, this gives
adequate reheating at early times while shutting off at late
times to preserve quintessence and satisfy nucleosynthesis
constraints.

We discuss a simple model which solves
the horizon, flatness, and ``why now'' problems. Without any
additional tuning of parameters, this model satisfies all
constraints from CMB, large scale structure, and nucleosynthesis.
The predictions for the 
inflationary spectral indices are $n_S = n_T = 1$.
In this model we are currently beginning the third cosmic epoch
of accelerated expansion. 

  
\end{abstract}

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%   98.80.Cq Particle-theory                                                    %
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%   98.70.Vc Background radiations                                              %
%   98.65.Dx Superclusters; large-scale structure of the Universe               %
%   95.35.+d Dark matter (stellar, interstellar, galactic, and cosmological)    %
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If we assume the correctness of the standard Friedmann equation
evolution, the existence of dark energy engenders two profound dilemmas.
The first is the cosmological constant problem, and the second is the
``why now?''\ problem. Quintessence models attempt to address the
second problem by introducing a very weakly coupled scalar field
whose potential and/or kinetic function have special properties.
One of the most successful approaches to 
quintessence \cite{Dodelson:2001fq,tracker}
is to combine tracking with an oscillating behavior in the quintessence
potential. In such models the equation of state parameter $w(z)$ has
a periodic component, leading to occasional periods of accelerated
expansion during epochs where $w(z)\simeq -1$.

It is natural in this context to ask whether the quintessence scalar
could replace the inflaton. The idea of quintessential
inflation has been examined by a number of 
authors \cite{Peebles:1998qn}-\cite{Sami:2004xk}.
The straightforward approach is to cobble together a
scalar potential which has both a flat, large vev portion
(for inflation) and a flat, small vev portion (for quintessence).
These features are connected by a steep step which corresponds
to a period of cosmic kination. As discussed 
in \cite{Dimopoulos:2001ix,Sami:2004xk}, such models suffer
from generic problems. First, they require significant
\textit{ad hoc} tuning to simultaneously produce the features
of inflation and quintessence. Second, they require a ``sterile''
inflaton, in order to avoid the decay of the putative
quintessence scalar at the end of
inflation. This in turn requires new mechanisms for reheating, such as
gravitational de Sitter phase particle production, leading
to difficulties in satisfying the constraints of CMB anisotropies
and of primordial nucleosynthesis.

\begin{figure}
    \includegraphics[width=7cm]{slinky.eps}%{figure 1} 
\vspace{2pt}
\includegraphics[width=7cm]{pot.eps}
\vspace{-5pt}
\caption{The upper (lower) panel shows the 
scalar potential $V(\theta )$ ($V(\phi )$) for $b=1$, plotted
on a logarithmic scale.}
\label{fig:slinky}
\end{figure}  
 
Our approach to quintessential inflation is to take advantage
of the tracking and oscillatory potential features that work
so well in addressing the ``why now'' problem of quintessence alone.
We will describe a model with a simple scalar lagrangian with exponential
and oscillatory features. The model uses the conventional reheating 
mechanism of new inflation \cite{Abbott:1982hn}, in which the
scalar decays to light fermions with a decay width that is
proportional to the scalar mass. We show that the scalar mass
is proportional to the Hubble rate. As a result, the model has
adequate reheating at early times while naturally shutting off at later
times.

We present a successful model with only three adjustable 
parameters. One parameter controls the period between 
inflationary epochs, a second controls the overall decay width,
and the third parametrizes our ignorance about the relative
fraction of matter versus radiation produced by reheating.
These three parameters are adjusted to produce sufficient
inflation along with the correct fractions
$\Omega_r/\Omega_{\Lambda}$, $\Omega_m/\Omega_{\Lambda}$ of
radiation, matter, and dark energy, as measured today.
Having thus fixed the model we find that we automatically
satisfy all constraints of primordial nucleosynthesis, CMB,
and large-scale structure.

\section{An oscillatory potential}
We start with a simple model with a single real
scalar quintessence field $\theta$. The action is
\begin{eqnarray}
\int d^4x \sqrt{-g} \left[
\frac{1}{2}f(\theta )g^{\mu\nu}\partial_{\mu}\theta
\partial_{\nu}\theta - V(\theta ) \right] \; ,
\label{eqn:piaction}
\end{eqnarray}  
where the kinetic function $f(\theta )$ and potential
$V(\theta )$ are given by:
\begin{eqnarray}\label{eqn:ourkin}
f(\theta ) &=& {3M_{\rm P}^2\over\pi b^2}\, {\rm sin}^2\,\theta\; ;\\
V(\theta ) &=& \rho_0\, {\rm cos}^2\,\theta\, {\rm exp}\left[{3\over b}\left(
2\theta - {\rm sin}\,2\theta \right) \right] \; ,
\label{eqn:ourpot} 
\end{eqnarray}
where $M_{\rm P}$ is the Planck mass: 
1.22 $\times 10^{19}$ GeV; $\rho_0$ is the dark energy
density observed today: $\simeq$ ($10^{-4}$ eV)$^4$; $b$ is a
dimensionless parameter which controls the periodic behavior.
A canonical kinetic term can be restored via a field redefinition
$\theta (x) \to \phi (x)$, where
\begin{eqnarray}
\phi (x) = \phi_0\, {\rm cos}\,\theta \; ,
\label{eqn:canonical}
\end{eqnarray}
with $\phi_0\equiv \sqrt{3M_{\rm P}^2/\pi b^2}$.

In the approximation where we ignore the energy density
of radiation and matter, 
and where the only friction is from the metric expansion,
the evolution of the model can be solved
analytically. Energy conservation requires:
\begin{eqnarray}
\dot{\rho}_{\theta} = -3H(1+w)\rho_{\theta} \; ,
\end{eqnarray}
where $H$ is the Hubble rate, $w$ is the equation of state
parameter, and $\rho_{\theta}$ is the dark energy density.
The solution to this equation as a function of 
the scale factor $a(t)$ is:
\begin{eqnarray}
\rho_{\theta}(a) = \rho_0 \, {\rm exp}\left[
-3\int_1^a {da\over a}(1+w(a)) \right] \; ,
\label{eqn:rhorel}
\end{eqnarray}
where $\rho_0$ is the dark energy density at $a=1$ (today).

We also know that
\begin{eqnarray}
V(\theta) = \frac{1}{2}(1-w)\rho_{\theta} \; .
\label{eqn:vrhorel}
\end{eqnarray}
Making the Ansatz 
\begin{eqnarray}
w(a) = -{\rm cos}\,2\theta (a) \; ,
\end{eqnarray}
one immediately gets a solution to the 
(flat) Friedmann
equation combined with the
relations (\ref{eqn:rhorel}-\ref{eqn:vrhorel}):
\begin{eqnarray}\label{eqn:ourtheta}
\theta (a) &=& -{b\over 2}\,{\rm ln}\,a \; ;\\
w(a) &=& -{\rm cos}\,[b\,{\rm ln}\,a] \; .
\label{eqn:ourw}
\end{eqnarray}

The expectation value
of the quintessence field $\theta$ evolves logarithmically with scale
factor from a positive initial value to zero today. 
The potential $V(\theta )$
has the ``Slinky'' form\footnote{A Slinky\texttrademark\ 
is a large spring with a very
weak spring constant. A Slinky will ``walk'' down a staircase, producing
in time-lapse a pattern resembling the top
panel of Figure \ref{fig:slinky}.}
shown in Figure \ref{fig:slinky}. 
Accelerated expansion
corresponds to epochs (such as today)
where $\theta$ is evolving through one of the
flat ``steps'' of the potential. From (\ref{eqn:ourkin}) we
see that the kinetic function is simultaneously suppressed
in this epochs, slowing the roll of the scalar field evolution.
Note also that our potential $V(\theta )\to 0$ as 
$\theta \to -\infty$, which corresponds to
$t\to\infty $; this is as desired for a quintessence model.

We can understand the same behavior by looking at the 
``canonical'' scalar
$\phi$ in (\ref{eqn:canonical}). The potential $V(\phi)$ 
resembles a series of descending ramps, with the value of $\phi$
bounded between $-\phi_0$ and $\phi_0$. Shifted around a generic vev
$\langle \phi \rangle$, the scalar has an effective mass-squared
given by
\begin{eqnarray}
{m_{\phi}^2\over H^2} = {b^2\over 4} + {3\over 2}b
\left[ {{\rm cos}^3\,\theta\over {\rm sin}\,\theta} -2{\rm sin}^2\,2\theta
\right]+{9\over 2}{\rm sin}^2\,2\theta\; ,
\label{eqn:meff}
\end{eqnarray}
where $H$ is the Hubble rate. Instead of a suppression
of the kinetic function when $\theta$ is a multiple of $\pi$,
the scalar $\phi$ has an effective mass that diverges when
$\phi = \pm\phi_0$, \textit{i.e.}, at each junction of the descending ramps.

The equation of state parameter $w(a)$ has the 
same periodic form assumed in two recent phenomenological 
analyses \cite{Barenboim:2004kz,Feng:2004ff}. The analysis 
in \cite{Barenboim:2004kz}
showed that, for $b=1$, $w(a)$ as given in (\ref{eqn:ourw})
is consistent with all current data from observations of
the CMB,
Type IA supernovae, and large scale structure.
It follows \textit{a fortiori} that our model with any choice
of $b$ less than one also agrees with this data.

To complete the model, we will assume that the quintessence
field $\phi$ has a weak perturbative coupling to light fermions.
This is the standard reheating mechanism of new 
inflation \cite{Abbott:1982hn}.
To avoid the strong constraints on long-range forces mediated 
by quintessence scalars \cite{Carroll:1998zi},
it is simplest to imagine that our
scalar only has a direct coupling to a sterile neutrino.
This is sufficient to hide the quintessence force from
Standard Model nonsinglet particles \cite{Fardon:2003eh},
while still allowing the 
generation of a radiative thermal bath of Standard Model
particles from quintessence decay.

With this assumption for reheating the evolution equations
for quintessence, radiation, and matter become:
\begin{eqnarray}
\dot{\rho}_{\theta} &=& -3H(1+w)\rho_{\theta} 
-k_0m_{\phi}(1+w)\rho_{\theta}
\; ;\nonumber\\
\dot{\rho}_r &=& -4H(1+w)\rho_r 
+(1\hspace*{-2pt}-\hspace*{-2pt}f_m)k_0m_{\phi}(1+w)\rho_{\theta}
\; ; \\
\dot{\rho}_m &=& -3H(1+w)\rho_m +f_mk_0m_{\phi}(1+w)\rho_{\theta}
\; ;\nonumber
\label{eqn:coupled}
\end{eqnarray}
where $k_0$ and $f_m$ are small dimensionless constants.
From (\ref{eqn:meff}) we see that, as long as $\theta$ is not near
a multiple of $\pi/2$, it is a reasonable approximation to make
the replacement
\begin{eqnarray}
k_0m_{\phi} \to kH \; ,
\label{eqn:makeghappy}
\end{eqnarray}
where $k$ is another small dimensionless parameter.
This replacement decouples the $\theta$ evolution equation
from the Friedmann equation, giving an immediate analytic
solution:
\begin{eqnarray}
\rho_{\theta}(a) = \rho_0 \, 
{\rm exp}\left[{1\over b}(3+k)\left(
2\theta - {\rm sin}\,2\theta \right) \right]
\; ,
\label{eqn:rhonew}
\end{eqnarray}
where $\theta (a)$ and $w(a)$ are still given by
(\ref{eqn:ourtheta})-(\ref{eqn:ourw}).

We have used this approximation in the solutions quoted
below. Clearly this approximation overestimates the
reheating effect for epochs where $\theta$ is close to an
odd multiple of $\pi/2$, however this is a small
effect since these are the epochs of maximum radiation
or matter domination. The approximation also breaks
down in the epochs of maximum inflation, \textit{i.e.}
when ${\rm sin}\,\theta \to 0$. However when
${\rm sin}\,\theta \to 0$ we also expect
strong-coupling physics to enter in a full
theory, providing a cutoff for the naive vanishing of the
kinetic function of $\theta$. Since we cannot compute
this effect without a full Planck-scale theory, we may as
well stay with the approximation (\ref{eqn:makeghappy}).

\begin{figure}
    \includegraphics[width=8cm]{history.eps}%{figure 2}    
\vspace*{-2pt}
\caption{History of a slinky
universe. Shown are the relative energy density
fractions in radiation (green), matter (blue), and 
dark energy (red), as a function of the logarithm
of the scale factor $a(t)$.}
\label{fig:history}
\end{figure}  

\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline\hline
log$_{10}$ $a $ & -42 & -40 & -38 & -36 & -34 & -32 \\ [0.5 ex]
\hline
$w(a)$ & -0.32 & -0.83 & -1 & -0.75 & -0.19 & 0.45 \\ \hline
$\Omega_\Lambda$ & 1 & 0.997 & 0.99998 & 0.996 &
0.975 & 0.78 \\ \hline
$\Omega_r$ & 0 & 0.003 & 2 10$^{-5}$ & 0.004 & 0.025 & 0.22 \\ \hline
$\Omega_m$ & 0 & 5 10$^{-14}$ & 3 10$^{-16}$ & 5 10$^{-14}$
& 5 10$^{-13}$ & 2 10$^{-11}$   \\\hline
log$_{10}$ $a $ & -30 & -28 & -26 & -24 & -22 & -20 \\\hline
$w(a)$ & 0.90 & 0.98 & 0.64 & 0.04 & -0.58 & -0.96 \\ \hline
$\Omega_\Lambda$ & 0.01 & 8 10$^{-7}$ & 4 10$^{-10}$ & 
2 10$^{-10}$ & 5 10$^{-7}$ & 0.062 \\\hline
$\Omega_r$ & 0.99 & 0.999999 & 0.99993 & 0.993 & 0.59 & 0.013 \\\hline
$\Omega_m$ & 7 10$^{-9}$ & 7 10$^{-7}$ & 7 10$^{-5}$
& 0.007 & 0.41 & 0.925   \\\hline
log$_{10}$ $a $ & -18 & -16 & -14 & -12 & -10 & -8 \\\hline
$w(a)$ & -0.94 & -0.52 & 0.11 & 0.69 & 0.99 & 0.87 \\ \hline 
$\Omega_\Lambda$ & 0.9992 & 0.99 & 0.94 & 0.32 & 0.0002
& 1 10$^{-8}$ \\\hline
$\Omega_r$ & 0.0008 & 0.01 & 0.06 & 0.68 & 0.9998 & 0.9999  \\\hline
$\Omega_m$ & 2 10$^{-5}$ & 6 10$^{-10}$ & 4 10$^{-11}$
& 6 10$^{-9}$ & 8 10$^{-7}$ & 8 10$^{-5}$ \\\hline
log$_{10}$ $a $ & -6 & -4 & -2 & 0 & 2 & 4 \\\hline
$w(a)$ & 0.39 & -0.24 & -0.79 & -1 & -0.79 & -0.25 \\ \hline
$\Omega_\Lambda$ & 1 10$^{-10}$ & 2 10$^{-9}$ & 
5 10$^{-6}$ & 0.67 & 0.997 & 0.98   \\\hline
$\Omega_r$ & 0.992 & 0.54 & 0.01 & 0.00005 & 0.003 & 0.02 \\\hline
$\Omega_m$ & 0.008 & 0.46 & 0.99 & 0.33 & 1 10$^{-6}$ & 8 10$^{-10}$ \\[1ex]
\hline\hline
\end{tabular}
\caption{The relative density fractions of dark energy, radiation,
and matter, as a function of the scale factor. The future projections
are only valid if the quintessence scalar is still decaying.}
\end{table}
\section{Results}

Figure \ref{fig:history} and Table 1 show the results obtained
from our model with $b=1/7$, $k=0.06$, and
$f_m = 10^{-11}$. Shown are the relative energy density
fractions in dark energy, radiation, and matter, as a function
of ${\rm log}\,a$. We have chosen to integrate the
evolution equations starting from $a= 10^{-42}$, which in
our model corresponds to a temperature of slightly
less than  $10^{16}$ GeV, and an initial comoving Hubble
radius of about 100 Planck lengths. 
For simplicity we
have also chosen the value of $w(a)$ now to be exactly $-1$. 
Neither of these choices corresponds to a necessary tuning.

Our three adjustable parameters have been chosen such
that the values of  
$\Omega_r/\Omega_{\Lambda}$, $\Omega_m/\Omega_{\Lambda}$
come out to their measured values at $a=1$, and such that
we have sufficient inflation. The latter is checked by
computing the ratio of the fully inflated size of the
intitial comoving Hubble radius to the current comoving
Hubble radius. This ratio is about 3 in our model, indicating
that the total amount of inflation is indeed enough to solve
the horizon problem. The flatness problem is solved because
the total $\Omega_r+\Omega_m+\Omega_{\Lambda} =1$ within
errors.

From Figure (\ref{fig:history}) we see that we are currently
beginning the third epoch of accelerated expansion. The first
epoch of inflation accumulated about 18 $e$-foldings. Quantum
fluctuations during this epoch produced the spatial inhomogeneities 
responsible for large scale structure and CMB anisotropies
observed today. 
Constraints on the physics responsible for the primodial power spectrum 
of these density fluctuations can be set with WMAP and 2dF data,
under the assumption that the Hubble rate is dominated by the
contribution from $\rho_{\phi}$
during the observable part of inflation \cite{Schwarz:2004tz}.
The scalar and tensor perturbations with comoving wavenumber $k$
are solutions of the equation
\begin{eqnarray}\label{eqn:perteq}
u_k^{\prime\prime} + \left( k^2 -{z^{\prime\prime}\over z}\right)u_k=0 \; ,
\end{eqnarray}
where prime denotes derivative with respect to conformal time $\tau$,
and $z = a\dot{\phi}/H$ for scalar perturbations while $z=a$ for the
tensor case. The power spectra have the form
\begin{eqnarray}\label{eqn:powsp}
{\cal P}_{\cal R}(k) ={H^2\over \epsilon}\,f_k\,;\quad
{\cal P}_h(k) = H^2\,f_k \,; \quad f_k = {k^3u_k^2\over a^2H^2} \; ,
\end{eqnarray}
where $\epsilon = -\dot{H}/H^2$.
For standard inflation models \cite{Stewart:1993bc}, the solutions of
(\ref{eqn:perteq}) have the property that $u_k^2 \to a^2H^2/k^3$  
for small wavenumbers $k/aH \to 0$, and thus $f_k \to $ constant.
This gives the standard formulae for the spectral indices
$n_S-1 = -4\epsilon -2\delta$ and $n_T-1 = -2\epsilon$, where
$\delta$ is defined by $\delta + \epsilon = \epsilon^{\prime}
/(2aH\epsilon )$.

In our model (\ref{eqn:perteq}) can be solved in the limit
$k/aH \to 0$
by changing variables from $\tau$ to $\theta$:
\begin{eqnarray}
&\hspace*{-10pt}
{d^2u_k\over d\theta^2}
-{2\over b}(1-3\,{\rm sin}^2\theta){du_k\over d\theta}
+c(\theta )u_k = 0 \; ;\\
&\hspace*{-10pt}
c(\theta ) = {6\over b}{\rm cot}\,\theta + 1 - {8\over b^2}
-{6\over b^2}({\rm cos}\,2\theta - 1) -{3\over b}{\rm sin}\, 2\theta \;; \\
&\hspace*{-10pt}
c(\theta ) = -{8\over b^2} + {12\over b^2}{\rm sin}^2\theta \; ,
\end{eqnarray}
where the first/second expression for $c(\theta )$ is for the 
scalar/tensor case. The solutions which match to
the correct large $k$ behavior are:
\begin{eqnarray}
u_k = {N\over \sqrt{k}}{a\over k}{\rm sin}\,\theta \; ;\quad
u_k = {N\over \sqrt{k}}{a\over k}\; ,
\end{eqnarray}
for the scalar and tensor cases respectively, where $N$ is a constant.

We can plug these solutions into the expressions
(\ref{eqn:powsp}) for the power spectra, using also that, for
our model, $\epsilon = 3\,{\rm sin}^2\theta$. Remarkably, one
finds that $n_S = n_T = 1$.

To compare with contraints from primordial nucleosynthesis,
we note that the second epoch of dark energy domination
ended well before $a \simeq 10^{-10}$, the time at which
nucleosynthesis occured. Indeed dark energy reheating effects
are completely negligible from $a \simeq 10^{-10}$ until today.
A possible exception is a small reheating effect occuring
at very late times in our current epoch of accelerated expansion;
this only occurs if the quintessence scalar is still managing to 
decay despite its incredibly small current mass of $\sim 10^{-36}$
eV. Even with such an effect, our model satisfies the constraints
of successful nucleosynthesis \cite{Bean:2002sm}. 

Figure (\ref{fig:history}) also shows one (very brief)
prior epoch of matter domination. This only occurs if there
is some suitably heavy and long-lived matter around to
go out of thermal equilibrium when $a \sim 10^{-22}$,
\textit{e.g.} a superheavy neutrino.

\section{Remarks}

It is fair to say that, compared to the simple model
presented above, the standard new inflation scenario looks
rather extreme. In the evolution history
portrayed in Figure \ref{fig:history}, the interplay between
inflationary expansion and reheating is much milder. In fact,
apart from a few very brief periods, reheating effects in
our model do not actually increase the temperature of the
thermal radiation bath. Instead, the temperature is almost
always decreasing, but it decreases more slowly than in
the standard $\Lambda$CDM evolution. 

Our particular realization of Slinky inflation must
be regarded as a toy model (pun intended),
since $\phi_0 > M_{\rm P}$ and
we have not invoked any consistent Planck scale framework
such as string theory. However the model itself
is surprisingly simple, and the physical picture 
which emerges from it has some compelling features.
These are worthy of further investigation. Also,
since the target space parametrized
by our scalar is $\mathbf{S^1}$, it would be
interesting to extend this scenario to a class of nonlinear
sigma models with other compact target spaces.



% If you have acknowledgments, this puts in the proper section head.
\begin{acknowledgments}
Fermilab is operated by Universities Research Association Inc.\ under
Contract No.\ DE-AC02-76CH03000 with the U.S.\ Department of Energy.
J.L. is grateful for the hospitality of Jos\'e Bernab\'eu
and the University of Valencia.
We thank Scott Dodelson, Olga Mena, Chris Quigg, Jos\'e Valle,
and Jochen Weller
for helpful discussions.
\end{acknowledgments}

% Create the reference section using BibTeX:
%\bibliography{basename of .bib file}
\begin{thebibliography}{99}

%\cite{Dodelson:2001fq}
\bibitem{Dodelson:2001fq}
S.~Dodelson, M.~Kaplinghat and E.~Stewart,
%``Tracking Oscillating Energy,''
Phys.\ Rev.\ Lett.\  {\bf 85}, 5276 (2000)
[arXiv:astro-ph/0002360].

\bibitem{tracker} P.J.E. Peebles \& B. Ratra, Phys. Rev. {\bf D37} 3406
(1988); I. Zlatev, L. Wang, \& P.J. Steinhardt, \prl {\bf 82}, 
896 (1998); P.J. Steinhardt, L. Wang, \& I. Zlatev, Phys. Rev. {\bf D59}, 
123504 (1999); A. Albrecht \& C. Skordis, astro-ph/9908085 (1999).

%\cite{Peebles:1998qn}
\bibitem{Peebles:1998qn}
P.~J.~E.~Peebles and A.~Vilenkin,
%``Quintessential inflation,''
Phys.\ Rev.\ D {\bf 59}, 063505 (1999)
[arXiv:astro-ph/9810509].

\bibitem{Peloso:1999dm}
  M.~Peloso and F.~Rosati,
  %``On the construction of quintessential inflation models,''
  JHEP {\bf 9912}, 026 (1999)
  [arXiv:hep-ph/9908271].

%\cite{Copeland:2000hn}
\bibitem{Copeland:2000hn}
  E.~J.~Copeland, A.~R.~Liddle and J.~E.~Lidsey,
  %``Steep inflation: Ending braneworld inflation by gravitational particle
  %production,''
  Phys.\ Rev.\ D {\bf 64}, 023509 (2001)
  [arXiv:astro-ph/0006421].

\bibitem{Kaganovich:2000fc}
  A.~B.~Kaganovich,
  %``Field theory model giving rise to 'quintessential inflation' without  the
  %cosmological constant and other fine tuning problems,''
  Phys.\ Rev.\ D {\bf 63}, 025022 (2001)
  [arXiv:hep-th/0007144].

%\cite{Dimopoulos:2001ix}
\bibitem{Dimopoulos:2001ix}
  K.~Dimopoulos and J.~W.~F.~Valle,
  %``Modeling quintessential inflation,''
  Astropart.\ Phys.\  {\bf 18}, 287 (2002)
  [arXiv:astro-ph/0111417].

\bibitem{Majumdar:2001mm}
A.~S.~Majumdar,
%``From brane assisted inflation to quintessence through a single scalar
%field,''
Phys.\ Rev.\ D {\bf 64}, 083503 (2001)
[arXiv:astro-ph/0105518].

%\cite{Huey:2001ae}
\bibitem{Huey:2001ae}
G.~Huey and J.~E.~Lidsey,
%``Inflation, braneworlds and quintessence,''
Phys.\ Lett.\ B {\bf 514}, 217 (2001)
[arXiv:astro-ph/0104006].

%\cite{Sami:2004xk}
\bibitem{Sami:2004xk}
M.~Sami and V.~Sahni,
%``Quintessential inflation on the brane and the relic gravity wave
%background,''
Phys.\ Rev.\ D {\bf 70}, 083513 (2004)
[arXiv:hep-th/0402086].

%\cite{Abbott:1982hn}
\bibitem{Abbott:1982hn}
  L.~F.~Abbott, E.~Farhi and M.~B.~Wise,
  %``Particle Production In The New Inflationary Cosmology,''
  Phys.\ Lett.\ B {\bf 117}, 29 (1982).
  %%CITATION = PHLTA,B117,29;%%

%\cite{Barenboim:2004kz}
\bibitem{Barenboim:2004kz}
  G.~Barenboim, O.~Mena and C.~Quigg,
  %``Undulant universe,''
  arXiv:astro-ph/0412010.

%\cite{Feng:2004ff}
\bibitem{Feng:2004ff}
B.~Feng, M.~Li, Y.~S.~Piao and X.~Zhang,
arXiv:astro-ph/0407432.

%\cite{Carroll:1998zi}
\bibitem{Carroll:1998zi}
  S.~M.~Carroll,
  %``Quintessence and the rest of the world,''
  Phys.\ Rev.\ Lett.\  {\bf 81}, 3067 (1998)
  [arXiv:astro-ph/9806099].

\bibitem{Fardon:2003eh}
R.~Fardon, A.~E.~Nelson and N.~Weiner,
%``Dark energy from mass varying neutrinos,''
JCAP {\bf 0410}, 005 (2004)
[arXiv:astro-ph/0309800].

\bibitem{Schwarz:2004tz}
D.~J.~Schwarz and C.~A.~Terrero-Escalante,
%``Primordial fluctuations and cosmological inflation after WMAP 1.0,''
JCAP {\bf 0408}, 003 (2004)
[arXiv:hep-ph/0403129].
%%CITATION = HEP-PH 0403129;%%

\bibitem{Stewart:1993bc}
  E.~D.~Stewart and D.~H.~Lyth,
  %``A More accurate analytic calculation of the spectrum of cosmological
  %perturbations produced during inflation,''
  Phys.\ Lett.\ B {\bf 302}, 171 (1993)
  [arXiv:gr-qc/9302019].
  %%CITATION = GR-QC 9302019;%% 
   
\bibitem{Kinney:2003uw}
W.~H.~Kinney, E.~W.~Kolb, A.~Melchiorri and A.~Riotto,
%``WMAPping inflationary physics,''
Phys.\ Rev.\ D {\bf 69}, 103516 (2004)
[arXiv:hep-ph/0305130].
%%CITATION = HEP-PH 0305130;%%

%\cite{Eidelman:2004wy}
\bibitem{Eidelman:2004wy}
S.~Eidelman {\it et al.}  [Particle Data Group],
%``Review of particle physics,''
Phys.\ Lett.\ B {\bf 592}, 1 (2004).

%\cite{Bean:2002sm}
\bibitem{Bean:2002sm}
  R.~Bean, S.~H.~Hansen and A.~Melchiorri,
  %``Constraining the dark universe,''
  Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 110}, 167 (2002)
  [arXiv:astro-ph/0201127].
  %%CITATION = ASTRO-PH 0201127;%%


\end{thebibliography}


\end{document}


astro-ph0504090.extracted_bib
\begin{thebibliography}{99}

%\cite{Dodelson:2001fq}
\bibitem{Dodelson:2001fq}
S.~Dodelson, M.~Kaplinghat and E.~Stewart,
%``Tracking Oscillating Energy,''
Phys.\ Rev.\ Lett.\  {\bf 85}, 5276 (2000)
[arXiv:astro-ph/0002360].

\bibitem{tracker} P.J.E. Peebles \& B. Ratra, Phys. Rev. {\bf D37} 3406
(1988); I. Zlatev, L. Wang, \& P.J. Steinhardt, \prl {\bf 82}, 
896 (1998); P.J. Steinhardt, L. Wang, \& I. Zlatev, Phys. Rev. {\bf D59}, 
123504 (1999); A. Albrecht \& C. Skordis, astro-ph/9908085 (1999).

%\cite{Peebles:1998qn}
\bibitem{Peebles:1998qn}
P.~J.~E.~Peebles and A.~Vilenkin,
%``Quintessential inflation,''
Phys.\ Rev.\ D {\bf 59}, 063505 (1999)
[arXiv:astro-ph/9810509].

\bibitem{Peloso:1999dm}
  M.~Peloso and F.~Rosati,
  %``On the construction of quintessential inflation models,''
  JHEP {\bf 9912}, 026 (1999)
  [arXiv:hep-ph/9908271].

%\cite{Copeland:2000hn}
\bibitem{Copeland:2000hn}
  E.~J.~Copeland, A.~R.~Liddle and J.~E.~Lidsey,
  %``Steep inflation: Ending braneworld inflation by gravitational particle
  %production,''
  Phys.\ Rev.\ D {\bf 64}, 023509 (2001)
  [arXiv:astro-ph/0006421].

\bibitem{Kaganovich:2000fc}
  A.~B.~Kaganovich,
  %``Field theory model giving rise to 'quintessential inflation' without  the
  %cosmological constant and other fine tuning problems,''
  Phys.\ Rev.\ D {\bf 63}, 025022 (2001)
  [arXiv:hep-th/0007144].

%\cite{Dimopoulos:2001ix}
\bibitem{Dimopoulos:2001ix}
  K.~Dimopoulos and J.~W.~F.~Valle,
  %``Modeling quintessential inflation,''
  Astropart.\ Phys.\  {\bf 18}, 287 (2002)
  [arXiv:astro-ph/0111417].

\bibitem{Majumdar:2001mm}
A.~S.~Majumdar,
%``From brane assisted inflation to quintessence through a single scalar
%field,''
Phys.\ Rev.\ D {\bf 64}, 083503 (2001)
[arXiv:astro-ph/0105518].

%\cite{Huey:2001ae}
\bibitem{Huey:2001ae}
G.~Huey and J.~E.~Lidsey,
%``Inflation, braneworlds and quintessence,''
Phys.\ Lett.\ B {\bf 514}, 217 (2001)
[arXiv:astro-ph/0104006].

%\cite{Sami:2004xk}
\bibitem{Sami:2004xk}
M.~Sami and V.~Sahni,
%``Quintessential inflation on the brane and the relic gravity wave
%background,''
Phys.\ Rev.\ D {\bf 70}, 083513 (2004)
[arXiv:hep-th/0402086].

%\cite{Abbott:1982hn}
\bibitem{Abbott:1982hn}
  L.~F.~Abbott, E.~Farhi and M.~B.~Wise,
  %``Particle Production In The New Inflationary Cosmology,''
  Phys.\ Lett.\ B {\bf 117}, 29 (1982).
  %%CITATION = PHLTA,B117,29;%%

%\cite{Barenboim:2004kz}
\bibitem{Barenboim:2004kz}
  G.~Barenboim, O.~Mena and C.~Quigg,
  %``Undulant universe,''
  arXiv:astro-ph/0412010.

%\cite{Feng:2004ff}
\bibitem{Feng:2004ff}
B.~Feng, M.~Li, Y.~S.~Piao and X.~Zhang,
arXiv:astro-ph/0407432.

%\cite{Carroll:1998zi}
\bibitem{Carroll:1998zi}
  S.~M.~Carroll,
  %``Quintessence and the rest of the world,''
  Phys.\ Rev.\ Lett.\  {\bf 81}, 3067 (1998)
  [arXiv:astro-ph/9806099].

\bibitem{Fardon:2003eh}
R.~Fardon, A.~E.~Nelson and N.~Weiner,
%``Dark energy from mass varying neutrinos,''
JCAP {\bf 0410}, 005 (2004)
[arXiv:astro-ph/0309800].

\bibitem{Schwarz:2004tz}
D.~J.~Schwarz and C.~A.~Terrero-Escalante,
%``Primordial fluctuations and cosmological inflation after WMAP 1.0,''
JCAP {\bf 0408}, 003 (2004)
[arXiv:hep-ph/0403129].
%%CITATION = HEP-PH 0403129;%%

\bibitem{Stewart:1993bc}
  E.~D.~Stewart and D.~H.~Lyth,
  %``A More accurate analytic calculation of the spectrum of cosmological
  %perturbations produced during inflation,''
  Phys.\ Lett.\ B {\bf 302}, 171 (1993)
  [arXiv:gr-qc/9302019].
  %%CITATION = GR-QC 9302019;%% 
   
\bibitem{Kinney:2003uw}
W.~H.~Kinney, E.~W.~Kolb, A.~Melchiorri and A.~Riotto,
%``WMAPping inflationary physics,''
Phys.\ Rev.\ D {\bf 69}, 103516 (2004)
[arXiv:hep-ph/0305130].
%%CITATION = HEP-PH 0305130;%%

%\cite{Eidelman:2004wy}
\bibitem{Eidelman:2004wy}
S.~Eidelman {\it et al.}  [Particle Data Group],
%``Review of particle physics,''
Phys.\ Lett.\ B {\bf 592}, 1 (2004).

%\cite{Bean:2002sm}
\bibitem{Bean:2002sm}
  R.~Bean, S.~H.~Hansen and A.~Melchiorri,
  %``Constraining the dark universe,''
  Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 110}, 167 (2002)
  [arXiv:astro-ph/0201127].
  %%CITATION = ASTRO-PH 0201127;%%


\end{thebibliography}