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\begin{preprint}
 \qquad\qquad\qquad \qquad\qquad\qquad\qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
USTC-ICTS-07-03
\end{preprint}

\title{A String-Inspired Quintom Model Of Dark Energy}

\author{Yi-Fu Cai$^a$, Mingzhe Li$^{b,c}$, Jian-Xin Lu$^d$, Yun-Song Piao$^e$, Taotao Qiu$^a$\footnote{qiutt@mail.ihep.ac.cn}, Xinmin Zhang$^a$}


\vspace{1.cm}
\address{$^a$Institute of High Energy Physics, Chinese Academy
of Sciences, P.O. Box 918-4, Beijing 100049, P. R. China
\vspace{.1in} \\
$^b$Institut f$\ddot{u}$r Theoretische Physik, Philosophenweg 16,
D-69120 Heidelberg, Germany
\vspace{.1in} \\
$^c$Fakult\"{a}t f\"{u}r Physik, Universit\"{a}t Bielefeld,
D-33615 Bielefeld, Germany
\vspace{.1in}\\
$^d$Interdisciplinary Center for Theoretical Study, University of
Science and Technology of China, Hefei, Anhui 230026, China
\vspace{.1in} \\
$^e$College of Physical Sciences, Graduate School of Chinese
Academy of Sciences, YuQuan Road 19A, Beijing 100049, China}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}

We propose in this paper a quintom model of dark energy with a
single scalar field $\phi$ given by the lagrangian ${\cal
L}=-V(\phi)\sqrt{1-\alpha^\prime\nabla_{\mu}\phi\nabla^{\mu}\phi
+\beta^\prime \phi\Box\phi}$. In the limit of
$\beta^\prime\rightarrow$0 our model reduces to the effective low
energy lagrangian of tachyon considered in the literature. We study
the cosmological evolution of this model, and show explicitly the
behaviors of the equation of state crossing the cosmological
constant boundary.
\end{abstract}

\maketitle


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

The current data from type Ia supernovae, cosmic microwave
background (CMB) radiation, and other cosmological
observations\cite{1998snia,Riess,Spergel,Seljak} have provided
strong evidences for a spatially flat and accelerated expanding
universe at the present time. Within the framework of the standard
cosmology, this acceleration can be understood by introducing a
mysterious component, dubbed dark energy (DE). The simplest
candidate for DE is a minor positive cosmological constant, but it
suffers from the fine-tuning and coincidence problems. As a possible
solution to these problems  various dynamical models of DE have been
proposed, such as quintessence. In the recent years with the
accumulated astronomical observational data it becomes possible to
probe the current and even early behavior of DE. Although the
current fits to the data in combination of the 3-year
WMAP\cite{3wmap}, the recently released 182 SNIa Gold
sample\cite{Riess06} and also other cosmological observational data
show the consistence of the cosmological constant, it is worth
noting that the dynamical DE models are not excluded and a class of
dynamical models with equation of state across $-1$, dubbed {\it
quintom}, is mildly favored (for recent references see e.g.
\cite{Lihong2006, Zhao2006}).

Theoretically it is a big challenge to the model building of the
quintom dark energy. The Ref.\cite{Quintom1} is the first paper
pointing out this challenge and showing explicitly the difficulty of
realizing $w$ crossing over $-1$ in the quintessence and phantom
like models. In general with a single fluid or a single scalar field
with a lagrangian of form ${\cal L}={\cal
L}(\phi,\partial_\mu\phi\partial^\mu\phi)$ it has been proved
\cite{GBZhaoperturb} (see also \cite{Caldwell2005}) that the dark
energy perturbation would be divergent as the equation of state
(EOS) $w$ approaches to $-1$. This ``no-go" theorem forbids the
dynamical models widely studied in the literature with a single
scalar field such as quintessence, phantom and k-essence to make the
EOS cross over the cosmological constant boundary. The quintom
scenario of dark energy is designed to understand the nature of dark
energy with $w$ across $-1$. The quintom models of dark energy
differ from the quintessence, phantom and k-essence and so on in the
determination of the cosmological evolution and the fate of the
universe.


To realize a viable quintom scenario of dark energy it needs to
introduce extra degree of freedom to the conventional theory with a
single fluid or a single scalar field. The first model of quintom
scenario of dark energy is given in Ref.\cite{Quintom1} with two
scalar fields. This model has been studied in detail later on. In
the recent years there have been active studies on models of quintom
like dark energy such as models with high derivative term\cite{lfz},
vector field\cite{Wei}, or even extended theory of
gravity\cite{extendgravity} and so on, see e.g. \cite{Quintomsum}.

In this paper, we propose a new type of quintom model inspired by
the string theory. We will demonstrate in this paper our model can
realize the equation of state crossing $-1$ naturally. This paper is
organized as follows. In section 2 we present our model and study
its properties especially on the conditions required for the model
parameters when $w$ crosses over $-1$. By solving numerically the
model we will study the evolution of the equation of state. The
section 3 is the summary of our paper.



\section{Our Model}


Our model is given by the following action\footnote{We have adopted
signature $(+, -, -, -)$ in this paper.}
\begin{eqnarray}\label{actionorigin}
S=\int d^4x\sqrt{-g}\left[-V(\phi)\sqrt{1-{\alpha}^\prime
\nabla_{\mu}\phi\nabla^{\mu}\phi+{\beta}^\prime\phi\Box\phi}\right].
\end{eqnarray}
This model generalizes the usual``Born-Infeld-type" action
 for the
effective description of tachyon dynamics by adding a term $\phi
\Box \phi$ to the usual $\nabla_\mu \phi\nabla^\mu \phi $ in the
square root. It is known that the 4D effective action of tachyon
dynamics to the lowest order in $ \nabla_\mu\phi\nabla^\mu \phi$
around the top of the tachyon potential can be obtained by the
stringy computations for either a D3 brane in bosonic theory
\cite{Gerasimov:2000zp, Kutasov:2000qp} or a non-BPS D3 brane in
supersymmetric theory \cite{Kutasov:2000aq}. To this order, we have
no need to include the operator $\phi \Box \phi$ in the action since
it is equivalent to the usual $\nabla_\mu \phi\nabla^\mu \phi $
term. However, we cannot exclude its existence in an action such as
the ``Born-Infeld-type" one when incorporating an infinite number of
higher derivative terms since now the two terms are in general
different dynamically. For example, we cannot simply replace the
$\phi \Box \phi$ term in the action (\ref{actionorigin}) by the
$\nabla_\mu \phi\nabla^\mu \phi $. Further this term in the above
generalized action has new cosmlogical consequence as will be shown
in this paper. Including an infinite number of higher derivative
terms that has a significant cosmological consequence has also been
discussed in the context of p-adic string recently in
\cite{Barnaby:2006hi}. The two parameters $\alpha'$ and $\beta'$ in
(\ref{actionorigin}) can also be made arbitrary when the background
flux is turned on \cite{Mukhopadhyay:2002en}. Without further
reasoning, we will take the action (\ref{actionorigin}) as our
starting point.

As will be demonstrated, the $\beta'$ term in (\ref{actionorigin})
is crucial to realize the $w$ across $-1$. There we have defined
$\alpha^\prime = \alpha/ M^4$ and $\beta^\prime = \beta/ M^4$ with
$\alpha$ and $\beta$ being the dimensionless parameters,
respectively and $M$ an energy scale used to make the ``kinetic
energy terms" dimensionless. $V(\phi)$ is the potential of scalar
field $\phi$ (e.g., a tachyon) with dimension of ${\rm [mass]}^4$
with an expected tachyon potential behavior in general, i.e.,
bounded and reaching its minimum asymptotically, unless specifically
stated. Note that,
$\Box=\frac{1}{\sqrt{-g}}\partial_{\mu}\sqrt{-g}g^{\mu\nu}\partial_{\nu}$,
therefore, in (\ref{actionorigin}) the terms
$\nabla_\mu\phi\nabla^\mu\phi$ and $\phi\Box\phi$ both involve two
fields and two derivatives.

The model with operator $\phi\Box\phi$ for the realization of $w$
crossing $-1$ has been proposed in \cite{lfz}. However in general
for a model with lagrangian as a sum of operators with a polynomial
function of the scalar field $\phi$ and its derivatives, the
operator $\phi\Box\phi$ can be rewritten as a total derivative term
which makes no contribution after integration and a term which
renormalizes the canonical kinetic term as discussed above. So if
one considers a renormalizable lagrangian, the operator
$\phi\Box\phi$ will not be included. Ref.\cite{lfz} considered a
dimension-6 operator as $(\Box \phi)^2$. However in the present
model, the operator $\phi\Box\phi$ appears at the same order as the
operator $\nabla_\mu\phi\nabla^\mu\phi$ does in the
``Born-Infeld-type" action. As discussed above, this model appears
more natural than the one used in \cite{lfz}.


With (\ref{actionorigin}) we obtain the equation of motion of the
scalar field $\phi$,
\begin{eqnarray}\label{Eqofmbasic}
\frac{\beta}{2}\Box(\frac{V\phi}{f})+
\alpha\nabla_{\mu}(\frac{V\nabla^{\mu}\phi}{f})+M^4V_{\phi}f+\frac{\beta
V}{2f}\Box\phi=0~,
\end{eqnarray}
where $f=\sqrt{1-{\alpha}^\prime
\nabla_{\mu}\phi\nabla^{\mu}\phi+{\beta}^\prime\phi\Box\phi}$ and
$V_\phi=dV /d\phi$. Following the convention in Ref.\cite{lfz}, the
energy-momentum tensor $T^{\mu\nu}$ is given by the standard
definition: $\delta_{g_{\mu\nu}}S\equiv-\int d^4 x
\frac{\sqrt{-g}}{2}T^{\mu\nu}\delta g_{\mu\nu}$,
\begin{eqnarray}\label{emtensor}
T^{\mu\nu}&=&g^{\mu\nu}[V(\phi)f+\nabla_{\rho}(\psi\nabla^{\rho}\phi)]+
\frac{\alpha}{M^4}\frac{V(\phi)}{f}\nabla^\mu\phi\nabla^\nu\phi-\nabla^\mu\psi\nabla^\nu\phi-\nabla^\nu\psi\nabla^\mu\phi~,
\end{eqnarray}
where $\psi\equiv\frac{\partial{\cal
L}}{\partial\Box\phi}=-\frac{V\beta\phi}{2M^4f}$.

For a flat Friedman-Robertson-Walker (FRW) universe and a homogenous
scalar field $\phi$, the equation of motion in (\ref{Eqofmbasic})
can be solved equivalently by the following two equations
\begin{eqnarray}\label{EqofmTT}
\ddot\phi+3H\dot\phi&=&\frac{\beta
V^2\phi}{4M^4\psi^2}-\frac{M^4}{\beta\phi}+\frac{\alpha}{\beta\phi}\dot\phi^2~,\\
\ddot\psi+3H\dot\psi&=&(2\alpha+\beta)(\frac{M^4\psi}{\beta^2\phi^2}-\frac{V^2}{4M^4\psi})-(2\alpha-\beta)\frac{\alpha\psi}{\beta^2\phi^2}\dot\phi^2-\frac{2\alpha}{\beta\phi}\dot\psi\dot\phi-\frac{\beta
V\phi}{2M^4\psi}V_{\phi}~,
\end{eqnarray}
where we have made use of the $\psi$ as defined before and the first
equation above is just the defining equation for $\psi$ in terms of
$\phi$ and its derivatives. $H = \dot a /a$ is the Hubble parameter.

 One can see from equations above the
$\beta$ term plays a role in determining the evolution of the scalar
field $\phi$. We can read the energy density from (\ref{emtensor})
as
\begin{eqnarray}\label{Energydensity}
\rho=Vf+\frac{d}{a^3dt}(a^3\psi\dot\phi)+\frac{\alpha}{M^4}\frac{V(\phi)}{f}\dot\phi^2-2\dot\psi\dot\phi~,
\end{eqnarray}
and similarly the pressure
\begin{eqnarray}\label{Pressure}
p=-Vf-\frac{d}{a^3dt}(a^3\psi\dot\phi)~.
\end{eqnarray}


With the Friedman equations $H^2=\frac{8\pi G}{3}\rho$ and $\dot
H=-4\pi G(\rho+p)$, we now study the cosmological evolution of
equation of state for the present model. Given $w=p/\rho$, we have
$\rho+p=(1+w)\rho$. To explore the possibility of the $w$ across
$-1$, we need to check if $\frac{d}{dt}(\rho+p)\neq0$ can be held
when $w \rightarrow -1$.

Using (\ref{Energydensity}) and (\ref{Pressure}) as well as  making
use of the defining equation for $\psi$, we have
\begin{eqnarray}\label{crosscondition1}
\rho+p=\frac{\alpha
V(\phi)\dot\phi^2}{M^4f}+\beta\dot\phi\frac{d}{dt}[\frac{\phi
V(\phi)}{M^4f}]~,
\end{eqnarray}
and from which,
\begin{eqnarray}\label{x}
\frac{d}{dt}(\rho+p)&=&\frac{\alpha}{M^4f^2}\{f\frac{d}{dt}
[V(\phi)\dot\phi^2]-\frac{df}{dt}
[\dot\phi^2{V}(\phi)]\}+\beta\ddot\phi\frac{d}{dt}[\frac{\phi
V(\phi)}{M^4f}]+\beta\dot\phi\frac{d^2}{dt^2}[\frac{\phi
V(\phi)}{M^4f}]~.
\end{eqnarray}
Eq. (\ref{crosscondition1}) implies that we have either
(i)$\dot\phi=0$ or (ii)$-\frac{\alpha
V(\phi)}{f}\dot\phi=\beta\frac{d}{dt}[\frac{\phi V(\phi)}{f}]$ when
$w \rightarrow -1$.


Let us assume $\dot\phi=0$ first when $w \to -1$. With this and Eq.
(\ref{x}), we have
$\frac{d}{dt}(\rho+p)=\beta\ddot\phi\frac{d}{dt}[\frac{\phi
V(\phi)}{M^4f}]=-\frac{V(\phi)}{2M^8f^3}\beta^2\phi^2\ddot\phi\frac{d}{dt}\Box\phi$.
Therefore the conditions for having the $w$ across over $-1$ are
(i-1)$\phi\neq0$; (i-2)$\ddot\phi\neq0$;
(i-3)$\frac{d}{dt}\Box\phi\neq0$ in addition to the $\dot \phi = 0$.
Since the characteristic behavior of $V(\phi)$ is to have a finite
maximum value at a finite $\phi$ and to reach its (exponentially)
vanishing minimum asymptotically\footnote{This is the behavior of a
tachyon potential.} where one expects $\ddot \phi = 0$ and
$\frac{d}{dt}\Box\phi = 0$ but a non-zero $\dot \phi$, so realizing
the above crossing over conditions must happen before reaching the
potential minimum asymptotically. This implies once the crossing
over conditions are met, the field $\phi$ must continue to run away
as it should be since we have $\ddot \phi \neq 0$.

Let us turn to the second case (ii) when $w \to -1$. We now have
$-\frac{\alpha V\dot\phi}{f}=\beta\frac{d}{dt}(\frac{\phi
V}{f})=\beta\dot\phi(\frac{V}{f})+\beta\phi\frac{d}{dt}(\frac{V}{f})$
which is equivalent to saying $Y \equiv (\alpha + \beta) \frac{V}{f}
\dot\phi + \beta\phi\frac{d}{dt}(\frac{V}{f}) = 0$. For now, we have
$\dot\phi \neq 0$. Then the above can be further expressed as
$\frac{d}{dt}[(\alpha+\beta)\ln\phi+\beta \ln(\frac{V}{f})]=0$ if
$\phi \neq 0$.\footnote{We can always choose $\phi \neq 0$ and the
$\phi = 0$ will give a degenerate case which will not be considered
here.} Then from (\ref{x}), the further condition for crossing over
is $\frac{d}{dt}(\rho+p)\neq0$ which implies $\dot Y \neq 0$ or
$\frac{d^2}{dt^2}[(\alpha+\beta)\ln\phi+\beta
\ln(\frac{V}{f})]\neq0$. In summary,  the following conditions are
required for the crossing over: ii-1)$Y = 0$ and ii-2) $\dot Y \neq
0$ in addition to the $\dot\phi \neq 0$ and $\phi \neq 0$.
%$\footnote{We use the $Y$ defined in the context not a simpler one
%such as $Y = \phi^{\alpha + \beta} (\frac{V}{f})^\beta$ because the
%$Y$ is well defined asymptotically and is useful for arguing the
%crossing over.}.
 Given the definition of the above $Y$ and the
characteristic behavior of $V(\phi)$ discussed previously, one
expects that both $Y$ and $\dot Y$ vanish asymptotically while $\dot
\phi$ can reach a non-vanishing value. This implies that the
crossing over if occurring at all must occur before the $V(\phi)$
reaches its minimum asymptotically as anticipated.
% In general, one
%expects that realizing the present case with $\dot\phi \neq 0$ is
%easier than the previous one with $\dot\phi = 0$ at the crossing
%over.

Before we demonstrate numerically that the crossing over of the
present model can indeed be realized using specific examples, let us
remark one salient feature of the present model that the $\beta$
term in the present model (\ref{actionorigin}) is the key for
realizing the crossing over. One can check from
(\ref{crosscondition1}) that if $\beta = 0$ the $\omega \to - 1$ is
possible only for $\dot\phi = 0$. Then from (\ref{x}), we will have
$\frac{d}{dt}(\rho+p) = 0$, the impossibility of crossing over.


The analysis above shows various possibilities of our model in
realizing the EOS $w$ crossing $-1$. Now we consider some specific
examples for numerical calculations of the evolution of the EOS. In
Figures \ref{fig:1bigrip} we take $V(\phi)=V_0 e^{-\lambda\phi^2}$
and plot the behavior of EOS. In the numerical calculations we have
normalized the values of the scalar field and $V_0$, respectively,
by the energy scale $M$. In Figure \ref{fig:1bigrip} our model
predicts the EOS crossing $-1$ during the evolution and a big-rip
singularity for the fate of the universe. Numerically we have
checked that $\dot \phi = 0$ when $w$ crosses over the cosmological
constant boundary.

\begin{figure}[htbp]
\includegraphics[scale=0.8]{Graph1n.eps}
\caption{Plot of the evolution of the EOS as a function of $\ln a$.
In the numerical calculation, we have taken $V_0=0.8$, $\lambda=1$,
$\alpha=1$, and $\beta=-0.8$. For the initial conditions we choose
$\phi_i=0.9$, $\dot\phi_i=0.6$,
$(\Box\phi)_i=\frac{d}{dt}(\Box\phi)_i=0$.} \label{fig:1bigrip}
\end{figure}


In Figure \ref{fig:2} we take a different potential for numerical
calculations. One can see that the EOS crosses over $-1$ during the
evolution. When we take $\beta$ positive Figure 3 shows the EOS
starts with $w<-1$, crosses over $-1$ into the region of $w>-1$,
then transits again to $w<-1$.

\begin{figure}[htbp]
\includegraphics[scale=0.8]{fig2.eps}
\caption{Plot of the evolution of the EOS as a function of $\ln a$.
In the numerical calculation we take
$V(\phi)=\frac{V_0}{e^{\lambda\phi}+e^{-\lambda\phi}}$, and
$V_0=0.5$. For the model parameters we choose $\lambda=1$,
$\alpha=1$, and $\beta=-0.8$. For the initial conditions we take
$\phi_i=0.9$, $\dot\phi_i=0.6$,
$(\Box\phi)_i=\frac{d}{dt}(\Box\phi)_i=0$.}\label{fig:2}
\end{figure}

\begin{figure}[htbp]
\includegraphics[scale=0.8]{fig3.eps}
\caption{Plot of the evolution of the EOS as a function of $\ln a$.
In the numerical calculation we take
$V(\phi)=\frac{V_0}{e^{\lambda\phi}+e^{-\lambda\phi}}$, and
$V_0=0.5$. For the model parameters we choose $\lambda=1$,
$\alpha=1$, and $\beta=0.8$. For the initial conditions we take
$\phi_i=0.9$, $\dot\phi_i=0.6$,
$(\Box\phi)_i=\frac{d}{dt}(\Box\phi)_i=0$.}\label{fig:3}
\end{figure}


The potentials used for the numerical calculations in Figures 1-3
are well motivated by the string theory. However as a
phenomenological study of our model as a quintom dark energy we in
Figure 4 plot the evolution of the EOS $w$ with a potential which is
a sum of $e^{\lambda \phi}$ and $e^{- \lambda \phi}$. This type of
potential does not have the general behavior of the tachyon
potential, however has been used for the study of phenomenological
models of dark energy. One can see from this Figure that the EOS
evolves from the region where $w> -1$ to $w<-1$, and stays there for
a period of time then comes back to $w>-1$. At late time our model
gives rise to the de-sitter phase.



\begin{figure}[htbp]
\includegraphics[scale=0.8]{fig4.eps}
\caption{Plot of the evolution of the EOS as a function of $\ln a$.
In the numerical calculation we take
$V(\phi)=V_0(e^{\lambda\phi}+e^{-\lambda\phi})$, and $V_0=0.5$. For
the model parameters we choose $\lambda=1$, $\alpha=1$, and
$\beta=-1.2$. For the initial conditions we take $\phi_i=0.9$,
$\dot\phi_i=0.6$, $(\Box\phi)_i=\frac{d}{dt}(\Box\phi)_i=0$.}
\label{fig:3}
\end{figure}



\section{Conclusion and Discussion}

The current cosmological observations indicate the possibility that
the acceleration of the universe is driven by dark energy with EOS
across $-1$, which if confirmed further in the future will challenge
the theoretical model building of the dark energy. In this paper we
have proposed a string-inspired model of dark energy through
modifying the usual effective ``Born-Infeld-type" description of
tachyon dynamics. As shown in the present work, this modification by
including a $\beta$ term in the action (\ref{actionorigin}) is the
key for the EOS crossing $-1$ during the evolution.\footnote{The
$\beta\rightarrow $0 limit reduces the present model to the
effective low energy Lagrangian of tachyon\cite{Sentachyon} which
has been considered to be a candidate for dark energy in the
literature\cite{TachyonDE}. This type of models for dark energy in
the absence of the $\beta$ term, however will not be possible to
realize the quintom scenario as shown in the present work.} Compared
to other models with $w$ across $-1$ in the literature so far the
present one has a motivation inspired from string theory
consideration and is also economical in the sense that it involves a
single scalar field such as a tachyon.

  % The classical stability is anticipated and will be discussed in a
  % forthcoming work. However, given the complexity of the
   %``Born-Infeld-type" action and the associated thorny problem of
   %studying its quantum behavior, the corresponding quantum
   %stability issue remains open.



\acknowledgments

We thank Bo Feng, Gongbo Zhao, Hong Li and Junqing Xia for useful
discussions. This work is supported in part by National Natural
Science Foundation of China under Grant Nos. 90303004, 10533010 and
19925523. The author M.L. would like to acknowledge the support by
Alexander von Humboldt Foundation. JXL acknowledges support by
grants from the Chinese Academy of Sciences and grants from the NSF
of China with Grant Nos: 10588503 and 10535060.

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}\fi \providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}


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%\cite{Gerasimov:2000zp}
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%\cite{Kutasov:2000qp}
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%\cite{Kutasov:2000aq}
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%\cite{Barnaby:2006hi}
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%\cite{Mukhopadhyay:2002en}
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\end{thebibliography}

\end{document}
