---
abstract: |
    In this paper, we develop a simple method for the reconstruction of the string-inspired dark energy model with the lagrangian $ {\cal L}
    = -V ( \phi ) \sqrt{ 1 - \alpha' {\nabla}_{\mu} \phi \nabla^{\mu}
    \phi + \beta' \phi \Box \phi} $ given by Cai et. al., which may allow the equation-of-state parameter cross the cosmological constant boundary $ ( w=-1 ) $. We reconstruct this model in the light of three forms of parametrization for dynamical dark energy.
author:
- Shuai Zhang
- Bin Chen
date: 'June 27, 2008'
title: 'Reconstructing a String-Inspired Quintom Model of Dark Energy'
---

Introduction {#1}
============

A number of recent cosmological data, including supernovae(SN) Ia [@SN], large-scale structure(LSS) [@LSS], and cosmic microwave background (CMB) anisotropy [@CMB], have provided strong evidences for a spatially flat and accelerated expanding universe at the present time. In the context of Friedmann-Robertson-Walker (FRW) cosmology, this acceleration is attributed to the domination of a mysterious component, dubbed dark energy (DE). Although a model of cosmological constant is consistent with the observations, it suffers serious difficulties, for example, the present value of the cosmological constant $\Lambda$ is $10^{123}$ times smaller than that predicted by particle physics on condition that the UV cutoff is at Plankscale, which is the so-called coincidence problem. [@Dynamics; @of; @dark; @energy] Hence, as a possible solution to these problems various dynamical models of DE have been proposed.

Up to now, a large number of scalar-field dark energy models has been proposed, such as quintessence [@quintessence], K-essence [@K-essence], tachyon [@tachyon], phantom [@phantom], quintom [@quintom] and ghost condensate [@ghost; @condensate1]. The analysis of the observational data of type Ia supernova, mainly the 157 “gold” data listed in [@SN], shows that the equation-of-state (EoS) of dark energy $w$ is likely to cross the cosmological constant boundary $-1$. Nevertheless, the dark energy models with a single scalar field with a lagrangian of form ${\cal L} = {\cal
L}(\phi,\partial_{\mu} \phi \partial^{\mu} \phi) $ such as quintessence and phantom only allow $w$ to be either larger or smaller than $-1$, according to the “no-go” theorem [@no-go]. This is why a class of dynamical models dubbed quintom has to require two fields in order to permit EoS across $-1$.

In this paper, we will discuss a string-inspired quintom model of dark energy proposed in Ref. [@a; @string-inspired; @quintom; @model; @of; @dark; @energy], of which the action is given by $$\label{eq:action}
S = \int{d^4x\sqrt{-g} \left[ -V ( \phi ) \sqrt{ 1 - \alpha'
{\nabla}_{\mu} \phi \nabla^{\mu} \phi + \beta' \phi \Box \phi}
\right]}~.$$ The signature $(+,-,-,-)$ is used in this paper. 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. $V (\phi)$ is a potential of scalar field $\phi$ with dimension of \[mass\]$^4$.

In a flat FRW spacetime, from the variation of the action (\[eq:action\]) with respect to $\phi$, we obtain the equations of motion of the homogenous scalar $\phi$, $$\begin{aligned}
\label{eq:motion1} \ddot{\phi}+3H\dot{\phi} &=& \frac{\beta V^2
\phi}{4M^4 \psi^2} -
\frac{M^4}{\beta\phi} + \frac{\alpha}{\beta}\dot{\phi}^2~,\\
\label{eq:motion2} \ddot{\psi}+3H\dot{\psi} &=&
(2\alpha+\beta)\left(\frac{M^4\psi}{\beta^2\phi^2} -
\frac{V^2}{4M^4\psi}\right) -
(2\alpha-\beta)\frac{\alpha\psi}{\beta^2\phi^2}\dot{\phi}^2-\frac{2
\alpha}{\beta \phi}\dot{\phi}\dot{\psi} - \frac{\beta V
\phi}{2M^4\psi}V_{\phi}~,\end{aligned}$$ where we have defined the new variables $\psi$ and $f$ to simplify the calculations, $$\begin{aligned}
\label{eq:psi}
\psi &\equiv& \frac{\partial \cal L}{\partial \Box \phi} = -
\frac{V
\beta \phi}{2 M^4 f}~, \\
\label{eq:f} f &=& \sqrt{1 - \alpha' {\nabla}_{\mu} \phi
\nabla^{\mu} \phi + \beta' \phi \Box \phi}~.\end{aligned}$$ Here it is defined that $\beta=M^4\beta'$, $\alpha=M^4\alpha'$, with $\alpha$ and $\beta$ being dimensionless parameters, and $M$ an energy scale used to make the “kinetic energy terms” dimensionless. And the variation of the action (\[eq:action\]) with respect to $g_{\mu \nu}$ gives the density and the pressure of dark energy: $$\begin{aligned}
\label{eq:density}
\rho &=& Vf + \frac{d}{a^3 dt}(a^3 \psi \dot{\phi}) + \frac{\alpha
V}{M^4 f}\dot{\phi}^2 - 2\dot{\psi}\dot{\phi}~,\\
\label{eq:pressure} p &=& -Vf - \frac{d}{a^3 dt}(a^3 \psi
\dot{\phi})~,\end{aligned}$$ then we obtain $$\label{eq:plus}
\rho+p = \frac{\alpha V}{M^4 f}\dot{\phi}^2 -
2\dot{\psi}\dot{\phi}~.$$ As is discussed in Ref. [@a; @string-inspired; @quintom; @model; @of; @dark; @energy], $w$ approaches $-1$ and then crosses over it on condition that (i)$ \dot{\phi}=0 ,~ \phi \neq 0 ,~ \ddot{\phi}\neq 0
,~ \frac{d}{dt} \Box \phi \neq 0$ or (ii) $ \phi \neq 0 ,~
\dot{\phi} \neq 0 ,~ Y=0 ,~ \dot{Y}\neq 0 $, where $ Y = (\alpha +
\beta)\frac{V}{f}\dot{\phi}+\beta \phi
\frac{d}{dt}\left(\frac{V}{f}\right)$.

In this paper, we will reconstruct this model in the light of three forms of parametrization. Next, in section \[2\], we present our program to reconstruct the model. In section \[3\] from three forms of parametrization of EoS, which fit observational data of the joint analysis of SNIa+CMB+LSS, we reconstruct the model. Finally, in Section \[4\], we end with a short summary of our results and some discussions.

Reconstruction Program {#2}
======================

In this section, we shall perform a reconstruction program of this model. The aim of this section is to express the physical quantities to be reconstructed in respect of redshift $z$. So once the form of parametrization of EoS is given, we can directly work out the relationships between any two of the physical quantities, including that of the potential $V$ and the scalar field $\phi$, which is what we expect eventually.

In terms of $\psi$, we can rewrite $$\begin{aligned}
\label{eq:rho}
\rho &=& -\frac{\beta V^2 \phi}{4 M^4 \psi}-\frac{\alpha \psi
\dot{\phi}^2}{\beta \phi}-\dot{\psi}\dot{\phi}-\frac{M^4
\psi}{\beta \phi}=\tilde{K}+\tilde{V}~,\\
\label{eq:pl} \rho+p &=& -2 \frac{\alpha \psi \dot{\phi}^2}{\beta
\phi}-2 \dot{\psi}\dot{\phi}=2 \tilde{K}~,\end{aligned}$$ where the effective kinetic energy term $\tilde{K}$ and the effective potential term $\tilde{V}$ are defined as $$\begin{aligned}
\label{eq:K}
\tilde{K} &=& -\frac{\alpha \psi \dot{\phi}^2}{\beta \phi}-
\dot{\psi}\dot{\phi}~,\\
\label{eq:V} \tilde{V} &=& -\frac{\beta V^2 \phi}{4 M^4
\psi}-\frac{M^4 \psi}{\beta \phi}~.\end{aligned}$$ Therefore, the Friedmann equations in a flat FRW spacetime are given by $$\begin{aligned}
\label{eq:H1} 3 M_{pl}^2 H^2&=&\rho_{m}+\rho
~=~ \rho_{m}+\tilde{K}+\tilde{V}~, \\
\label{eq:H2} 2 M_{pl}^2 \dot{H} &=& -\rho_{m}-\rho-p ~=~
-\rho_{m}-2\tilde{K},\end{aligned}$$ where $M_{pl}^2 \equiv (8 \pi G)^{-1/2}$ is the reduced Planck mass, and $\rho_{m}$ is the energy density of dust matter. The EoS of dark energy is $$\label{eq:EoS}
w\equiv\frac{p}{\rho}=-1+\frac{2}{1+\tilde{V} / \tilde{K}}.$$ From Eq. (\[eq:EoS\]), it is obviously that $w>-1$, when $
\frac{\tilde{V}}{\tilde{K}}>-1$, while $w<-1$, when $\frac{\tilde{V}}{\tilde{K}}<-1$. And when $\tilde{V}+\tilde{K}=0$, the transition takes place.

From Eq. (\[eq:H1\]) and Eq. (\[eq:H2\]), we can easily get $$\label{eq:V''}
\tilde{V}=-\frac{1}{2} \rho_{m}+3 M_{pl}^2 H^2+M_{pl}^2 \dot{H},$$ $$\label{eq:K''}
\tilde{K}=-\frac{1}{2} \rho_{m}-M_{pl}^2 \dot{H}.$$ As in our model, the dark energy fluid does not couple to the background fluid, the expression of the energy density of dust matter in respect of redshift $z$ is $$\label{eq:density-m}
\rho_{m}=3 M_{pl}^2 H_{0}^2 \Omega_{m0} (1+z)^3,$$ where $\Omega_{m}$ represent the ratio density parameter of matter fluid and the subscript “0” indicates the present value of the corresponding quantity. By using the relationship $$\label{eq:dz-dt}
\frac{d}{dt}=-H(1+z)\frac{d}{dz},$$ $\tilde{V}$ and $\tilde{K}$ in respect of $z$ can be expressed as $$\begin{aligned}
\tilde{V}(z) &&=-\frac{3}{2} M_{pl}^2 H_{0}^2
\Omega_{m0}(1+z)^3-\frac{1}{2} M_{pl}^2 H_{0}^2 r'(1+z)+3M_{pl}^2
H_{0}^2 r,
\label{eq:V'}\\
\tilde{K}(z) &&=-\frac{3}{2} M_{pl}^2 H_{0}^2
\Omega_{m0}(1+z)^3+\frac{1}{2} M_{pl}^2 H_{0}^2 r'(1+z)
\label{eq:K'}\end{aligned}$$ where $$\label{eq:r}
r \equiv \frac{H^2}{H_{0}^2}.$$

In this manner, once the expression of $r$ is given in respect of $z$, $\tilde{V}(z)$ and $\tilde{K}(z)$ can be reconstructed from Eq. (\[eq:V’\]) and Eq. (\[eq:K’\]). And by using Eq. (\[eq:K\]) and Eq. (\[eq:V\]), $\tilde{K}$ and $\tilde{V}$ can also be expressed in respect of $\phi$ or $\psi$, on condition that another relationship among $\phi$, $\psi$ and $z$ is given. And it is not hard for us to get the very relationship from equations (\[eq:motion1\]), (\[eq:V\]), (\[eq:dz-dt\]) and (\[eq:r\]), which is $$\phi'' r H_{0}^2 (1+z)^2 -2\phi' r H_{0}^2 (1+z) +
\frac{1}{2}r'H_{0}^2 (1+z)^2 \phi' + 2 \frac{M^4}{\beta
\phi}-\frac{\alpha}{\beta}\phi'^2 rH_{0}^2 (1+z)^2 +
\frac{\tilde{V}}{\psi}=0.
\label{eq:phi-psi}\\$$

Now it is explicit that the expression of $\tilde{V}$ and $\tilde{K}$ in respect of $z$ are the same as those in [@Hessence]. So the reconstructing procedure will be similar to that in [@Hessence], except that the expression of $\tilde{V}$ in respect of the fields $\phi$ and $\psi$ will be given by the inverse function of Eq. (\[eq:V\]) $$\label{eq:inverse}
V^2 = -\frac{4M^4 \psi}{\beta\phi} \tilde{V}-\frac{4M^8
\psi^2}{\beta^2 \phi^2},$$ and the expression of $\phi$ in respect of z is also different from that in [@Hessence]. It will never escape our notice that as $V^2>0$, Eq. (\[eq:inverse\]) may restrict $\phi$ within an interval other than the whole real number set. Next, in the following section, we will obtain the relationship between the potential $V$ and the scalar field $\phi$ in the light of three parametrizations of $w(z)$(or $r(z)$, as $r(z)$ and $w(z)$ are related by Eq. (\[eq:w\]) and Eq. (\[eq:r-w\])).

Reconstruction Results {#3}
======================

As has been discussed above, the two examples considered in the third section of  [@Hessence], where parameterizations for $r(z)$ is fitted to the latest 182 SNe Ia Gold dataset [@182], are also suitable for this model. In this paper, to provide the base for the reconstruction of the string-inspired quintom model of dark energy, we will use three forms of parametrization that has been discussed by Gong and Wang [@parameter]. These forms of parametrization have also been used in [@ghost; @condensate], and two of which is the very two forms used in [@Hessence]. The final results are derived from these three parametrizations.

The three forms of parametrization are investigated uniformly here, in order that it is convenient for us to compare one with another. The three forms are

-   Parametrization 1:$$\label{eq:parameter1}
    w(z) = w_{0} + \frac{w_{a}z}{(1+z)},$$ which was suggested by Chevallier $\&$ Polarski [@C] and Linder [@L], to avoid the divergence problem effectively;

-   Parametrization 2:$$\label{eq:parameter2}
    w(z) = w_{0} + \frac{w_{b}z}{(1+z)^2},$$ which was suggested by Jassal, Bagla $\&$ Padmanabhan [@J];

-   Parametrization 3:$$\label{eq:parameter3}
    r(z) = \Omega_{m0}(1+z)^3+A_{0}+A_{1}(1+ z) + A_{2}(1 + z)^2,$$ which was suggested by Alam, Sahni, Saini $\&$ Starobinsky [@S]. It is an interpolating fit for $r(z)$ having the right behavior for both small and large redshifts.

The similar results for the “parametrization 1” and “parametrization 3” were also independently obtained in Ref. [@N] and in Ref. [@A] respectively.

Using the new measurement of the CMB shift parameter [@W], together with LSS data (the BAO measurement from the SDSS LRGs) [@E] and SNIa data (182 “gold” data released recently) [@182], the authors of Ref. [@parameter] constrained the parameters, whose fit results are summarized as follows: for “parametrization 1”, $\Omega_{\rm m0}=0.29\pm 0.04$, $w_0=-1.07^{+0.33}_{-0.28}$ and $w_a=0.85^{+0.61}_{-1.38}$; for “parametrization 2”, $\Omega_{\rm m0}=0.28^{+0.04}_{-0.03}$, $w_0=-1.37^{+0.58}_{-0.57}$ and $w_b=3.39^{+3.51}_{-3.93}$; for “parametrization 3”, $\Omega_{\rm m0}=0.30\pm 0.04$, $A_1=-0.48^{+1.36}_{-1.47}$ and $A_2=0.25^{+0.52}_{-0.45}$.

From Eq. (\[eq:H1\]), Eq. (\[eq:H2\]) and Eq. (\[eq:density-m\]) we obtain the model-independent expression of the EoS $w$ and the deceleration parameter $q$ : $$\begin{aligned}
w(z) &&= \frac{p}{\rho} = \frac{(1+z)r'-3r}{3r-3\Omega_{m0}
(1+z)^3},
\label{eq:w} \\
r(z) &&= \Omega_{m0} (1+z)^3 + (1-\Omega_{m0}) \exp \left(3
\int_{0}^{z} \frac{1+w(\tilde{z})}{1+\tilde{z}}d\tilde{z}\right),
\label{eq:r-w}
\\
q(z) &&= -1-\frac{\dot{H}}{H^2}=-1-\frac{r'(1+z)}{2r}. \label{eq:q}\end{aligned}$$

![ The evolutions of the EoS in respect of redshift $z$. The red, yellow and blue solid line represent parametrization 1, parametrization 2, and parametrization 3 respectively. The best-fit values of the joint analysis of SNIa+CMB+LSS [@parameter] is accepted here, namely, for parametrization 1, $\Omega_{\rm m0}=0.29$, $w_0=-1.07$ and $w_a=0.85$; for parametrization 2, $\Omega_{\rm m0}=0.28$, $w_0=-1.37$ and $w_b=3.39$; for parametrization 3, $\Omega_{\rm
m0}=0.30$, $A_1=-0.48$ and $A_2=0.25$. []{data-label="fig:w"}](w.eps)

![ The evolutions of the deceleration parameter in respect of redshift $z$. The red, yellow and blue solid line represent parametrization 1, parametrization 2, and parametrization 3 respectively. The best-fit values of the joint analysis of SNIa+CMB+LSS is accepted here. []{data-label="fig:q"}](q.eps)

The evolution of $w(z)$ and $q(z)$ are plotted in Fig. \[fig:w\] and Fig. \[fig:q\] respectively. In Fig. \[fig:w\] we find that all the three forms of parametrization require a model that permits EoS to cross cosmological constant boundary $ ( w=-1 ) $. And Fig. \[fig:q\] tells us our universe is experiencing an accelerating expansion now.

According to Eq. (\[eq:V’\]), Eq. (\[eq:K’\]) and the three parametrizations, the evolutions of $\tilde{K}(z)$ and $\tilde{V}(z)$ are shown in Fig. \[fig:K\] and Fig. \[fig:V\] respectively.

![ The reconstructed $\tilde{K}$ in respect of $z$. The red, yellow and blue solid line represent parametrization 1, parametrization 2, and parametrization 3 respectively. The best-fit values of the joint analysis of SNIa+CMB+LSS is accepted here. []{data-label="fig:K"}](K.eps)

![ The reconstructed $\tilde{V}$ in respect of $z$. The red, yellow and blue solid line represent parametrization 1, parametrization 2, and parametrization 3 respectively. The best-fit values of the joint analysis of SNIa+CMB+LSS is accepted here. []{data-label="fig:V"}](V.eps)

As $\tilde{K}(z)$ and $\tilde{V}(z)$ are not unknown any more, by solving the differential equations (\[eq:K\]) and (\[eq:phi-psi\]), we get the evolution of $\phi$ in respect of $z$, which is plotted in Fig. \[fig:phi\].

![ The reconstructed scalar field $\phi$ in respect of $z$.The red, yellow and blue solid line represent parametrization 1, parametrization 2, and parametrization 3 respectively. The best-fit values of the joint analysis of SNIa+CMB+LSS is accepted here. And we set $\alpha=\beta=M=1$, $\psi(0)=\phi(0)=1$ and $\phi'(0)=1$. []{data-label="fig:phi"}](phi.eps)

Since now that $V$ and $\phi$ are all related to $z$, we can directly obtain the relationship between the potential $V$ and the scalar field $\phi$, which is plotted in Fig. \[fig:Vphi\].

![ The reconstructed potential $V$ in respect of $\phi$.The red, yellow and blue solid line represent parametrization 1, parametrization 2, and parametrization 3 respectively. The best-fit values of the joint analysis of SNIa+CMB+LSS is accepted here. []{data-label="fig:Vphi"}](Vphi.eps)

Here we notice that some range of $\phi$ is forbidden by Eq. (\[eq:inverse\]).

By now, we have completed reconstructing the form of the potential $V(\phi)$ of our model on the basis of observational data. The result also tells us that this model is a viable candidate to describe the dynamics of dark energy.

Conclusions and Discussions {#4}
===========================

As is discussed in the first section, we need a model consisted of two scalar fields to describe the dynamics of dark energy to make it possible that EoS may cross the cosmological-boundary $w=-1$. However, a general quintom model with double fields [@Cai:2006dm] is not easy to be reconstructed, because there is three physical quantities to be expressed in respect of redshift $z$, which is the potential $V$, the scalar fields $\phi$ and $\psi$, while only two equations are viable, namely Friedmann equations. But for a quintom with the potential related to only one of the two fields and has nothing to do with another, such as Hessence proposed in Ref. [@Hessence'] and the string-inspired model that we have just discussed, one of the equations of motion does not contain derivative terms of $V$ with respect of $\phi$ or $\psi$. So the very equation gives another relationship among $V$, $\phi$ and $\psi$. As a result, it is possible for us to reconstruct such models according to the observational data.

In fact, to some extent, we must confess that the quintom with a potential related to only one scalar field is more or less a “pseudo quintom”, because we can always represent one field with an expression of another field by solving some equations. Our model and Hessence (See Eq. (12) in Ref. [@Hessence]) are both examples. And in this way, the lagrangian can be expressed with just one scalar field. As a matter of fact, at the beginning of this paper, Eq. (\[eq:action\]) does express the lagrangian with only one field. It is Eq. (\[eq:psi\]) that defines a new scalar field $\psi$ and make our model a quintom model, but, at the same time, it gives the relationship between $\psi$ and $\phi$, which makes the model possible to be reconstructed. We notice that “no-go” theorem just forbids the dark energy models with a lagrangian of form ${\cal L} = {\cal L}(\phi,\partial_{\mu}
\partial^{\mu} \phi) $ from crossing $w=-1$. But Eq. (\[eq:action\]) does not belong to this class, for it has the term $\beta' \phi \Box \phi$ in the square root. Finally, it is possible for a reconstructable quintom model to be expressed as a model constructed by a single field. A string-inspired quintom has provided a good example. From the analysis of this paper, we find that the result of the reconstruction is able to give us the expressions of $\phi$ and $\psi$ in respect of $z$, which result in $\psi=\psi(\phi)$ and so will eliminates one of the fields in the action. In this sense, we might say a quintom model that fits the observational data is equivalent to a model consisted of one scalar field with a lagrangian other than ${\cal L} = {\cal L}(\phi,\partial_{\mu}
\partial^{\mu} \phi) $ that allows $w$ to cross $-1$.

Acknowledgments {#acknowledgments .unnumbered}
===============

This work was partially supported by NSFC. Grant. No. 10535060, 10775002.

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