# Crossing the phantom divide

Hongsheng Zhang111Electronic address<EMAIL_ADDRESS>Shanghai United
Center for Astrophysics (SUCA), Shanghai Normal University, 100 Guilin Road,
Shanghai 200234,China  Korea Astronomy and Space Science Institute, Daejeon
305-348, Korea  Department of Astronomy, Beijing Normal University, Beijing
100875, China

###### Abstract

The cosmic acceleration is one of the most significant cosmological
discoveries over the last century. Following the more accurate data a more
dramatic result appears: the recent analysis of the observation data
(especially from SNe Ia) indicate that the time varying dark energy gives a
better fit than a cosmological constant, and in particular, the equation of
state parameter $w$ (defined as the ratio of pressure to energy density)
crosses $-1$ at some low redshift region. This crossing behavior is a serious
challenge to fundamental physics. In this article, we review a number of
approaches which try to explain this remarkable crossing behavior. First we
show the key observations which imply the crossing behavior. And then we
concentrate on the theoretical progresses on the dark energy models which can
realize the crossing $-1$ phenomenon. We discuss three kinds of dark energy
models: 1. two-field models (quintom-like), 2. interacting models (dark energy
interacts with dark matter), and 3. the models in frame of modified gravity
theory (concentrating on brane world).

###### pacs:

95.36.+x 04.50.+h

##  I  The universe is accelerating

Cosmology is an old and young branch of science. Every nation had his own
creative idea about this subject. However, till the 1920s about the unique
observation which had cosmological significance was a dark sky at night. On
the other hand, we did not prepare a proper theoretical foundation till the
construction of general relativity. Einstein’s 1917 paper is the starting
point of modern cosmology e1 . The next mile stone was the discovery of cosmic
expansion, that was the recession of galaxies and the recession velocity was
proportional to the distance to us.

Except some rare cases, our researches are always based on the cosmological
principle, which says that the universe is homogeneous and isotropic. In the
early time, this is only a supposition to simplify the discussions. Now we
have enough evidences that the universe is homogeneous and isotropic at the
scale larger than 100 Mpc. The cosmological principle requires that the metric
of the universe is FRW metric,

$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)(dr^{2}+r^{2}d\Omega_{2}^{2});$ (1)
$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)(dr^{2}+\sin(r)^{2}d\Omega_{2}^{2});$
(2) $\displaystyle
ds^{2}=-dt^{2}+a^{2}(t)(dr^{2}+\sinh(r)^{2}d\Omega_{2}^{2}),$ (3)

depending on the spatial curvature, which can be Euclidean, spherical or
pseudo-spherical. Here, $t$ is the cosmic time, $a$ denotes the scale factor,
$r$ represents the comoving radial coordinate of the maximal symmetric
3-space, and $d\Omega_{2}^{2}$ stands for a 2-sphere. Which geometry serves
our space is decided by observations. FRW metric describes the kinetic
evolution of the universe. To describe the dynamical evolution of the
universe, that is, the function of $a(t)$, we need the gravity theory which
ascribes the space geometry to matter. The present standard gravity theory is
general relativity. In 1922 and 1924, Friedmann found that there was no static
cosmological solution in general relativity, that is to say, the universe is
either expanding or contracting f1 . To get a static universe, Einstein
introduce the cosmological constant. However, even in the Einstein universe,
where the contraction of the dust is exactly counteracted by the repulsion of
the cosmological constant, the equilibrium is only tentative since it is a
non-stationary equilibrium. Any small perturbation will cause it to contract
or expand. Hence, in some sense we can say that general relativity predicts an
expanding (or contracting) universe, which should be regarded as one of the
most important prediction of relativity.

In almost 70 years since the discovery of the cosmic expansion in 1929 h1 ,
people generally believe that the universe is expanding but the velocity is
slowing down. People try to understand via observation that the universe will
expand forever or become contracting at some stage. A striking result appeared
in 1998, which demonstrated that the universe is accelerating rather than
decelerating. Now we show how to conclude that our universe is accelerating.
We introduce the standard general relativity,

$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi Gt_{\mu\nu},$ (4)

where $G_{\mu\nu}$ is Einstein tensor, $G$ is (4-dimensional) Newton constant,
$t_{\mu\nu}$ denotes the energy-momentum tensor, $g_{\mu\nu}$ stands for the
metric of a spacetime, $\mu,\nu$ run from 0 to 3. Throughout this article, we
take a convention that $c=\hbar=1$ without special notation. Define a new
energy momentum $T_{\mu\nu}$,

$8\pi GT_{\mu\nu}=8\pi Gt_{\mu\nu}-\Lambda g_{\mu\nu}.$ (5)

$T_{\mu\nu}$ has included the contribution of the cosmological constant, whose
effect can not be distinguished with vacuum if we only consider gravity.

The $00$ component of Einstein equation (4) is called Friedmann equation,

$H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3}\rho,$ (6)

where $H=\frac{\dot{a}}{a}$ is the Hubble parameter, an overdot stands for the
derivative with respect to the cosmic time, $k$ is the spatial curvature of
the FRW metric, for (1), $k=0$; for (2), $k=1$; for (3), $k=-1$,
$\rho=-T^{0}_{0}$. Throughout this article, we take the signature
$(-,+,...,+)$. The spatial component of Einstein equation can be replaced by
the continuity equation, which is much more convenient,

$\dot{\rho}+3H(\rho+p)=0,$ (7)

where $p=T_{1}^{1}=T_{2}^{2}=T_{3}^{3}$. Here we use a supposition that the
source of the universe $T_{\mu}^{\nu}$ is in perfect fluid form. Using (6) and
(7), we derive the condition for acceleration,

$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p).$ (8)

We see that the universe is accelerating when $\rho+3p<0$. We know that the
galaxies and dark matter inhabit in the universe long ago. They are dust
matters with zero pressure. Thus, if the universe is accelerating, there must
exist an exotic matter with negative pressure or we should modify general
relativity. The cosmological constant is a far simple candidate for this
exotic matter, or dubbed dark energy. For convenience we separate the
contribution of the cosmological constant (dark energy) from other sectors in
the energy momentum, the Friedmann equation becomes,

$\frac{H^{2}}{H^{2}_{0}}=\Omega_{\Lambda
0}+\Omega_{m0}(1+z)^{3}+\Omega_{k0}(1+z)^{2},$ (9)

where $z$ is the redshift, the subscript 0 denotes the present value of a
quantity,

$\Omega_{\Lambda 0}=\frac{\Lambda}{8\pi H_{0}^{2}G},$ (10)
$\Omega_{m0}=\frac{\rho_{m0}}{8\pi H_{0}^{2}G},$ (11)
$\Omega_{k0}=-\frac{k}{H_{0}^{2}a_{0}^{2}}.$ (12)

The type Ia supernova is a most powerful tool to probe the expanding rate of
the universe. In short, type Ia supernova is a supernova which just reaches
the Chandrasekhar limit (1.4 solar mass) and then explodes. Hence they have
the same local luminosity since they have roughly the same mass and the same
exploding process. They are the standard candles in the unverse. We can get
the distance of a type Ia supernova through its apparent magnitude. A sample
of type Ia supernovae will generates a diagram of Hubble parameter versus
distance, through which we get the information of the expanding velocity in
the history of the universe. In 1998, two independent groups found that the
universe is accelerating using the observation data of supernovae acce . After
that the data accumulate fairly quickly. The famous sample includes Gold04
gold04 , Gold06 gold06 , SNLS snls , ESSENCE essence , Davis07 davis07 , Union
union , Constitution constitution . Here, we show some results of one of the
most recent sample, Union union , which is plotted by $\chi^{2}$ statistics.

Figure 1: The figure displays the counter constrained by SNe Ia (in blue), by
BAO (in green), by CMB (in yellow), and the joint constraint by all the three
kinds of observation (in black and white). $\Omega_{\Lambda}$ is
$\Omega_{\Lambda 0}$ in (10), $\Omega_{m}$ is $\Omega_{m0}$ in (11). This
figure is borrowed from union .

From fig 1, we see that: 1. the universe is almost spatially flat, that is the
curvature term $\Omega_{k0}$ is very small. 2. the present universe is
dominated by cosmological constant (dark energy), whose partition is
approximately 70%, and the partition of dust is 30%. We introduce a
dimensionless parameter, the deceleration parameter $q$,

$q=-\frac{\ddot{a}a}{\dot{a}^{2}}.$ (13)

A negative deceleration parameter denotes acceleration. In a universe with
dust and cosmological constant (which is called $\Lambda$CDM model), by
definition

$q=\frac{1}{2}\Omega_{m}-\Omega_{\Lambda},$ (14)

whose present value $q_{0}=-0.55$. Hence the present universe is accelerating.

The previous result depends on a special cosmological model, $\Lambda$CDM
model in frame of general relativity. How about the conclusion if we only
consider the kinetics of the universe?

Figure 2: Kinetic universe vs dynamic universe. The fitting results of sudden
transition model, linear expansion model, and $\Lambda$CDM model. The solid
line is the best-fit of the sudden transition model (the deceleration
parameter jumps at some redshift); the long-dashed line denotes the best-fit
of linear expansion model ($q=q_{0}+q_{1}z$, $q_{1}$ is a constant) gold04 ;
the short-dashed line represents the best-fit of $\Lambda$CDM model. From sha
.

The simplest kinetic model is a sudden transition model, in which the
deceleration parameter is a constant in some high redshift region and jumps to
another constant at a critical redshift. The other simple choice is that the
deceleration parameter is a linear function of $z$. We show the two kinetic
models with the dynamic model $\Lambda$CDM in fig 2. It is clear that the
universe accelerates in the present epoch in all the three models. A more
rigorous analyze shows that the evidence for an accelerating universe is
fairly strong (more than 5 $\sigma$) sha . So we should investigate it
seriously.

$\Lambda$CDM is the most simple model for the acceleration, which is a
concordance model of several observations. As we shown in fig 1, the counters
of CMB, BAO, and SNe Ia have cross section, which almost laps over result of
the joint fittings. However, $\Lambda$CDM has its own theoretical problems.
Furthermore, it is found that a dynamical dark energy model fits the
observation data better. Especially, there are some evidences that the
equation of state (EOS) of dark energy may cross $-1$, which is a serious
challenge to the foundation of theoretical physics.

In the next section we shall study some problems of $\Lambda$CDM model and
display that a dynamical dark energy model is favored by observations. We’ll
focus on the crossing behavior implied by the observation. In section III, we
study 3 kinds of models with a crossing phantom divide dark energy. In section
IV, we present the conclusion and more references of this topic.

##  II A dark energy with crossing $-1$ EOS is slightly favored by
observations

###  II.1  The problems of $\Lambda$CDM

$\Lambda$CDM has two famous theoretical problems.

The first is the finetune problem. The effect of the vacuum energy can not be
distinguished from the cosmological constant in gravity theory. We can
calculate the vacuum energy by a well-constructed theory, quantum field theory
(QFT), which says that the vacuum energy should be larger than the observed
value by 122 orders of magnitude, if QFT works well up to the Planck scale. In
supersymmetric (SUSY) theory, the vacuum energy of the Bosons exactly
counteracts the vacuum energy of Fermions, such that we obtain a zero vacuum
energy. However, SUSY must break at the electro-weak scale. At that scale, the
vacuum energy is still large than the observed value by 60 orders of
magnitude. So for getting a vacuum energy we observed, we should introduce a
bare cosmological constant $\Lambda_{\rm bare}$. The effective vacuum energy
$\rho_{\rm effect}$ then becomes,

$\rho_{\rm effect}=\frac{1}{8\pi G}\Lambda_{\rm bare}+\rho_{\rm vacuum}.$ (15)

$\frac{1}{8\pi G}\Lambda_{\rm bare}$ and $\rho_{\rm vacuum}$ have to almost
counteract each other but do not exactly counteract each other, leaving a tiny
tail which is smaller than the $\rho_{\rm vacuum}$ by 60 orders of magnitude.
Which mechanism can realize such a miraculous counteraction?

The second problem is coincidence problem, which says that the cosmological
constant keeps a constant while the density of the dust evolves as $(1+z)^{3}$
in the history of the universe, then why do they approximately equal each
other at “our era”? Different from the first problem, the second problem says
the present ratio of dark energy and dark matter is sensitively depends on the
initial conditions. Essentially, the coincidence problem is the problem of an
unnatural initial condition. The densities of different species in the
universe redshift with different rate in the evolution of the universe, so if
their densities coincidence in $our~{}era$, their density ratio must be a
specific, tiny number in the $early~{}universe$. It is also a finetune
problem, but a finetune problem of the initial condition.

Except the above theoretical problems, $\Lambda$CDM also suffers from
observation problem, especially when faced to the fine structure of the
universe, including galaxies, clusters and voids. Some specific observations
differ from the predictions of $\Lambda$CDM (with standard partitions of dust
and cosmological constant) at a level of 2$\sigma$ or higher. Six observations
are summarized in problem lcdm : 1\. scale velocity flows is much larger than
the prediction of $\Lambda$CDM, 2\. Type Ia Supernovae (Sne Ia) at High
Redshift are brighter than what $\Lambda$CDM indicates, 3. the void seems more
empty than what $\Lambda$CDM predicts, 4\. the cluster haloes look denser than
what $\Lambda$CDM says, 5\. the density function of galaxy haloes is smooth,
while $\Lambda$CDM indicates a cusp in the core, 6\. there are too much disk
galaxies than the prediction of $\Lambda$CDM. We do not fully understand the
dynamics and galaxies and galaxy clusters, that is, the gravitational
perturbation theory at the small scale. The agreement may approve when we
advance our perturbation theory with a cosmological constant and the
simulation methods at the small scale. However, in the cosmological scale,
there are also some evidences that the dark energy is dynamical, including no.
2 of the previous 6 problems.

###  II.2 crossing $-1$

With data accumulation, observations which favor dynamical dark energy become
more and accurate. Now we loose the condition that $p=-\rho$ for the exotic
matter (dark energy) which accelerates the universe. We go beyond the
$\Lambda$CDM model. We permit that the EOS of dark energy is not exactly equal
to $-1$, but still a constant. The fitting results by different samples of SNe
Ia are displayed in fig 3. We see that although a cosmological constant is
permitted, the dark energy whose EOS $<-1$ is favored by SNe Ia. The essence
whose EOS is less than $-1$ is called phantom, which can be realized by a
scalar field with negative kinetic term. The action for phantom $\psi$ is

$S_{\rm ph}=\int
d^{4}x\sqrt{-g}\left(\frac{1}{2}\partial_{\mu}\psi\partial^{\mu}\psi-U(\psi)\right),$
(16)

where $\sqrt{-g}$ is the determinate of the metric, the lowercase Greeks run
from 0 to 3, $U(\psi)$ denotes the potential of the phantom. In an FRW
universe, the density and pressure of the phantom (16) reduce to

$\rho=-\frac{1}{2}\dot{\psi}^{2}+U,$ (17) $p=-\frac{1}{2}\dot{\psi}^{2}-U,$
(18)

respectively. Now the EOS of the phantom $w$ is given by,

$w=\frac{p}{\rho}=\frac{-\frac{1}{2}\dot{\psi}^{2}-U}{-\frac{1}{2}\dot{\psi}^{2}+U},$
(19)

which is always less than $-1$ for a positive $U$. It seems that phantom is
proper candidate for the dark energy whose EOS less than $-1$ phantom . It is
very famous that a phantom field is unstable when quantized since the energy
has no lower bound. It will transit to a lower and lower energy state. In this
article we have no time to discuss this important topic for phantom dark
energy. We would point out the basic idea for this issue, which requires the
life time of the phantom is much longer than the age of the universe such that
we still have no chance to observe the decay of the phantom, though it is
fundamentally unstable. For references, see stablephan . A completely regular
quantum stress with $w<-1$ is suggested in phanquan .

Figure 3: The EOS of dark energy fitted by SNe Ia in a spatially flat
universe, the contours display 68.3 %, 95.4 % and 99.7% confidence level on
$w$ and $\Omega_{m0}$ ($\Omega_{m}$ in the figure). The results of the Union
set are shown as filled contours. The empty contours, from left to right, show
the results of the Gold sample, Davis 07, and the Union without SCP nearby
data. From union .

If the dark energy really behaves as phantom at some low redshift region, it
is an unusual discovery. But the dark energy may be more fantastic. In some
model-independent fittings, the EOS of dark energy crosses $-1$, which is a
really remarkable property and a serious challenge to our present theory of
fundamental physics.

Pioneer results of the crossing $-1$ of EOS of dark energy appeared in cross0
; cross1 . Fig 4 illuminates that the EOS of dark energy may cross $-1$ in
some low redshift. In fig 4, the Gold04 data are applied, a uniform prior of
$0.22\leq\Omega_{m0}\leq 0.38$ is assumed, and a spatially flat universe is
the working frame.

Figure 4: Panel (a): Uncorrelated band-power estimates of the EOS $w(z)$ of
dark energy by SNe Ia (Gold set gold04 ). Vertical error bars show the 1 and
2-$\sigma$ error bars (in blue and green, respectively). The horizontal error
bars denote the data bins used in cross1 . Panel (b): The window functions for
each bin from low redshift to high redshift. Panel (c): the likelihoods of
$w(z)$ in the bins from low redshift to high redshift. From cross1 .

The perturbation of the dark energy will growth if its EOS is not exactly $-1$
in the evolution history of the universe. Hence to fit a model with dynamical
dark energy with observation, the perturbation of the dark matter should be
considered in principle. Such a study was presented in cross2 , in which a
parametrization of the EOS of the dark energy with two constant
$w_{0},~{}w_{1}$ was applied,

$w=w_{0}+w_{1}\frac{z}{1+z}.$ (20)

The result is shown in fig 5.

Figure 5: Constraints on the EOS of w(z) by WMAP3 wmap3  and Gold04 gold04 .
The light grey region denotes 2 $\sigma$ constraint, while the dark grey for
1$\sigma$ constraint. The left panel shows the constraint with dark energy
perturbation, while the right displays the result without dark energy
perturbation. From cross2 .

We see from fig 5 that there is a mild tendency that the EOS of the dark
energy cross $-1$. For a more general parametrization of EOS for dark energy,
see zhupara .

With more and accurate data, the possibility of crossing $-1$ (phantom divide)
seems a little more specific, see for example crossmany . This crossing
behavior is a significant challenge for theoretical physics. It was proved
that the EOS of dark energy can not cross the phantom divide if 1. a dark
energy component with an arbitrary scalar-field Lagrangian, which has a
general dependence on the field itself and its first derivatives, 2\. general
relativity holds and 3. the spatially flat Friedmann universe nogo , for a
more detailed proof, wee the appendix of nogo2 . Thus realizing such a
crossing is not a trivial work. In the next section we investigate the
theoretical progresses for this extraordinary phenomenon.

##  III three roads to cross the phantom divide

To cross the phantom divide, we must break at least one of the conditions in
nogo . Now that the dark energy behaves as quintessence at some stage , while
evolves as phantom at the other stage, a natural suggestion is that we should
consider a 2-field model, a quintessence and a phantom. The potential is
carefully chosen such that the quintessence dominates the universe at some
stage while the phantom dominates the universe at the other stage. It was
invented a name for such 2-field model, “quintom” . There are also some
varieties of quintom, such as hessence. We introduce these 2-field models in
the first subsection. The next road is to consider an interacting model, in
which the dark energy interacts with dark matter. The interaction can realize
the crossing behavior which is difficult for independent dark energy. We shall
study the interacting models in subsection B. The other possibility is that
general relativity fails at the cosmological scale. The ordinary dark energy
candidates, such as quintessence or phantom, can cross the phantom divide in a
modified gravity theory. We investigate this approach in subsection C.

###  III.1 2-field model

A typical 2-field model is the quintom model, which was proposed in quintom1 ,
and was widely investigated later quintom2 . Generally , the action of a
universe with quintom dark energy $S$ is

$S=\int d^{4}x\sqrt{-g}\left(\frac{R}{16\pi G}+{\cal L_{\rm stuff}}\right),$
(21)

where $R$ is the Ricci scalar, $\cal L_{\rm stuff}$ encloses all kinds of the
stuff in the universe, for instance the dust matter, radiation, and quintom.
At the late universe, the radiation can be negligible. So, often we only
consider the dark energy, here quintom $\cal L_{\rm quintom}$, and dust matter
$\cal L_{\rm dm}$,

${\cal L_{\rm
quintom}}=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}\partial_{\mu}\psi\partial^{\mu}\psi-W(\phi,~{}\psi).$
(22)

In (22) the first term is the kinetic term of an ordinary scalar, the second
term is the kinetic term of a phantom, and $W(\phi,~{}\psi)$ is an arbitrary
function of $\phi$ and $\psi$. In an FRW universe, the density and pressure of
the quintom are

$\rho=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}+W,$ (23)
$p=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}-W.$ (24)

Hence, the EOS of the quintom $w$ is

$w=\frac{\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}-W}{\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}+W}.$
(25)

$w=-1$ requires

$\dot{\phi}^{2}=\dot{\psi}^{2}.$ (26)

We see that in a quintom model, we do not require a static field (a field with
zero kinetic term or a field at ground state) to get a cosmological constant.
We only need that $\psi$ and $\phi$ evolves in the same step. $w<-1$ implies,

$\dot{\phi}^{2}-\dot{\psi}^{2}<0,$ (27)

if

$\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}+W>0;$ (28)

and

$\dot{\phi}^{2}-\dot{\psi}^{2}>0,$ (29)

if

$\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\psi}^{2}+W<0.$ (30)

(30) yields an unnatural physical result,that is, the density of dark energy
is negative. However, this is not as serious as the first glance, since we
have little knowledge of the dark energy besides its effect of gravitation.
Several evidences imply that we should go beyond the standard model of the
particle physics when we describe dark energy. There are a few dark energy
models permit density of the dark energy, or a component of it is negative (at
the same time keep the total density positive), for example, see negden1 ;
negden2 . But, for a model with only two components, a dust and a quintom, it
is difficult to set a negative density dark energy. In that case we need too
much dust than we observed or a big curvature term. In the following text of
this section, we only consider a dark energy with positive density. So $w>-1$
implies,

$\dot{\phi}^{2}-\dot{\psi}^{2}>0.$ (31)

In summary, if the kinetic term of the quintessence dominates that of phantom,
the quintom behaves as quintessence; else it behaves as phantom. We should
select a proper potential to make quintessence and phantom dominate
alternatively such that we can realize the crossing behavior.

A simple choice of the potential is that the quintessence and the phantom do
not interact with each other, which requires, $W(\phi,\psi)=V(\phi)+U(\psi)$.
The exponential potential is an important example which can be solved exactly
in the quintessence model (a toy universe only composed by quintessence). In
addition, we know that such exponential potentials of scalar fields occur
naturally in some fundamental theories such as string/M theories. We introduce
a model with such potentials in quintomguo , in which the potential
$V(\phi,\psi)$ is given by

$W(\phi,\psi)=V(\phi)+U(\psi)=A_{\phi}e^{-\lambda_{\phi}\kappa\phi}+A_{\psi}e^{-\lambda_{\psi}\kappa\psi},$
(32)

where $A_{\phi}$ and $A_{\psi}$ are the amplitude of the potentials,
$\kappa^{2}=8\pi G$, $\lambda_{\phi}$ and $\lambda_{\psi}$ are two constants.
Since there is no direct couple between the quintessence and the phantom, the
equations of motion of the quintessence and the phantom are two independent
equations,

$\ddot{\phi}+3H\dot{\phi}+\frac{dV}{d\phi}=0,$ (33)

$\ddot{\psi}+3H\dot{\psi}-\frac{dU}{d\psi}=0.$ (34)

The continuity equation of the dust reads,

$\rho_{\rm dust}+3H\rho_{\rm dust}=0,$ (35)

where $\rho_{\rm dust}$ denotes the density of the dust. The method of
dynamical system has been widely used in cosmology. This method can offer a
clear history of the cosmic evolution, especially the final states of the
university. For applying this method, first we define the following
dimensionless variables,

$\displaystyle x_{\phi}\equiv\frac{\kappa\dot{\phi}}{\sqrt{6}H}$ ,
$\displaystyle\quad y_{\phi}\equiv\frac{\kappa\sqrt{V_{\phi}}}{\sqrt{3}H},$
$\displaystyle x_{\psi}\equiv\frac{\kappa\dot{\psi_{i}}}{\sqrt{6}H}$ ,
$\displaystyle\quad y_{\psi}\equiv\frac{\kappa\sqrt{V_{\psi}}}{\sqrt{3}H},$
(36) $\displaystyle z\equiv\frac{\kappa\sqrt{\rho_{\rm dust}}}{\sqrt{3}H}$ ,

the evolution equations (33)-(35) become,

$\displaystyle x^{\prime}_{\phi}$ $\displaystyle=$
$\displaystyle-3x_{\phi}\left(1+x_{\phi}^{2}-x_{\psi}^{2}-\frac{1}{2}z^{2}\right)+\lambda_{\phi}\frac{\sqrt{6}}{2}y_{\phi}^{2}\,,$
(37) $\displaystyle y^{\prime}_{\phi}$ $\displaystyle=$ $\displaystyle
3y_{\phi}\left(-x_{\phi}^{2}+x_{\psi}^{2}+\frac{1}{2}z^{2}-\lambda_{\phi}\frac{\sqrt{6}}{6}x_{\phi}\right),$
(38) $\displaystyle x^{\prime}_{\psi}$ $\displaystyle=$
$\displaystyle-3x_{\psi}\left(1+x_{\phi}^{2}-x_{\psi}^{2}-\frac{1}{2}z^{2}\right)-\lambda_{\psi}\frac{\sqrt{6}}{2}y_{\psi}^{2}\,,$
(39) $\displaystyle y^{\prime}_{\psi}$ $\displaystyle=$ $\displaystyle
3y_{\psi}\left(-x_{\phi}^{2}+x_{\psi}^{2}+\frac{1}{2}z^{2}-\lambda_{\psi}\frac{\sqrt{6}}{6}x_{\psi}\right),$
(40) $\displaystyle z^{\prime}$ $\displaystyle=$ $\displaystyle
3z\left(-x_{\phi}^{2}+x_{\psi}^{2}+\frac{1}{2}z^{2}-\frac{1}{2}\right),$ (41)

in which a prime denotes derivative with respect to $\ln a$. Generally, $z$ in
the above set will not be confused with redshift. The five equations in this
system are not independent. They are constrained by Fridemann equation,

$H^{2}=\frac{\kappa^{2}}{3}\left(\frac{1}{2}\dot{\phi}^{2}+V-\frac{1}{2}\dot{\psi}^{2}+U+\rho_{\rm
dust}\right),$ (42)

which becomes

$x_{\phi}^{2}+y_{\phi}^{2}-x_{\psi}^{2}+y_{\psi}^{2}+z^{2}=1.$ (43)

with the dimensionless variables defined before. The critical points dwell at
$x_{\phi}^{\prime}=y_{\phi}^{\prime}=x_{\psi}^{\prime}=y_{\psi}^{\prime}=z^{\prime}=0$.
We present the result in table 1.

Label | $x_{\psi}$ | $y_{\psi}$ | $x_{\phi}$ | $y_{\phi}$ | z | Stability
---|---|---|---|---|---|---
$K$ | $-x_{\psi}^{2}+x_{\phi}^{2}=1$ | 0 |  | 0 | 0 | unstable
$P$ | $-\frac{\lambda_{\psi}}{\sqrt{6}}$ | $\sqrt{(1+\frac{\lambda_{\psi}^{2}}{6})}$ | 0 | 0 | 0 | stable
$S$ | 0 | 0 | $\frac{\lambda_{\phi}}{\sqrt{6}}$ | $\sqrt{(1-\frac{\lambda_{\phi}^{2}}{6})}$ | 0 | unstable
$F$ | 0 | 0 | 0 | 0 | 1 | unstable
$T$ | 0 | 0 | $\frac{3}{\sqrt{6}\lambda_{\phi}}$ | $\frac{\sqrt{{3}}}{\lambda_{\phi}}$ | $\sqrt{1-\frac{3}{\lambda_{\phi}^{2}}}$ | unstable
Table 1:  The critical points, from quintomguo

For detailed discussion of the critical points, see quintomguo . We would like
to show a numerical example in which the EOS of the quintom crosses the
phantom divide. Fig 6 illuminates that the EOS crosses $-1$.

Figure 6: The evolution of the effective equation of state of the phantom and
normal scalar fields with $W(\phi,\sigma)$ for the case $\lambda_{\phi}=1$.
From quintomguo .

The previous quintom model includes two fields, which are completely
independent and rather arbitrary. We can impose some symmetry in the quintom
model. An interesting model with an internal symmetry between the two fields
which work as dark energy is hessence hessence . Rather than two uncorrelated
fields, we consider one complex scalar field with internal symmetry between
the real and the imaginary parts,

$\Phi=\phi_{1}+i\phi_{2},$ (44)

with a Lagrangian density

${\cal L}_{\rm
hess}=-\frac{1}{4}\left[(\partial_{\mu}\Phi)^{2}+(\partial_{\mu}\Phi^{*})^{2}\right]-V(\xi,\Phi^{\ast})=-\frac{1}{2}\left[\,(\partial_{\mu}\xi)^{2}-\xi^{2}(\partial_{\mu}\theta)^{2}\,\right]-V(\xi),$
(45)

which is invariant under the transformation,

$\displaystyle\phi_{1}\to\phi_{1}\cos\alpha-i\phi_{2}\sin\alpha,$ (46)
$\displaystyle\phi_{2}\to-i\phi_{1}\sin\alpha+\phi_{2}\cos\alpha,$ (47)

if the potential is only a function of $\Phi^{2}+(\Phi^{*})^{2}$. For
convenience, in (45) we have introduced two new variables $(\xi,\theta)$,

$\phi_{1}=\xi\cosh\theta,~{}~{}~{}~{}~{}~{}~{}\phi_{2}=\xi\sinh\theta,$ (48)

which are defined by

$\xi^{2}=\phi_{1}^{2}-\phi_{2}^{2},~{}~{}~{}~{}~{}~{}~{}\coth\theta=\frac{\phi_{1}}{\phi_{2}}.$
(49)

The equations of motion of $\xi$ and $\theta$ are

$\ddot{\xi}+3H\dot{\xi}+\xi\dot{\theta}^{2}+\frac{dV}{d\xi}=0,$ (50)
$\xi^{2}\ddot{\theta}+(2\xi\dot{\xi}+3H\xi^{2})\dot{\theta}=0.$ (51)

Clearly, $\xi$ and $\theta$ couple to each other. The pressure and density of
the hessence read,

$p_{\rm
hess}=\frac{1}{2}\left(\dot{\xi}^{2}-\xi^{2}\dot{\theta}^{2}\right)-V(\xi),$
(52) $\rho_{\rm
hess}=\frac{1}{2}\left(\dot{\xi}^{2}-\xi^{2}\dot{\theta}^{2}\right)+V(\xi),$
(53)

respectively. The EOS of hessence, playing as dark energy,

$w=\frac{\frac{1}{2}\left(\dot{\xi}^{2}-\xi^{2}\dot{\theta}^{2}\right)-V(\xi)}{\frac{1}{2}\left(\dot{\xi}^{2}-\xi^{2}\dot{\theta}^{2}\right)+V(\xi)}.$
(54)

Qualitatively, hessence evolves as quintessence when
$\dot{\xi}^{2}\geq\xi^{2}\dot{\theta}^{2}$, while as phantom when
$\dot{\xi}^{2}<\xi^{2}\dot{\theta}^{2}$. The Lagrangian (45) does not include
$\theta$, hence the canonical momentum $\pi_{\theta}^{\mu}$ corresponding to
the cyclic coordinate $\theta$ are conserved quantities,

$\pi_{\theta}^{\mu}=\frac{\partial({\cal L}_{\rm
hess}\sqrt{-g})}{\partial(\partial_{\mu}\theta)}.$ (55)

In an FRW universe, only $\pi_{\theta}^{0}$ exists. We define a conserved
quantity $Q$ which is proportional to $\pi_{\theta}^{0}$,

$Q=a^{3}\xi^{2}\dot{\theta}.$ (56)

With this conserved quantity, the EOS becomes,

$w=\frac{\frac{1}{2}\dot{\xi}^{2}-\frac{Q^{2}}{2a^{6}\xi^{2}}-V(\xi)}{\frac{1}{2}\dot{\xi}^{2}-\frac{Q^{2}}{2a^{6}\xi^{2}}+V(\xi)},$
(57)

which is only a function of $\xi$. The Friedmann equations read as

$\displaystyle H^{2}=\frac{8\pi G}{3}\left[\rho_{\rm
dust}+\frac{1}{2}\left(\dot{\xi}^{2}-\xi^{2}\dot{\theta}^{2}\right)+V(\xi)\right],$
(58)

where $\rho_{\rm dust}$ is the energy density of dust. The continuity equation
of dust is (35). The continuity equations of hessence are identical to the
equations of motion (50) and (51). Then the system is closed and we present a
numerical example in fig 7. Evidently, the EOS of hessence, playing the role
of dark energy, crosses $-1$ at about $a=0.95$ ($z=0.06$).

Figure 7: The EOS of hessence $w$ as a function of scale factor with the
potential $V(\xi)=\lambda\xi^{4}$. The parameters for this plot are as
follows: $\Omega_{m0}=\rho_{m0}/(3H_{0}^{2})=0.3$, $\lambda=5.0$, $Q=1.0$,
$a_{0}=1$ and the unit $8\pi G=1$. From hessence .

After the presentation of hessence model, several aspects of this model have
been investigated, including to avoid the big rip wei2 , attractor solutions
for general hessence hess3 , reconstruction of hessence by recent observations
hess4 , dynamics of hessence in frame of loop quantum cosmology hess5 , and
holographic hessence modelhess6 .

###  III.2 interacting model

Two-field model is a natural and obvious construction to realize the crossing
$-1$ behavior of dark energy. However, there are two many parameters in the
set-up, though we can impose some symmetries to reduce the parameters to a
smaller region. One symmetry decrease one parameter, but we have little clue
to impose the symmetries since we have no evidence in the ground labs.

Interaction is a universal phenomenon in the physics world. An interaction
term is helpful to cross the phantom divide. To illuminate this point, we
first carefully analyze the previous observations which imply the crossing.
Both the results of cross1  and cross2 , which are shown in fig 4 and fig 5
respectively, are derived with a presupposition, that is, the dark energy
evolves freely. In fact, what we observed is the effective EOS of the dark
energy in the sense of gravity at the cosmological scale. When we suppose it
evolves freely, we find that its EOS may cross the phantom divide. We can
demonstrate for an essence with (local) EOS$<-1$, the cosmological effective
EOS can cross $-1$ by aids of an interacting term. For the case with
interaction, the continuity equation for dark energy becomes,

$\dot{\rho}_{\rm de}+3H(\rho_{\rm de}+p_{\rm de})=-\Gamma,$ (59)

or

$\dot{\rho}_{\rm de}+3H(\rho_{\rm de}+p_{\rm de}+\frac{\Gamma}{3H})=0.$ (60)

Here $\rho_{\rm de}$ is the density of dark energy, $p_{\rm de}$ denotes the
local pressure measured in the lab (if we can measure), $\Gamma$ stands for
the interaction term, and $p_{\rm eff}=p_{\rm de}+\frac{\Gamma}{3H}$ is the
effective pressure in the cosmological sense. In a universe without expanding
or contracting, $H=0$, the interaction does no effect on the continuity
equation, or energy conservation law, and thus does not yield surplus pressure
222One may think that $\Gamma/H$ is meaningless when $H=0$. But in fact, in
most realistic cases, we always assume that $\Gamma$ is proportional to $H$..
Two special cases are interesting: 1\. $\frac{\Gamma}{3H}$ is a constant,
under which the interaction term contributes a constant pressure throughout
the history of the universe. 2. $\frac{\Gamma}{3H\rho_{de}}$ is a constant,
under which the interaction term contributes a constant EOS in the history of
the universe. In frame of a quintessence or phantom dark energy, the
interaction term $\frac{\Gamma}{3H\rho_{de}}$ only shifts the EOS up or down
by a constant distance in the $w-z$ plane, without changing the profile of the
curve of $w$. While the term $\frac{\Gamma}{3H}$ shifts the pressure, which
can change the EOS significantly since the density $\rho_{\rm de}$ is a
variable in the history of the universe.

If the dark energy can couple to some stuff of the universe, the dark matter
is the best candidate. Although non-minimal coupling between the dark energy
and ordinary matter fluids is strongly restricted by the experimental tests in
the solar system will , due to the unknown nature of the dark matter as part
of the background, it is possible to have non-gravitational interactions
between the dark energy and the dark matter components, without conflict with
the experimental data. The continuity equation for dust-like dark matter
reads,

$\dot{\rho}_{\rm dm}+3H\rho_{\rm dm}=\Gamma.$ (61)

Based on the previous discussion, we assume a most simple case

$\Gamma=H\delta\rho_{\rm dm},$ (62)

where $\delta$ is constant guointer ; weiinter ; amen . This interaction term
shifts a constant to the EOS of $dark~{}matter$, that is, it is no longer
evolving as $(1+z)^{3}$. We uniformly deal with quintessence and phantom,
which are often labeled by $X$, with a constant EOS $w_{X}$. So, the
continuity equation of dark energy can be written as,

$\dot{\rho}_{\rm X}+3H(\rho_{\rm X}+w_{X}\rho_{\rm X})=-H\delta\rho_{\rm dm}.$
(63)

Integrating (61), we derive

$\rho_{\rm dm}=\rho_{\rm dm0}a^{-3+\delta}=\rho_{\rm dm0}(1+z)^{3-\delta}.$
(64)

Substituting to (63), we reach

$\rho_{X}=\rho_{X0}(1+z)^{3(1+w_{X})}+\rho_{dm0}\frac{\delta}{\delta+3w_{X}}\left[(1+z)^{3(1+w_{X})}-(1+z)^{3-\delta}\right].$
(65)

Only from the above equation, we can extract the effective EOS of the dark
energy. To see this point, we make a short discussion. In a dynamical universe
with interaction, the effective EOS of dark energy reads,

$w_{de}=\frac{p_{\rm eff}}{\rho_{\rm de}}=\frac{p_{\rm
de}+\Gamma/3H}{\rho_{\rm de}}=-1+\frac{1}{3}\frac{d\ln\rho_{de}}{d\ln(1+z)}.$
(66)

Clearly, if $\frac{d\ln\rho_{de}}{d\ln(1+z)}$ is greater than 0, dark energy
evolves as quintessence; if $\frac{d\ln\rho_{de}}{d\ln(1+z)}$ is less than 0,
it evolves as phantom; if $\frac{d\ln\rho_{de}}{d\ln(1+z)}$ equals 0, it is
just cosmological constant. In a more intuitionistic way, if $\rho_{de}$
decreases and then increases with respect to redshift (or time), or increases
and then decreases, which implies that EOS of dark energy crosses phantom
divide. So, some time we directly use the evolution of density of dark energy
to describe the EOS of it. There is a more important motivation to use the
density directly: the density is more closely related to observables, hence is
more tightly constrained for the same number of redshift bins used wangyun .

The derivative of $\rho_{X}$ with respect to (1+z) reads,

$\frac{d\rho_{X}}{d(1+z)}=3(1+w_{X})\rho_{X0}(1+z)^{2+3w_{X}}+\rho_{dm0}\frac{\delta}{\delta+3w_{X}}\left[3(1+w_{X})(1+z)^{2+3w_{X}}-(3-\delta)(1+z)^{2-\delta}\right].$
(67)

If $\frac{d\rho_{X}}{d(1+z)}=0$ at some redshift $z=z_{c}$, the effective EOS
crosses $-1$. The result is illuminated by fig 8, in which we set $z_{c}=0.3$
as an example. This figure displays the corresponding $w_{X}$ when one fixes a
$\delta$, or vice versa if we require the EOS crosses $-1$ at $z_{c}=0.3$.
This is an original figure plotted for this review article.

Figure 8:  $w_{X}$ vs $\delta$ under the condition
$\frac{d\rho_{X}}{d(1+z)}=0$.

Then the Friedmann equation reads,

$\frac{H^{2}}{H_{0}^{2}}=\Omega_{X0}(1+z)^{3(1+w_{X})}+\frac{1-\Omega_{X0}}{\delta+3w_{X}}\left[\delta(1+z)^{3(1+w_{X})}+3w_{X}(1+z)^{3-\delta}\right],$
(68)

where $\Omega_{\rm X0}=\kappa^{2}\rho_{\rm X0}/(3H_{0}^{2})$, and we have used
$\Omega_{\rm dm0}+\Omega_{\rm X0}=1$. Thus we need to constrain the three
parameters $\delta,\Omega_{\rm X0},w_{\rm X}$. The constraint result by SNLS
data is shown in fig 9. $\delta=0$ and $w_{X}=-1$ are indicated by the
horizontal and vertical dashed lines, which represent the non-interacting XCDM
model and interacting $\Lambda$CDM model, respectively.

Figure 9: Constraints of ($w_{X},\delta$) by SNLS data at 68.3%, 95.4% and
99.7% confidence levels marginalized over $\Omega_{X0}$ with priors
$\Omega_{X0}=0.72\pm 0.04$ and $\delta<3$. From guointer

From fig 8 and 9, we see that the observations leave enough space for the
parameters ($\delta$, $w_{X}$) to cross the phantom divide.

In the previous interacting model, we consider a phenomenological interaction,
which is put in “by hand”. We should find a more sound physical foundation for
the interactions. We will deduce an interaction term from the low energy limit
of string/M theory in the scenario of the interacting Chaplygin gas model
negden1 .

The Chaplygin gas model was suggested as a candidate of a unified model of
dark energy and dark matter cp . The Chaplygin gas is characterized by an
exotic equation of state

$p_{ch}=-A/\rho_{ch},$ (69)

where $A$ is a positive constant. The above equation of state leads to a
density evolution in the form

$\rho_{ch}=\sqrt{A+\frac{B}{a^{6}}},$ (70)

where $B$ is an integration constant. The attractive feature of the model is
that it naturally unifies both dark energy and dark matter. The reason is
that, from (70), the Chaplygin gas behaves as dust-like matter at early stage
and as a cosmological constant at later stage.

Though Chaplygin gas has such a nice property, it is a serious flaw when one
studies the fluctuation growth in Chaplygin gas model. It is found that
Chaplygin gas produces oscillations or exponential blowup of the matter power
spectrum, which is inconsistent with observations antiudm . So we turn to a
model that the Chaplygin gas only plays the role of dark energy. To cross the
phantom divide we consider a model in which the Chaplygin gas couples to dark
mater.

Although non-minimal coupling between the dark energy and ordinary matter
fluids is strongly restricted by the experimental tests in the solar system
will , due to the unknown nature of the dark matter as part of the background,
it is possible to have non-gravitational interactions between the dark energy
and the dark matter components, without conflict with the experimental data.
Thus, the observation constrain the only proper candidate to be coupled to
Chaplygin gas is dark matter.

We consider the original Chaplygin gas, whose pressure and energy density
satisfy the relation, $p_{ch}=-A/\rho_{ch}$. By assuming the cosmological
principle the continuity equations are written as

$\dot{\rho}_{ch}+3H\gamma_{ch}\rho_{ch}=-\Gamma,$ (71)

and

$\dot{\rho}_{dm}+3H\gamma_{dm}\rho_{dm}=\Gamma,$ (72)

where the subscript $dm$ denotes dark matter, and $\gamma$ is defined as

$\gamma=1+\frac{p}{\rho}=1+w,$ (73)

in which $w$ is the parameter of the state of equation, and $\gamma_{dm}=1$
throughout the evolution of the universe, whereas $\gamma_{ch}$ is a variable.

$\Gamma$ is the interaction term between Chaplygin gas and dark matter. Since
there does not exist any microphysical hint on the possible nature of a
coupling between dark matter and Chaplygin gas (as dark energy), the
interaction terms between dark energy and dark matter are rather arbitrary in
literatures inter . Here we try to present a possible origin from fundamental
field theory for $\Gamma$.

Whereas we are still lack of a complete formulation of unified theory of all
interactions (including gravity, electroweak and strong), there at present is
at least one hopeful candidate, string/M theory. However, the theory is far
away from mature such that it is still not known in a way that would enable us
to ask the questions about space-time in a general manner, say nothing of the
properties of realistic particles. Instead, we have to either resort to the
effective action approach which takes into account stringy phenomena in
perturbation theory, or we could study some special classes of string
solutions which can be formulated in the non-perturbative regime. But the
latter approach is available only for some special solutions, most notably the
BPS states or nearly BPS states in the string spectrum: They seems to have no
relation to our realistic Universe. Especially, there still does not exist a
non-perturbative formulation of generic cosmological solutions in string
theory. Hence nearly all the investigations of realistic string cosmologies
have been carried out essentially in the effective action range. Note that the
departure of string-theoretic solutions away from general relativity is
induced by the presence of additional degrees of freedom which emerge in the
massless string spectrum. These fields, including the scalar dilaton field,
the torsion tensor field, and others, couple to each other and to gravity non-
minimally, and can influence the dynamics significantly. Thus such an
effective low energy string theory deserve research to solve the dark energy
problem. There a special class of scalar-tensor theories of gravity is
considered to avoid singularities in cosmologies in st . The action is written
below,

$\displaystyle S_{st}=\int d^{4}x\sqrt{-g}\left[\frac{1}{16\pi
G}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{q(\phi)^{2}}L_{\rm
dm}(\xi,\partial\xi,q^{-1}g_{\mu\nu})\right],$ (74)

where $G$ is the Newton gravitational constant, $\phi$ is a scalar field,
$L_{\rm dm}$ denotes Lagrangian of matter , $\xi$ represents different matter
degrees of matter fields, $q$ guarantees the coupling strength between the
matter fields and the dilaton. With action (74), the interaction term can be
written as follow st ,

$\Gamma=H\rho_{\rm dm}\frac{d\ln q^{\prime}}{d\ln a}.$ (75)

Here we introduce new variable $q(a)^{\prime}\triangleq q(a)^{(3w_{n}-1)/2}$,
where $a$ is the scale factor in standard FRW metric. By assuming

$q^{\prime}(a)=q_{0}e^{3\int c(\rho_{\rm dm}+\rho_{\xi})/\rho_{\rm dm}d\ln
a},$ (76)

where $\rho_{\rm dm}$ and $\rho_{\xi}$ are the densities of dark matter and
the scalar field respectively, one arrive at the interaction term,

$\Gamma=3Hc(\rho_{\rm dm}+\rho_{\phi}).$ (77)

With this interaction form we study the equation set (71) and (72). Set
$s=-\ln(1+z)$, $\Gamma=3Hc(\rho_{ch}+\rho_{dm})$,
$u=(3H_{0}^{2})^{-1}(3\mu^{2})^{-1}\rho_{dm}$,
$v=(3H_{0}^{2})^{-1}(3\mu^{2})^{-1}\rho_{ch}$,
$A^{\prime}=A(3H_{0}^{2})^{-2}(3\mu^{2})^{-2}$, where $c$ is a constant
without dimension. Using these variables, (71) and (72) reduce to

$\frac{du}{ds}=-3u+3c(u+v),$ (78) $\frac{dv}{ds}=-3(v-A^{\prime}/v)-3c(u+v).$
(79)

We note that the variable time does not appear in the dynamical system (78)
and (79) because time has been completely replaced by redshift $s=-ln(1+z)$.
The critical points of dynamical system (78) and (79) are given by

$\frac{du}{ds}=\frac{dv}{ds}=0.$ (80)

The solution of the above equation is

$\displaystyle u_{c}=\frac{c}{1-c}v_{c},$ (81) $\displaystyle
v_{c}^{2}=(1-c)A^{\prime}.$ (82)

We see the final state of the model contains both Chaplygin gas and dark
matter of constant densities if the singularity is stationary. The final state
satisfies perfect cosmological principle: the universe is homogeneous and
isotropic in space, as well as constant in time. Physically $\Gamma$ in (72)
plays the role of matter creation term $C$ in the theory of steady state
universe at the future time-like infinity. Recall that $c$ is the coupling
constant, may be positive or negative, corresponds the energy to transfer from
Chaplygin gas to dark matter or reversely. $A^{\prime}$ must be a positive
constant, which denotes the final energy density if $c$ is fixed. Also we can
derive an interesting and simple relation between the static energy density
ratio

$c=\frac{r_{s}}{1+r_{s}},$ (83)

where

$r_{s}=\lim_{z\to-1}\frac{\rho_{dm}}{\rho_{ch}}.$ (84)

To investigate the properties of the dynamical system in the neighbourhood of
the singularities, impose a perturbation to the critical points,

$\displaystyle\frac{d(\delta u)}{ds}=-3\delta u+3c(\delta u+\delta v),$ (85)
$\displaystyle\frac{d(\delta v)}{ds}=-3(\delta
v+\frac{A^{\prime}}{v_{c}^{2}}\delta v)-3c(\delta u+\delta v).$ (86)

The eigen equation of the above linear dynamical system $(\delta u,\delta v)$
reads

$(\lambda/3)^{2}+(2+\frac{1}{1-c})\lambda/3+2-2c^{2}=0,$ (87)

whose discriminant is

$\Delta=[(1-c)^{4}+(3/2-c)^{2}]/(1-c)^{2}\geq 0.$ (88)

Therefore both of the two roots of eigen equation (87) are real, consequently
centre and focus singularities can not appear. Furthermore only
$r_{s}\in(0,\infty)$, such that $c\in(0,1)$, makes physical sense. Under this
condition it is easy to show that both the two roots of (87) are negative.
Hence the two singularities are stationary. However it is only the property of
the linearized system (85) and (86), or the property of orbits of the
neighbourhoods of the singularities, while global Poincare-Hopf theorem
requires that the total index of the singularities equals the Euler number of
the phase space for the non-linear system (78) and (79). So there exists other
singularity except for the two nodes. In fact it is a non-stationary saddle
point at $u=0,~{}v=0$ with index $-1$. This singularity has been omitted in
solving equations (78) and (79). The total index of the three singularities is
$1$, which equals the Euler number of the phase space of this plane dynamical
system. Hence there is no other singularities in this system. From these
discussions we conclude that the global outline of the orbits of this non-
linear dynamical system (78) and (79) is similar to the electric fluxlines of
two negative point charges. Here we plot figs 10 and 11 to show the properties
of evolution of the universe controlled by the dynamical system (78) and (79).
As an example we set $c=0.2,~{}~{}A^{\prime}=0.9$ in figs 10-12.

Figure 10: The plane v versus u. (a) left panel: We consider the evolution of
the universe from redshift $z=e^{2}-1$. The initial condition is taken as
$u=0,~{}v=400$; $u=50,~{}v=350$; $u=100,~{}v=300$; $u=120,~{}v=280$ on the
four orbits, from the left to the right, respectively. It is clear that there
is a stationary node, which attracts most orbits in the first quadrant. At the
same time the orbits around the neibourhood of the singularity is not shown
clearly. (b) right panel: Orbit distributions around the node
$u_{c}=v_{c}c/(1-c),~{}v_{c}=\sqrt{(1-c)A^{\prime}}$. From negden1

Figure 11: The plane v versus u. (a) left panel: To show the global properties
of dynamical system (78) and (79) we have to include some “unphysical ”
initial conditions, such as $u=-100,~{}v=-300$, except for physical initial
conditions which have been shown in figure 10. (b) right panel: Orbits
distributions around the nodes. The two nodes
$u_{c}=v_{c}c/(1-c),~{}v_{c}=\sqrt{(1-c)A^{\prime}}$ and
$u_{c}=v_{c}c/(1-c),~{}v_{c}=-\sqrt{(1-c)A^{\prime}}$ keep reflection symmetry
about the original point. Just as we have analyzed, we see that the orbits of
this dynamical system are similar to the electric fluxlines of two negative
point charges. From negden1

Further, to compare with observation data we need the explicit forms of $u(x)$
and $v(x)$, especially $v(x)$. We need the properties of $\gamma_{ch}$ in our
model, which is contained in $v(x)$, to compare with observations. Eliminate
$u(x)$ by using (78) and (79) we derive

$\displaystyle\frac{1}{3c}\frac{d^{2}v}{ds^{2}}+[1+(1+A^{\prime}/v^{2})/c]\frac{dv}{ds}+3cv+3(1-c)\left\\{v+\left[\frac{dv}{ds}+3(v-A^{\prime}/v)\right]/(3c)\right\\}=0,$
(89)

which has no analytic solution. We show some numerical solutions in figure 12.
We find that for proper region of parameter spaces, the effective equation of
state of Chaplygin gas crosses the phantom divide successfully.

Figure 12: v versus s. The evolution of $v$ with different initial conditions
$u(-2)=0,~{}v(-2)=400$; $u(-2)=50,~{}v(-2)=350$; $u(-2)=100,~{}v(-2)=300$
reside on the blue, red, and yellow curves, respectively. Obviously the energy
density of Chaplygin gas rolls down and then climbs up in some low redshift
region. So the Chaplygin gas dark energy can cross the phantom divide $w=-1$
in a fitting where the dark energy is treated as an independent component to
dark matter. From negden1 , this figure has been re-plotted.

Up to now all of our results do not depend on Einstein field equation. They
only depend on the most sound principle in physics, that is, the continuity
principle, or the energy conservation law. Different gravity theories
correspond to different constraints imposed on our previous discussions. Our
improvements show how far we can reach without information of dynamical
evolution of the universe.

(78) illuminates that the dark matter in this interacting model does not
behaves as dust. Qualitatively, the dark matter gets energy from dark energy
for a positive $c$, and becomes soft, ie, its energy density decreases slower
than $(1+z)^{3}$ in an expanding universe. The parameter which carries the
total effects of cosmic fluids is the deceleration parameter $q$. From now on
we introduce the Friedmann equation of the standard general relativity. As a
simple case we study the evolution of $q$ in a spatially flat universe. So $q$
reads

$q=-\frac{\ddot{a}a}{\dot{a}^{2}}=\frac{1}{2}\left(\frac{u+v-3A^{\prime}/v^{2}}{u+v}\right),$
(90)

and density of Chaplygin gas $u$ and density of dark matter $v$ should satisfy

$u(0)+v(0)=1.$ (91)

And then Friedmann equation ensures the spatial flatness in the whole history
of the universe. Before analyzing the evolution of $q$ with redshift, we first
study its asymptotic behaviors. When $z\to\infty$, $q$ must go to $1/2$
because both Chaplygin gas and dark matter behave like dust , while when
$z\to-1$ $q$ is determined by

$\lim_{z\to-1}q=\frac{1}{2}\left(\frac{u_{c}+v_{c}-3A^{\prime}/v_{c}^{2}}{u_{c}+v_{c}}\right).$
(92)

One can finds the parameters $c=0.2,~{}A^{\prime}=0.9$ are difficult to
content the previous constraint Friedmann constraint (91). Here we carefully
choose a new set of parameter which satisfies Friedmann constraint (91), say,
$A^{\prime}=0.4,~{}c=0.06$. Therefore we obtain

$\lim_{z\to-1}q=-1.95,$ (93)

by using (81) and (82). Then we plot figure 13 to clearly display the
evolution of $q$. One can check
$u(0)=0.25,~{}v(0)=0.75;~{}u(0)=0.28,~{}v(0)=0.72;~{}u(0)=0.3,~{}v(0)=0.7$,
respectively on the curves $v(-2)=273;~{}v(-2)=250;~{}v(-2)=233$. One may find
an interesting property of the deceleration parameter displayed in fig 13: the
bigger the proportion of the dark energy, the smaller the absolute value of
the deceleration parameter. The reason roots in the extraordinary state of
Chaplygin gas (69), in which the pressure $p_{ch}$ is inversely proportional
to the energy density $\rho_{ch}$.

Figure 13: q versus s. The evolution of $q$ with different initial conditions
$u(-2)=0,~{}v(-2)=273$; $u(-2)=15,~{}v(-2)=250$; $u(-2)=25,~{}v(-2)=233$,
reside on the blue, red, and yellow curves, respectively. Evidently the
deceleration parameter $q$ of Chaplygin gas rolls down and crosses $q=0$ in
some low redshift region. The transition from deceleration phase to
acceleration phase occurs at $z=0.18;~{}z=0.21;~{}z=0.23$ to the curves
$u(-2)=0,~{}v(-2)=273$; $u(-2)=15,~{}v(-2)=250$; $u(-2)=25,~{}v(-2)=233$,
respectively. One finds $-q\thickapprox 0.5\sim 0.6$ at $z=0$, which is well
consistent with observations. From negden1 , this figure has been re-plotted.

Also we note that maybe an FRW universe with non-zero spatial curvature fits
deceleration parameter better than spatially flat FRW universe. This point
deservers to research further.

After the presentation of the original interacting Chaplygin gas model, there
are several generalizations. For details of these generalizations, see
generalcp .

###  III.3 model in frame of modified gravity

The judgement that there exists an exotic component with negative pressure, or
dark energy, which accelerates the universe, is derived in frame of general
relativity. The validity of general relativity has been well tested from the
scale of millimeter to the scale of the solar system. Beyond this scale, the
evidences are not so sound. So we should not be surprised if general
relativity fails at the scale of the Hubble radius. Surely, any new gravity
theory must reduce to general relativity at the scale between millimeter to
the solar system. In frame of the new gravity theories, the cosmic
acceleration may be a natural result even we only have dust in the universe.

There are various suggestions on how to modify general relativity. In this
brief review we concentrates on the brane world theory. Inspired by the
developments of string/M theory, the idea that our universe is a 3-brane
embedded in a higher dimensional spacetime has received a great deal of
attention in recent years. In this brane world scenario, the standard model
particles are confined on the 3-brane, while the gravitation can propagate in
the whole space. In this picture, the gravity field equation gets modified at
the left hand side (LHS) in (4), while the dark energy is a stuff put at the
right hand side (RHS) in (4). In the modified gravity model, the surplus
geometric terms respective to the Einstein tensor play the role of the dark
energy in general relativity.

We consider a 3-brane imbedded in a 5-dimensional bulk. The action includes
the action of the bulk and the action of the brane,

$S=S_{\rm bulk}+S_{\rm brane}.$ (94)

Here

$S_{\rm bulk}=\int_{\cal M}d^{5}X\sqrt{-{g_{5}}}{\cal L}_{\rm bulk},$ (95)

where $X=(t,z,x^{1},x^{2},x^{3})$ is the bulk coordinate, $x^{1},x^{2},x^{3}$
are the coordinates of the maximally symmetric space. ${\cal M}$ denotes the
bulk manifold. The bulk Lagrangian can be

${\cal L}_{\rm bulk}={1\over 2\kappa_{5}^{2}}\left[R_{5}+\alpha
F(R_{5})\right]+{\cal L}_{\rm m}+\Lambda_{5},$ (96)

where ${g_{5}}$, $\kappa_{5}$, $R_{5}$, ${\cal L}_{\rm m}$, denote the bulk
manifold, the determinant of the bulk metric, the 5-dimensional Newton
constant, the 5-dimensional Ricci scalar, and the bulk matter Lagrangian,
respectively. $F(R_{5})$ denotes the higher order term of scalar curvature
$R_{5}$, the Ricci curvature $R_{5{\rm AB}}$, the Riemann curvature $R_{5{\rm
ABCD}}$.

There are too much possibilities and rather arbitrary to choose the higher
order terms. Generally the resulting equations of motion of such a term give
more than second derivatives of metric and the resulting theory is plagued by
ghosts. However there exists a combination of quadratic terms, called Gauss-
Bonnet term, which generates equation of motion without the terms more than
second derivatives of metric and the theory is free of ghosts earlygb .
Another important property of Gauss-Bonnet term is that, just like Hilbert
Lagrangian is a pure divergence in 2 dimensions and Einstein tensor identifies
zero in 1 and 2 dimensions, we have that in 4 or less dimension the Gauss-
Bonnet Lagrangian is a pure divergence. We see the dilemma of quadratic term
in 4 dimensional theory: if we include it with non pure divergence we shall
confront ghosts; if we want to remove ghosts we get a pure divergence term. So
only in theories in more than 4 dimensional Gauss-Bonnet combination provides
physical effects. Moreover the Gauss-Bonnet term also appears in both low
energy effective action of Bosonic string theory stringgb1  and low energy
effective action of Bosonic modes of heterotic and type II super string theory
stringgb2 . An investigation into the effects of a Gauss-Bonnet term in the 5
dimensional bulk of brane world models is therefore well motivated. The Gauss-
Bonnet term in 5 dimension reads,

$F(R_{5})=R_{5}^{2}-4R_{5{\rm AB}}R_{5}^{\rm AB}+R_{5{\rm ABCD}}R_{5}^{\rm
ABCD}.$ (97)

The action of the brane can be written as,

$S_{\rm brane}=\int_{M}d^{4}x\sqrt{-g}\left({\kappa_{5}^{-2}}K+L_{\rm
brane}\right),$ (98)

where $M$ indicates the brane manifold, $g$ denotes the determinant of the
brane metric, $L_{\rm brane}$ stands for the Lagrangian confined to the brane,
and $K$ marks the trace of the second fundamental form of the brane.
$x=(\tau,x^{1},x^{2},x^{3})$ is the brane coordinate. Note that $\tau$ is not
identified with $t$ if the the brane is not fixed at a position in the extra
dimension $z=$constant. We will investigate the cosmology of a moving brane
along the extra dimension $z$ in the bulk, and such that $\tau$ is different
from $t$.

We set the Lagrangian confined to the brane as follows,

$L_{\rm brane}=\frac{1}{16\pi G}R-\lambda+L_{\rm m},$ (99)

where $\lambda$ is the brane tension and $L_{\rm m}$ denotes the ordinary
matter, such as dust and radiation, located at the brane. $R$ denotes the 4
dimensional scalar curvature term on the brane, which is an important one
except a Gauss-Bonnet term in the bulk. This induced gravity correction arises
because the localized matter fields on the brane, which couple to bulk
gravitons, can generate via quantum loops a localized four-dimensional world-
volume kinetic term for gravitons dgp .

Assuming there is a mirror symmetry in the bulk, we have the Friedmann
equation on the brane combinecos , see also cov ,

$\displaystyle{4\over r_{c}^{2}}\left[1+\frac{8}{3}\alpha\left(H^{2}+{k\over
a^{2}}+{U\over 2}\right)\right]^{2}\left(H^{2}+{k\over
a^{2}}-U\right)=\left(H^{2}+{k\over a^{2}}-\frac{8\pi
G}{3}(\rho+\lambda)\right)^{2},$ (100)

where

$U=-\frac{1}{4\alpha}\pm\frac{1}{4\alpha}\sqrt{1+4\alpha\left(\frac{\Lambda_{5}}{6}+\frac{M\kappa_{5}^{2}}{4\pi
a^{4}}\right)},$ (101) $r_{c}=\kappa_{5}^{2}\mu^{2}.$ (102)

Here $M$ is a constant, standing for the mass of bulk black hole. For various
limits of (100), see selfgbde .

For convenience, we introduce the following new variables and parameters,

$\displaystyle
x\equiv\frac{H^{2}}{H_{0}^{2}}+\frac{k}{a^{2}H_{0}^{2}}=\frac{H^{2}}{H_{0}^{2}}-\Omega_{k0}(1+z)^{2},$
$\displaystyle u\equiv\frac{8\pi\
^{(4)}G}{3H_{0}^{2}}(\rho+\lambda)=\Omega_{m0}(1+z)^{3}+\Omega_{\lambda},$
$\displaystyle m\equiv\frac{8}{3}\alpha H_{0}^{2},$ $\displaystyle
n\equiv\frac{1}{H_{0}^{2}r_{c}^{2}},$ $\displaystyle
y\equiv\frac{1}{2}UH_{0}^{-2}=\frac{1}{3m}\left(-1+\sqrt{1+\frac{4\alpha\Lambda_{5}}{6}+{\frac{8\alpha
M{}^{(5)}G}{a^{4}}}}\right)$
$\displaystyle~{}~{}~{}=\frac{1}{3m}\left(-1+\sqrt{1+m\Omega_{\Lambda_{5}}+m\Omega_{M0}(1+z)^{4}}\right),$
(103)

and we have assumed that there is only pressureless dust in the universe. As
before, we have used the following notations

$\Omega_{k0}=-\frac{k}{a_{0}^{2}H_{0}^{2}},~{}~{}\Omega_{m0}=\frac{8\pi
G}{3}\frac{\rho_{m0}}{H_{0}^{2}},~{}~{}\Omega_{\lambda}=\frac{8\pi
G}{3}\frac{\lambda}{H_{0}^{2}},~{}~{}\Omega_{\Lambda_{5}}=\frac{3\Lambda_{5}}{8H_{0}^{2}},~{}~{}\Omega_{M0}=\frac{3M\kappa_{5}^{2}}{8\pi
a_{0}^{4}H_{0}^{2}}.$ (104)

With these new variables and parameters, (100) can be rewritten as

$4n(x-2y)[1+m(x+y)]^{2}=(x-u)^{2}.$ (105)

This is a cubic equation of the variable $x$. According to algebraic theory it
has 3 roots. One can explicitly write down three roots. But they are too
lengthy and complicated to present here. Instead we only express those three
roots formally in the order given in Mathematica

$\displaystyle x_{1}=x_{1}(y,u|m,n),$ $\displaystyle x_{2}=x_{2}(y,u|m,n),$
$\displaystyle x_{3}=x_{3}(y,u|m,n),$ (106)

where $y$ and $u$ are two variables, $m$ and $n$ stand for two parameters. The
root on $x$ of the equation (105) gives us the modified Friedmann equation on
the Gauss-Bonnet brane world with induced gravity. From the solutions given in
(III.3), this model seems to have three branches. In addition, note that all
parameters introduced in (III.3) and (104) are not independent of each other.
According to the Friedmann equation (III.3), when all variables are taken
current values, for example, $z=0$, the Friedmann equation will give us a
constraint on those parameters,

$1=f(\Omega_{k0},~{}\Omega_{m0},~{}\Omega_{M0},~{}\Omega_{\Lambda_{5}},~{}\Omega_{\lambda},~{}m,~{}n).$
(107)

To compare with observation, we introduce the concept “equivalent dark energy”
or “virtual dark energy” in the modified gravity models, since almost all the
properties of dark energy are deduced in the frame of general relativity with
a dark energy.

The Friedmann equation in the four dimensional general relativity can be
written as

$H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3}(\rho+\rho_{de}),$ (108)

where the first term of RHS of the above equation represents the dust matter
and the second term stands for the dark energy. Generally speaking the Bianchi
identity requires,

$\frac{d\rho_{de}}{dt}+3H(\rho_{de}+p_{de})=0,$ (109)

we can then express the equation of state for the dark energy as

$w_{de}=\frac{p_{de}}{\rho_{de}}=-1-\frac{1}{3}\frac{d\ln\rho_{de}}{dlna}.$
(110)

Note that we can rewrite the Friedmann equation (III.3) in the form of (108)
as

$xH_{0}^{2}=\frac{8\pi G}{3}\rho+\left(H_{0}^{2}x(y,u|m,n)-\frac{8\pi
G}{3}\rho\right)=\frac{8\pi G}{3}(\rho+Q),$ (111)

where $\rho$ is the energy density of dust matter on the brane and the term

$Q\equiv\frac{3H_{0}^{2}}{8\pi G}x(y,u|m,n)-\rho$ (112)

corresponds to $\rho_{de}$ in (108).

In fig 14 we show the equation of state for the virtual dark energy when we
take $m=1.036$ and $n=0.04917$. In this case, from the constraint equation
(128), one has $\Omega_{M0}=2.08$. From the figure we see that $w_{eff}<-1$ at
$z=0$ and

$\left.\frac{dw_{eff}}{dz}\right|_{z=0}<0.$

Therefore the equation of state for the virtual dark energy can indeed cross
the phantom divide $w=-1$ near $z\sim 0$.

Figure 14: The equation of state $w_{eff}$ with respect to the red shift
$1+z$, with $\Omega_{m0}=0.28$ and $\Omega_{A}=2.08$. From selfgbde .

Fig 14 illuminates that the behavior of the virtual dark energy seems rather
strange. However we should remember that it is only virtual dark energy, not
actual stuff. The whole evolution of the universe is described by the Hubble
parameter. We plot the Hubble parameter $H$ corresponding to fig 14 in fig 15.

Figure 15: $H^{2}/H_{0}^{2}$ versus $1+z$, with $\Omega_{m0}=0.28$ and
$\Omega_{A}=2.08$. From selfgbde .

Fig 15 displays that the universe will eventually becomes a de Sitter one. For
more figures with different parameters, see selfgbde . The constraint of this
brane model with induced scalar term on the brane and Gauss-Bonnet term in the
bulk has been investigated in DGPGBcon .

The above is an example of “pure geometric” dark energy. We can also consider
some mixed dark energy model, ie, the cosmic acceleration is driven by an
exotic matter and some geometric effect in part. Why such an apparently
complicated suggestion? There are many interesting models are proposed to
explain the cosmic acceleration, including dark energy and modified gravity
models. However, several influential and hopeful models, such as quintessence
and DGP model, fundamentally can not account for the crossing $-1$ behavior of
dark energy. By contrast, some hybrid model of the dark energy and modified
gravity may realize such a crossing. As an example we study the quintessence
and phantom in frame of DGP self3 .

Our starting point is still action (94). In a DGP model with a scalar, (96)
becomes a pure Einstein-Hilbert action,

${\cal L}_{\rm bulk}={1\over 2\kappa_{5}^{2}}R_{5},$ (113)

and we add a scalar term in (99),

$L_{\rm brane}=\frac{1}{16\pi G}R+L_{\rm m}+L_{\rm scalar}.$ (114)

Here the scalar term can be ordinary scalar (quintessence) or phantom (scalar
with negative kinetic term). The Lagrangian of a quintessence reads,

$L_{\phi}=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi),$ (115)

and for phantom,

$L_{\psi}=\frac{1}{2}\partial_{\mu}\psi\partial^{\mu}\psi-U(\psi).$ (116)

In an FRW universe we have

$\displaystyle\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (117)
$\displaystyle p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi).$ (118)

The exponential potential is an important example which can be solved exactly
in the standard model. Also it has been shown that the inflation driven by a
scalar with exponential potential can exit naturally in the warped DGP model
hs1 . It is therefore quite interesting to investigate a scalar with such a
potential in late time universe on a DGP brane. Here we set

$V=V_{0}e^{-\lambda_{1}\frac{\phi}{\mu}}.$ (119)

Here $\lambda_{1}$ is a constant and $V_{0}$ denotes the initial value of the
potential.

The Friedmann equation (100) becomes

$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3\mu^{2}}\left[\rho+\rho_{0}+\theta\rho_{0}(1+\frac{2\rho}{\rho_{0}})^{1/2}\right],$
(120)

where

$\rho_{0}=\frac{6\mu^{2}}{r_{c}^{2}}.$ (121)

Similar to the previous case, we derive the virtual dark energy by comparing
(120) and (108),

$\rho_{de}=\rho_{\phi}+\rho_{0}+\theta\rho_{0}\left[\rho+\rho_{0}+\theta\rho_{0}(1+\frac{2\rho}{\rho_{0}})^{1/2}\right].$
(122)

From (110), we calculate the derivation of effective density of dark energy
with respective to $\ln(1+z)$ for a ordinary scalar,

$\displaystyle\frac{d\rho_{de}}{d\ln(1+z)}=3[\dot{\phi}^{2}+\theta(1+\frac{\dot{\phi}^{2}+2V+2\rho_{dm}}{\rho_{0}})^{-1/2}(\dot{\phi}^{2}+\rho_{dm})].$
(123)

If $\theta=1$, both terms of RHS are positive, hence it never goes to zero at
finite time. But if $\theta=-1$, the two terms of RHS carry opposite sign,
therefore it is possible that the EOS of dark energy crosses phantom divide.
In a scalar-driven DGP, we only consider the case of $\theta=-1$.

For convenience, we define some dimensionless variables,

$\displaystyle y_{1}$ $\displaystyle\triangleq$
$\displaystyle\frac{\dot{\phi}}{\sqrt{6}\mu H},$ (124) $\displaystyle y_{2}$
$\displaystyle\triangleq$ $\displaystyle\frac{\sqrt{V}}{\sqrt{3}\mu H},$ (125)
$\displaystyle y_{3}$ $\displaystyle\triangleq$
$\displaystyle\frac{\sqrt{\rho_{m}}}{\sqrt{3}\mu H},$ (126) $\displaystyle
y_{4}$ $\displaystyle\triangleq$
$\displaystyle\frac{\sqrt{\rho_{0}}}{\sqrt{3}\mu H}.$ (127)

The Friedmann equation (120) becomes

$y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}-y_{4}^{2}\left(1+2\frac{y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}}\right)^{1/2}=1.$
(128)

The stagnation point, that is, $d\rho_{de}/d\ln(1+z)=0$ dwells at

$\frac{y_{4}}{\sqrt{2}+y_{4}}\left(2+\frac{y_{3}^{2}}{y_{1}^{2}}\right)=2,$
(129)

which can be derived from (123) and (128).

One concludes from the above equation that a smaller $r_{c}$, a smaller
$\Omega_{m0}$ (Recall that it is defined as the present value of the energy
density of dust matter over the critical density), or a larger $\Omega_{ki}$
(which is defined as the present value of the kinetic energy density of the
scalar over the critical density) is helpful to shift the stagnation point to
lower redshift region. We show a concrete numerical example of this crossing
behaviours in fig 16. For convenience we introduce the dimensionless density
and rate of change with respect to redshift of dark energy as below,

$\displaystyle\beta=\frac{\rho_{de}}{\rho_{c}}=\frac{\Omega_{r_{c}}}{b^{2}}\left[y_{1}^{2}+y_{2}^{2}+y_{4}^{2}-y_{4}^{2}(1+2\frac{y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}})^{1/2}\right],$
(130)

where $\rho_{c}$ denotes the present critical density of the universe, and

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{1}{\rho_{c}}\frac{y_{4}^{2}}{\Omega_{r_{c}}}\frac{d\rho_{de}}{ds}$
(131) $\displaystyle=$ $\displaystyle
3\left[(1+2\frac{y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}})^{-1/2}(2y_{1}^{2}+y_{3}^{2})-2y_{1}^{2}\right].$

A significant parameters from the viewpoint of observations is the
deceleration parameter $q$, which carries the total effects of cosmic fluids.
We plot $q$ in these figures for corresponding density curve of dark energy.
In the fig 16 we set $\Omega_{m}=0.3$. $\Omega_{r_{c}}$ is defined as the
present value of the energy density of $\rho_{0}$ over the critical density
$\Omega_{r_{c}}={\rho_{0}}/{\rho_{c}}$.

Figure 16: For this figure, $\Omega_{ki}=0.01$, $\Omega_{r_{c}}=0.01$,
$\lambda_{1}=0.5$. (a) The left panel: $\beta$ and $\gamma$ as functions of
$s$, in which $\beta$ resides on the solid line, while $\gamma$ dwells at the
dotted line. The EOS of dark energy crosses $-1$ at about $s=-0.22$, or
$z=0.25$. (b) The right panel: the corresponding deceleration parameter, which
crosses 0 at about $s=-0.40$, or $z=0.49$. From self3 .

Now, we turn to the evolution of a universe with a phantom (116) in DGP. In an
FRW universe the density and pressure of a phantom can be written as (17),
(18).

To compare with the results of the ordinary scalar, here we set a same
potential as before,

$U=U_{0}e^{-\lambda_{2}\frac{\psi}{\mu}}.$ (132)

The ratio of change of density of virtual dark energy with respective to
$\ln(1+z)$ becomes,

$\displaystyle\frac{d\rho_{de}}{d\ln(1+z)}=3[-\dot{\psi}^{2}+\theta(1+\frac{-\dot{\psi}^{2}+2U+2\rho_{dm}}{\rho_{0}})^{-1/2}(-\dot{\psi}^{2}+\rho_{dm})].$
(133)

To study the behaviour of the EOS of dark energy, we first take a look at the
signs of the terms of RHS of the above equation. $(-\dot{\psi}^{2}+\rho_{dm})$
represents the total energy density of the cosmic fluids, which should be
positive. The term $(1+\frac{-\dot{\psi}^{2}+2U+2\rho_{dm}}{\rho_{0}})^{-1/2}$
should also be positive. Hence if $\theta=-1$, both terms of RHS are negative:
it never goes to zero at finite time. Contrarily, if $\theta=1$, the two terms
of RHS carry opposite sign: the EOS of dark energy is able to cross phantom
divide. In the following of the present subsection we consider the branch of
$\theta=1$.

Now the Friedmann constraint becomes

$-y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}+y_{4}^{2}\left(1+2\frac{-y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}}\right)^{1/2}=1.$
(134)

Again, one will see that in reasonable regions of parameters, the EOS of dark
energy crosses $-1$, but from below $-1$ to above $-1$.

The stagnation point of $\rho_{de}$ inhabits at

$\frac{y_{4}}{\sqrt{2}-y_{4}}\left(-2+\frac{y_{3}^{2}}{y_{1}^{2}}\right)=2,$
(135)

which can be derived from (133) and (134). One concludes from the above
equation that a smaller $r_{c}$, a smaller $\Omega_{m}$, or a larger
$\Omega_{ki}$ is helpful to shift the stagnation point to lower redshift
region, which is the same as the case of an ordinary scalar. Then we show a
concrete numerical example of the crossing behaviour of this case in fig 17.
The dimensionless density and rate of change with respect to redshift of dark
energy become,

$\displaystyle\beta=\frac{\rho_{de}}{\rho_{c}}=\frac{\Omega_{r_{c}}}{b^{2}}\left[-y_{1}^{2}+y_{2}^{2}+y_{4}^{2}+y_{4}^{2}(1+2\frac{-y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}})^{1/2}\right],$
(136)

and

$\gamma=3\left[-(1+2\frac{-y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}{y_{4}^{2}})^{-1/2}(-2y_{1}^{2}+y_{3}^{2})+2y_{1}^{2}\right].$
(137)

Similarly, the deceleration parameter is plotted in the figure 17 for
corresponding density curve of dark energy. In this figures we also set
$\Omega_{m}=0.3$.

Figure 17: For this figure, $\Omega_{ki}=0.01$, $\Omega_{r_{c}}=0.01$,
$\lambda=0.01$. (a) The left panel: $\beta$ and $\gamma$ as functions of $s$,
in which $\beta$ resides on the solid line, while $\gamma$ dwells at the
dotted line. The EOS of dark energy crosses $-1$ at about $s=-1.25$, or
$z=1.49$. (b) The right panel: The corresponding deceleration parameter, which
crosses 0 at about $s=-0.50$, or $z=0.65$. From self3 .

Fig 17 explicitly illuminates that the EOS of virtual dark energy crosses
$-1$, as expected. At the same time the deceleration parameter is consistent
with observations.

##  IV summary

The recent observations imply that the EOS of dark energy may cross $-1$. This
is a remarkable phenomenon and attracts much theoretical attention.

We review three typical models for the crossing behavior. They are two-field
model, interacting model, and modified gravity model.

There are several other interesting suggestions in or beyond the three
categories mentioned above. We try to list them here for the future
researches. We are apologized for this incomplete reference list on this
topic.

Almost in all dark energy models the dark energy is suggested as scalar.
However, 3 orthogonal vectors can also play this role. For the interacting
vector dark energy and phantom divide crossing, see weivector . For the
suggestion of crossing the phantom divide with a spinor, see spin . Multiple
k-essence sources are helpful to fulfil the condition for phantom divide
crossing multike . Phantom divide crossing can be realized by non-minimal
coupling and Lorentz invariance violation lorevio . An exact solution of a
two-field model for this crossing has been found in exacttwo .

For previous interacting $X$ (quintessence or phantom) models with crossing
$-1$, see internew . The cosmology of interacting $X$ in loop gravity has been
studied in Xloop .

Interacting holographic dark energy is a possible mechanism for the phantom
divide crossing ihd . And the thermodynamics of interacting holographic dark
energy with phantom divide crossing is investigated in thermalihd . An
explicit model of $F(R)$ gravity in which the dark energy crosses the phantom
divide is reconstructed in f(R) . The phantom-like effects in a DGP-inspired
$F(R,\phi)$ gravity model is investigated in phandgp . Based on the recent
progress in studies of source of Taub space taubsource , a new braneworld in
the sourced-Taub background is proposed taubbrane , while the previous brane
world models are imbedded in AdS (RS) or Minkowski (DGP). In this model the
EOS for the virtual dark energy of a dust brane in the source region can cross
the phantom divide. For other suggestions in brane world model, see branecorss
.

Similar to the coincidence problem of dark energy, we can ask why the EOS
crosses $-1$ recently? This problem is studied in wei2coin .

On the observational side, the present data only mildly favor the crossing
behavior. We need more data to confirm or exclude it.

Theoretically, we should find more natural model which has less parameters. We
must go beyond the standard model of particle physics. The problem cosmic
acceleration is a pivotal problem to access new physics. To study the problem
of crossing $-1$ EOS will impel the investigation to the new Laws of nature.

Acknowledgments We thank to all the original authors who permit us to use
their figures. After submission of this invited review, Our special
thankfulness goes to Y. Cai, who informed me that their review on quintom
cosmology would appear soon. And Y Cai give us several beneficial suggestions.
Hence this article is slightly different from the published version.

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