# Cosmological evolution of interacting phantom (quintessence) model in Loop
Quantum Gravity

Puxun Wu<EMAIL_ADDRESS>and Shuang Nan Zhang 1,3,4 1Department of
Physics and Tsinghua Center for Astrophysics, Tsinghua University, Beijing
100084, China
2School of Sciences and Institute of Math-Physics, Central South University of
Forestry and Technology, Changsha, Hunan 410004, China
3 Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences, P.O. Box 918-3, Beijing 100049, China
4 Physics Department, University of Alabama in Huntsville, Huntsville, AL
35899, USA

###### Abstract

The dynamics of interacting dark energy model in loop quantum cosmology (LQC)
is studied in this paper. The dark energy has a constant equation of state
$w_{x}$ and interacts with dark matter through a form
$3cH(\rho_{x}+\rho_{m})$. We find for quintessence model ($w_{x}>-1$) the
cosmological evolution in LQC is the same as that in classical Einstein
cosmology; whereas for phantom dark energy ($w_{x}<-1$), although there are
the same critical points in LQC and classical Einstein cosmology, loop quantum
effect reduces significantly the parameter spacetime ($c,w_{x}$) required by
stability. If parameters $c$ and $w_{x}$ satisfy the conditions that the
critical points are existent and stable, the universe will enter an era
dominated by dark energy and dark matter with a constant energy ratio between
them, and accelerate forever; otherwise it will enter an oscillatory regime.
Comparing our results with the observations we find at $1\sigma$ confidence
level the universe will accelerate forever.

###### pacs:

98.80.Cq

##  I Introduction

Many cosmological observations show that our universe is undergoing an
accelerating expansion and now mainly consists of two dark components: dark
matter and dark energy. The dark matter is a clumpy component that traces the
baryonic matter and accounts for about $23\%$ of present total cosmic energy;
the dark energy is an exotic energy with negative pressure and accounts for
about $72\%$ of total cosmic energy today. The simplest candidate of dark
energy is the cosmological constant Constant , however it suffers from two
problems. One is the cosmological constant problem: why is the inferred value
of cosmological constant so tiny (120 orders of magnitude lower) compared to
the typical vacuum energy values predicted by the quantum field theory? The
other is the coincidence problem: why is the dark energy density comparable to
the matter density right now? Therefore a dynamical scalar field: quintessence
Quint  is proposed as an alternative of dark energy, but it can not explain
the region of the equation of state less than $-1$, which is favored by
observations Nesseris . Later Caldwell Cald  proposed a phantom field to
explain the present cosmic accelerating expansion. This field possesses of a
negative kinetic energy and so has a super negative equation of state. In the
Einstein gravity it is found that if the universe is dominated by the phantom
energy, it will end with a big rip, i.e., a future singularity Cald2 . Many
works have been done trying to avoid this singularity McInne . There are many
other scalar field models: such as quintom Quintom  and hessence Hessence .
However these scale field dark energy models still suffer from the coincidence
problem. A possible alleviation for this problem is to assume the existence of
an interaction between dark matter and dark energy Chimento2003 .

Recent investigations have shown that there are some new nice features
appearing in Loop Quantum Cosmology (LQC) (see LQC ; LQC2  for recent
reviews), such as: easier inflation Easy  and correspondence between LQC and
braneworld cosmology Copeland2006 . The LQC is the application in the
cosmology context of the Loop Quantum Gravity (LQG) (see LQG  for recent
reviews), which is a theory trying to quantize the gravity with a non-
perturbative and background-independent method. Due to the loop quantum effect
the standard Friedmann equation can be modified by adding a correction term
LQC2 ; Fried1 ; Fried2 ; Fried3 ,

$\displaystyle
H^{2}=\frac{1}{3}\rho\bigg{(}1-\frac{\rho}{\rho_{c}}\bigg{)}\;,$ (1)

where $H$ is the Hubble parameter, $8\pi G\equiv 1$, $\rho$ is the total
cosmic energy density,
$\rho_{c}\equiv\frac{\sqrt{3}}{16\pi^{2}\gamma^{3}G^{2}\hbar}$ denotes the
critical loop quantum density and $\gamma$ is the dimensionless Barbero-
Immirzi parameter (it is suggested that $\gamma=0.2375$ by the black hole
thermodynamics in LQG Ashtekar ). Since this modified equation is correct
under the condition that the quantum state is semiclassical, this condition is
assumed to be satisfied forever in this paper. In addition we assume the
quantum correlations do not build up during long-term evolution of cosmology,
otherwise there are additional correction terms from LQC which become
important Bojowald2008 . The correction term appearing in Eq. (1) essentially
encodes the discrete quantum geometric nature of spacetime. When this
correction term becomes dominant, the universe begins to bounce and then
expands backwards. By studying the early universe inflation and the fate of
future singularity in LQC, it is found that the big bang singularity, the big
rip and other future singularities can be avoided LQC2 ; Fried1 ; Fried2 ;
Samart . Recently Samart and Gumjudpai Samart , Wei and Zhang Hao  studied the
dynamics of phantom, quintom and hessence dark energy models in LQC, and found
the results are different from that obtained in classical Einstein cosmology.
In this paper we will investigate the evolution of our universe dominated by a
scalar field in LQC, which has constant equation of state and interacts with
dark matter, and then investigate whether there are some interesting features
arising from the loop quantum gravity effect.

##  II The interacting model

We consider a spatially flat universe in which there are only dark matter and
dark energy with a constant equation of state $w_{x}$. Apparently $w_{x}>-1$
corresponds a quintessence model and $w_{x}<-1$ is a phantom case. In addition
we assume that between the dark matter and dark energy there is an interaction
term $\Gamma$. Thus the conservation equations for dark matter and dark energy
can be expressed as

$\displaystyle\dot{\rho}_{x}+3H(1+w_{x})\rho_{x}=-\Gamma\;,$ (2)
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=\Gamma\;,$ (3)

where $\rho_{x}$ and $\rho_{m}$ correspond to the energy densities of dark
energy and dark matter respectively, and a dot denotes the derivative with
respect to cosmic time $t$. The interacting term $\Gamma$ is assumed to be
$\Gamma=3Hc(\rho_{x}+\rho_{m})$, where $c$ is a coupling constant denoting the
transfer strength. A positive $c$ corresponds to energy transferred from dark
energy to dark matter and the other way around for a negative one. In this
paper we constrain our discussion in the case of $c>0$. This type of
interaction, motivated by analogy with dissipation of cosmological fluids, has
been introduced to solve the coincidence problem Chimento2003 , and has been
studied in the context of quintessence Zimdahl , phantom Guo  and the
(generalized) Chaplygin gas model Zhang . In addition the observational
constraints for this type interaction dark energy model have been studied in
Refs. Olivares ; Wang .

In LQC, using the conservation equation of cosmic total energy
$\dot{\rho}+3H\left(\rho+p\right)=0$, where $\rho=\rho_{x}+\rho_{m}$, one can
easily obtain the effective modified Raychaudhuri equation

$\displaystyle\dot{H}=-\frac{~{}1}{~{}2}\left(\rho+p\right)\left(1-2\frac{\rho}{\rho_{\rm
c}}\right),$ (4)

where $p$ is the total pressure ($p=w_{x}\rho_{x}$ in this paper).

To analyze the dynamical system, we set

$\displaystyle N=\ln a,\qquad u=\frac{\rho_{x}}{3H^{2}},\qquad
v=\frac{\rho_{m}}{3H^{2}},$ (5)

where $a_{0}=1$ is assumed. Using Eqs.(2, 3, 4), one can obtain

$\displaystyle
u^{\prime}=-3u(1+w_{x})-3c(u+v)+3u(u+w_{x}u+v)\big{(}-1+\frac{2}{u+v}\big{)}\;,$
(6) $\displaystyle
v^{\prime}=-3v+3c(u+v)+3v(u+w_{x}u+v)\big{(}-1+\frac{2}{u+v}\big{)}\;,$ (7)

where the prime denotes a derivative with respect to $N$. As discussed in
Refs. Chimento2003 ; Guo ; Olivares  this interacting model can solve, or at
least ameliorate the coincidence problem in classical Einstein cosmological
since in the dynamical system there is a late time attractor solution with a
constant energy ratio between dark energy and dark matter. Therefore,
regardless the initial conditions, the universe evolves to a final state
characterized by a constant dark matter to dark energy ratio. Here we will
discuss in LQC whether the dynamics system of interacting model exists the
attractor solution, and then study the cosmic evolutions within different
conditions. In order to obtain the possible attractor for the system given by
Eqs. (6, 7), we should firstly solve these equations with $u^{\prime}=0$ and
$v^{\prime}=0$ to get the critical points:

$\displaystyle
Point\;A:u_{c}=\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{4c}{w_{x}}},\qquad
v_{c}=1-u_{c}.$ (8) $\displaystyle
Point\;B:u_{c}=\frac{1}{2}-\frac{1}{2}\sqrt{1+\frac{4c}{w_{x}}},\qquad
v_{c}=1-u_{c}\;.$ (9)

Both two critical points correspond to the era dominated by dark matter and
dark energy with a constant energy ratio between them and exist for
$c\leq\frac{-w_{x}}{4}$. Apparently these critical points are the same as that
obtained in classical Einstein cosmology Olivares . If the critical point is
stable, it is a late time attractor; otherwise the solution is oscillatory. In
order to investigate the stability of the critical point, we linearize the
system near the critical point and arrive at

$\displaystyle\delta u^{\prime}$ $\displaystyle=$
$\displaystyle-\bigg{[}3c+3(-1+2u_{c}+v_{c})+3w_{x}+6w_{x}u_{c}\bigg{(}1-\frac{u_{c}+2v_{c}}{(u_{c}+v_{c})^{2}}\bigg{)}\bigg{]}\delta
u$ (10)
$\displaystyle-\bigg{[}3(c+u_{c})+\frac{6w_{x}u_{c}^{2}}{(u_{c}+v_{c})^{2}}\bigg{]}\delta
v$ $\displaystyle\delta v^{\prime}$ $\displaystyle=$
$\displaystyle\bigg{[}3c+3(1+w_{x})(-v_{c}+\frac{2v_{c}}{u_{c}+v_{c}})-6\frac{v_{c}(u_{c}+w_{x}u_{c}+v_{c})}{(u_{c}+v_{c})^{2}}\bigg{]}\delta
u$ (11)
$\displaystyle-\bigg{[}3+3c-3(1+w_{x})u_{c}-6v_{c}+\frac{6w_{x}u_{c}^{2}}{(u_{c}+v_{c})^{2}}\bigg{]}\delta
v$

Apparently there are two eigenvalues of the coefficient matrix of the above
equations. If the real parts of two eigenvalues for a critical point are all
negative, this critical point is stable and is an attractor; otherwise it is
unstable and thus oscillatory. We find the point B is always unstable, however
the point A is an attractor if the equation of state for dark energy $w_{x}$
and the coupling factor $c$ satisfy the conditions $\frac{1+w_{x}}{w_{x}}\leq
c\leq\frac{-w_{x}}{4}$ and $w_{x}>-2$. Comparing our results with that
obtained in Einstein cosmology where point A is stable only under condition
$c\leq\frac{-w_{x}}{4}$ Olivares , we find that for quintessence dark energy
$w_{x}>-1$ the results in LQC are the same as that in Einstein cosmology if a
positive $c$ is considered since $\frac{1+w_{x}}{w_{x}}<0$. However for
phantom dark energy $w_{x}<-1$ the conclusions seem to be different: in region
$c<\frac{1+w_{x}}{w_{x}}$ or $w_{x}<-2$, the point A is stable in Einstein
cosmology, but it is unstable in LQC, that is, the quantum correction effect
will break the stability of point A if the interaction factor $c$ is smaller
than $\frac{1+w_{x}}{w_{x}}$ or the equation of state for phantom is less than
$-2$. In Fig. (1) we show the stability regions of $(c,w_{x})$ parameter
space. Regions I$+$II are allowed for Einstein cosmology; however in LQC only
region II is allowed to obtain a stable solution.

Since in LQC the interacting quintessence model has the same dynamics as that
in Einstein cosmology, thereafter we will only discuss the case of phantom
$w_{x}<-1$. In Figs. (2, 3), we plot the numerical results for $c$ and $w_{x}$
satisfying the conditions $\frac{1+w_{x}}{w_{x}}\leq c\leq\frac{-w_{x}}{4}$
and $w_{x}>-2$. Fig. 2 shows the evolutionary properties of the universe
controlled by the interacting phantom energy with $w=-1.2$, $c=0.2$ and
different initial conditions. Apparently the trajectories converge to the same
final state determined by parameter $c$ and $w_{x}$. Since in the final state
$\Omega_{x}=u_{c}$ and $\Omega_{m}=1-u_{c}$, our universe will contain both
dark matter and phantom energy, and the energy ratio between them approaches a
constant. Fig. (3) shows the evolutionary curve of the equation of state for
total cosmic energy $w=p_{x}/(\rho_{x}+\rho_{m})$ with $w_{x}=-1.2$ and
$c=0.2$, we find in the final state the equation of state is a constant and
$w>-1$, which means that the total energy density decreases with the cosmic
expansion but the universe accelerates forever. Therefore if
$\frac{1+w_{x}}{w_{x}}\leq c\leq\frac{-w_{x}}{4}$ and $w_{x}>-2$, regardless
the initial conditions, the universe will enter a final state with a constant
energy ratio between dark energy and dark matter and accelerate forever.

In the following we will give the numerical results for the cases
$c<\frac{1+w_{x}}{w_{x}}$, $w_{x}<-2$ and $c>\frac{-w_{x}}{4}$ according to
the Eqs. (2, 3, 4) with $\rho_{c}=1.5$. The cases $c<\frac{1+w_{x}}{w_{x}}$
and $w_{x}<-2$ are allowed for the stable solution in classical Einstein
cosmology but ruled out by loop quantum effect. In Figs. (4, 5) we plot the
evolutionary curves of $H(t)$ and $\rho(t)$ for the case
$c<\frac{1+w_{x}}{w_{x}}$ with $w_{x}=-1.2$ and $c=0.1$. In Figs (6, 7) we
give the results for the case $w_{x}<-2$ with $w_{x}=-2.5$ and $c=0.25$. It is
easy to find from these figures that at beginning the phantom energy density
increases with time, which then leads to the increase of total cosmic energy
density. When the total energy density equals to $\rho_{c}/2$, $H$ takes the
maximum value. When $\rho$ reaches its maximum value $\rho_{max}\sim\rho_{c}$,
$H=0$ and then the universe undergos contraction until bounce. Therefore the
universe will oscillate forever.

Figs (8, 9) show the results for the case $c>\frac{-w_{x}}{4}$ with
$w_{x}=-1.2$ and $c=0.35$ which corresponds to the case that the critical
points do not exist. Comparing these figures with Figs. (4, 5, 6, 7), we find,
although the universe finally also enters an oscillating regime, the process
is different from that obtained with the $c<\frac{1+w_{x}}{w_{x}}$ or
$w_{x}<-2$. It is found that in this case the energy densities of dark energy
and dark matter have the same evolution with time, and the $H$ changes from
positive to negative (or inverse) when $\rho=0$ or $\rho\approx\rho_{c}$,
while in case $c<\frac{1+w_{x}}{w_{x}}$ or $w_{x}<-2$, $H$ changes from
positive to negative (or inverse) only at $\rho\approx\rho_{c}$

##  III Conclusion

In this paper the cosmological evolution with the interacting phantom or
quintessence dark energy in loop quantum cosmology is studied. We consider the
case of dark energy with a constant equation of state $w_{x}$ and the
interaction term with the form $\Gamma=3Hc(\rho_{x}+\rho_{m})$. It is found
that in LQC the dynamic of interacting quintessence model is the same as that
obtained in classical Einstein cosmological; whereas for interacting phantom
model, the loop quantum effect reduces significantly the parameter space, in
which the attractor solution exists. In LQC we obtain the critical point is
existent and stable under the conditions $\frac{1+w_{x}}{w_{x}}\leq
c\leq\frac{-w_{x}}{4}$ and $w_{x}>-2$; however in classical Einstein cosmology
only the condition $c\leq\frac{-w_{x}}{4}$ is required. If the coupling
parameter $c$ satisfies the stable and existent conditions for stable tracking
solution, our universe will enter an ere dominated by both dark energy and
dark matter with a constant energy ratio between them, and accelerate forever,
although the total energy density decreases with cosmic expansion, otherwise
the universe will enter an oscillatory regime.

Recently using the WMAP3 WMAP3  data Olivares et al. Olivares  obtained that
at $1\sigma$ confidence level $c\leq 0.0023$. More recently in Ref. Wang  by
combining the Gold Sne Ia, BAO and CMB data the authors found that at
$1\sigma$ confidence level $-0.99<w<-0.83$ and $c=0.0057_{-0.0026}^{+0.0030}$,
which show that at $1\sigma$ confidence level our universe will enter a final
stable state and can not oscillate. Letting $c=0.0057$ we find in LQC if
$-1.006\leq w<-1$ the universe with an interaction between dark matter and
dark energy will accelerate forever; whereas if $w<-1.006$ it will enter an
oscillatory regime. Therefore it is clear that at $2\sigma$ confidence level
the current observations seem to be unable to predict the late time evolution
of our universe with the interacting dark energy in LQC .

###### Acknowledgements.

We are grateful to Professor Hongwei Yu for helpful discussions. P. Wu is
partially supported by the National Natural Science Foundation of China under
Grant No. 10705055, the Youth Scientific Research Fund of Hunan Provincial
Education Department under Grant No. 07B085, and the Hunan Provincial Natural
Science Foundation of China under Grant No. 08JJ4001. S. N. Zhang is supported
in part by the Ministry of Education of China, Directional Research Project of
the Chinese Academy of Sciences under project No. KJCX2-YW-T03 and by the
National Natural Science Foundation of China under project Nos. 10521001,
10733010 and 10725313.

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Figure 1:  The stable regions of ($w_{x}$, $c$) parameter space. In the region
II, critical point A is a late time attractor in LQC. In Einstein cosmology
critical point A is a late time attractor in the region I+II. III represents
the region of the solution without physical meaning. Figure 2:  The phase
diagram of the interacting phantom dark energy in LQC with $w_{x}=-1.2$ and
$c=0.2$.  Figure 3:  The evolution of $w$ for total cosmic energy with
$w_{x}=-1.2$ and $c=0.2$.  Figure 4:  The evolution of $H$ with time under the
condition of $c<\frac{1+w_{x}}{w_{x}}$. Parameters are set as $w=-1.2$,
$c=0.1$ and $\rho_{c}=1.5$. Figure 5:  The evolution of cosmic energy density
with time under the condition of $c<\frac{1+w_{x}}{w_{x}}$. The solid, dashed
and dotted curves correspond to $\rho_{x}+\rho_{m}$, $\rho_{x}$ and $\rho_{m}$
respectively. Parameters are set as $w=-1.2$, $c=0.1$ and $\rho_{c}=1.5$.
Obviously $\rho_{x}$ triggers the recollapses, while $\rho_{m}$ triggers the
bounces. Figure 6:  The evolution of $H$ with time under the condition of
$w_{x}<-2$. Parameters are set as $w=-2.5$, $c=0.25$ and $\rho_{c}=1.5$.
Figure 7:  The evolution of cosmic energy density with time under the
condition of $c>\frac{-w_{x}}{4}$. The solid, dashed and dotted curves
correspond to $\rho_{x}+\rho_{m}$, $\rho_{x}$ and $\rho_{m}$ respectively.
Parameters are set as $w=-2.5$, $c=0.25$ and $\rho_{c}=1.5$. Figure 8:  The
evolution of $H$ with time under the condition of $c>\frac{-w_{x}}{4}$.
Parameters are set as $w=-1.2$, $c=0.35$ and $\rho_{c}=1.5$. Figure 9:  The
evolution of cosmic energy density with time under the condition of
$c>\frac{-w_{x}}{4}$. The solid, dashed and dotted curves correspond to
$\rho_{x}+\rho_{m}$, $\rho_{x}$ and $\rho_{m}$ respectively. Parameters are
set as $w=-1.2$, $c=0.35$ and $\rho_{c}=1.5$.