Dark Energy Density and IS(Israel-Stewart) Bulk Viscosity Model
S. Davood Sadatian1,2, A. Saburi1
1Research Department of astronomy $\&$ cosmology, University of Neyshabur, P.
O. Box 9319774446, Neyshabur, IRAN
2 Department of Physics, Faculty of Basic Sciences, University of Neyshabur ,
P. O. Box 9319774446, Neyshabur, Iran
email<EMAIL_ADDRESS>

###### Abstract

We investigate the thermodynamics of a dark energy bulk viscosity model as a
cosmic fluid. In this regard, the two theories of Eckart and Israel-Stewart
(IS) are the basis of our work. Therefore, we first investigate the
thermodynamics of cosmic fluids in the dark energy bulk viscosity model and
the general relationships. Then, we express the thermodynamic relationships of
Eckart’s theory. Due to the basic equations of Eckart’s theory and Friedmann’s
equations, we consider two states, one is $p=-\rho$ (standard) and the other
is $p\neq-\rho$ (non-standard). In the standard state, we define the pressure
$(p)$, energy density $(\rho)$ and bulk viscosity coefficient $(\xi)$ of the
cosmic fluid in terms of cosmic time and we obtain its relations. We also
mention that in this standard state, because of $p=-\rho$, the value of $a(t)$
is zero, so $a(t)$ is not defined in this state. But in the non-standard case
$(p\neq-\rho)$ the bulk viscosity coefficient $(\xi)$ is zero and only the
scale factor and pressure and energy density of the cosmic fluid is defined.
We also consider two states of constant and variable bulk viscosity
coefficients and obtain three Hubble constant parameters and scale factor in
terms of cosmic time, and energy density in terms of scale factor. In the
state of variable bulk viscosity coefficient, we consider the viscosity
coefficient as the power-law from energy density $(\xi=\alpha\rho^{s})$, which
is $\alpha>0$ and a constant. Following, we discuss about the dissipative
effects of cosmic fluids and examine the effects of energy density for dark
energy in the Israel-Stewart(IS) theory. The results are comprehensively
presented in two tables (1) and (2).

Keywords: Dark Energy, Viscosity, Effective Equation of State.

PACS: 04.50.kd, 95.36.+x

##  1 Introduction

Studies of the red shift of type Ia supernova in far galaxies [1,2] showed
that the rate of expansion of the universe not only did not decrease but also
increases, meaning that we are in an accelerating universe. In addition, the
expansion of the universe can be studied through the field of cosmic microwave
radiation [3]. One of the factors expressing the concept of the accelerating
universe is dark energy that contains more than 70% of the total energy of the
universe and has a negative pressure. In fact the universe is filled with a
mysterious energy field whose effect on the structure of large scales [4].
Dark energy uses in various models to study the expansion of the universe
including : quintessence [5,6], phantom [7], K-essence [8], chaplygin gas [9],
modified gravity [10,11], quantum model [12], holographic model [13], dynamic
models [14] and etc. Given recent advances in this area, however, the nature
of dark energy is still incomprehensible to us. Another model that is
considered for dark energy is the perfect fluids model, this means, they
consider the dark energy of the universe as a perfect fluid in an isotropic
world that has a certain density, pressure, and temperature. One of the
characteristics of fluids is the viscosity of fluids. In this regard, there
are important reviews on Dark Energy from viscous fluids [15,16]. Authors in
[15] presented the review of a number of popular dark energy models, such as
the $\Lambda$CDM model, Little Rip and Pseudo-Rip scenarios, the phantom and
quintessence cosmologies with the four types (I, II, III and IV) of the
finite-time future singularities and non-singular universes filled with dark
energy. Also, Brevik et al considered the important implications and the
capabilities of the incorporation of viscosity, which makes viscous cosmology
a good candidate for the description of Nature [16]. We know that viscosity
divided into two types of bulk and shear viscosity, shear viscosity is lost
due to the isotropic principle of the universe [17,18]. However, the perfect
fluid model for dissipative processes was first proposed by Eckart [19]. Due
to this theory was faced with limitations such as non-causality and the
propagation of unlimited dissipation perturbations, Israel and Stewart
proposed a generalized theory that did not have these limitations [20] and
this two theory investigate the effects of bulk viscosity fluid as dark energy
on the expansion of the universe. These kind of dark energy models show that
the accelerated expansion of the universe can be studied without considering
the cosmological constant [21]. In the bulk viscosity model, it can be
examined early and late in the cosmic time. The condition of accelerated
expansion of the universe is a violation of strong energy conditions, ie the
sum of energy density and three times the pressure of each component of the
universe be negative $(\sum_{i}{{\rho_{i}}+{3p_{i}}}<0)$. Due to this point
and also that dark matter does not emit radiation and is without pressure,
then $\rho_{i}/{3}>0$ in which case there must be a source of negative
pressure for the expansion of the universe which is called dark energy. In the
standard cosmological model, dark energy is showed as a fluid with negative
pressure and with the equation of state $p=\omega\rho$ with a constant
parameter $\omega$ that is $\omega=-1$, so the corrections observed in the
equation show that $\omega$ can also change dynamically [22] and be
$\omega<-1$ (crossing the phantom dividing line) [23]. If the parameter
$\omega$ is assumed to be dynamic, it shows some thermodynamic problems relate
to the equation of state, such as positive entropy, chemical potential and
temperature [24,25]. One way to prevent these thermodynamic problems is to
assume dark energy as a perfect fluid with bulk viscosity(actually called
imperfect fluid). Bulk viscosity means that dissipative processes occur that
violation the predominant energy conditions, $p+\rho<0$ , and in this case
dark energy does not become phantom necessary [26,27]. In this case there is a
pressure called the effective pressure which is denoted by $P_{eff}=p+\Pi$,
where $p$ barotropic pressure and $\Pi$ viscosity pressure which is proposed
in [28] by considering the negative effective pressure in the bulk viscosity
model of cosmic fluids. Equilibrium fluids have no entropy and frictional
heating because they are reversible and without dissipative but real fluids
are irreversible. As mentioned earlier, dissipation processes in perfect
fluids were first described by Eckart, he modeled the effective pressure of
fluid as $\Pi=-3H\xi$ ($\xi$ is a function in terms of cosmic time and energy
density, and $H$ is Hubble’s parameter ). Crossing the phantom dividing line
[29] , big rip singularities using different values of the state equation
parameter ($\omega$) in bulk viscosity [30] , dark fluid cosmology [31] are
among the applications of Eckart theory. It is noteworthy that most dark
energy models are tuned to cosmic data and observations and therefore do not
require other experiments, so most dark energy models can explain the
expansion of the universe well. However, due to the unknown nature of dark
energy, dark energy models that could be studied as cosmic fluid must be
within the framework of known laws of physics and thermodynamic laws. Several
works on dark energy thermodynamics have been described in
[18,32,33,34,35,36,37,38].
Therefore, in the first part of this paper, we express the thermodynamics of
cosmic fluids in general with a constant and variable state parameter under
the two theories of Eckart and Israel-Stewart. In the next section, we examine
the dissipative effects of perfect cosmic fluids and finally examine the
effects of energy density for dark energy in the Israel - Stewart theory.

##  2 Some aspects of Cosmic fluids in Eckart and Israel-Stewart Theories

Let us first discuss a review of the cosmic fluids thermodynamics in general.
When we consider dark energy as a cosmic fluid, a state equation is defined
for it as

$p=\omega\rho$ (1)

where $p$ is the cosmic fluid pressure and $\rho$ is the cosmic fluid energy
density considered for dark energy and $\omega$ is a state equation parameter
that can be both constant and variable (relative to cosmic time or any other
parameter). On the other hand, the Friedman equation for a homogeneous,
isotropic and flat universe based on the Friedman-Lematre-Robertson-Walker
(FLRW) parameter is defined as follows:

$\dot{H}+H^{2}=-\frac{1}{6}[\rho+3p]$ (2)

where $H$ is the Hubble parameter and $H=\frac{\dot{a}}{a}$ ,
$H^{2}=\frac{1}{3}\rho$ and $a$ is the scale factor for such a universe, also
we consider the natural units $8\pi G=c=1$. According to Friedman’s equation,
due to the effect of dark energy density on the expansion of the universe, in
the cosmic fluid model, the equation of conservation of fluid energy density
is defined as

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

In the last two equations, the derivatives are in terms of cosmic time. If we
want to consider the first law of thermodynamics in the dark energy cosmic
fluid model, we must first know that the internal energy of this fluid is
equal to $U=\rho V$, which $V$ is the physical volume of the fluid based on
the time scale factor, $V=V_{0}a^{3}$ is defined ( $V_{0}$ is the volume of
the fluid at the present time), so the first law of thermodynamics for cosmic
fluid will be equal to [37]:

$TdS=dU+\rho dV-\mu dN$ (4)

where $S$ is fluid entropy, $T$ fluid temperature, $U$ fluid internal energy,
$V$ physical volume of fluid, $\rho$ fluid energy density, $\mu$ chemical
potential and $N$ number of particles in the fluid.

###  2.1 Thermodynamics of Eckart theory

Now we express the thermodynamics of cosmic fluids in Eckart theory. Due to
the investigation of the bulk viscosity of the cosmic fluid in Eckart theory,
the bulk viscosity pressure of the fluid under Eckart theory is defined as
follows [42]

$\Pi=-3H\xi$ (5)

where $\xi$ is the viscosity coefficient and can be a function in terms of
cosmic time or a constant. On the other hand, the equation of conservation of
cosmic fluids in Eckart’s theory is equal to:

$\dot{\rho}+3H(\rho+P_{eff})=0$ (6)

where $P_{eff}$ is called effective pressure which is used in dissipative
processes for bulk viscosity and is equal to:

$P_{eff}=p+\Pi$ (7)

Where $p$ is called the barotropic equilibrium pressure, so according to
equations (5), (6) and (7) we have:

$\dot{\rho}+3H(\rho+p)-9H^{2}\xi=0.$ (8)

Equation (8) is the equation of conservation of cosmic fluids in Eckart’s
theory that if we want to get another form of this according to the parameter
of the equation of state $\omega$ we have:

$\dot{\rho}+3H(1+\omega)\rho-9H^{2}\xi=0.$ (9)

Now we have to get an equation in terms of the Hubble parameter and its
derivatives (to determine the effects of dark energy and Bulk-Eckart viscosity
on the expansion of the universe), according to Friedman’s equation in
relation (2) and $H^{2}=\frac{1}{3}\rho$ we have:

$2\dot{H}+3H^{2}(1+\frac{p}{\rho})=3H\xi$ (10)

Now, if we assume the state parameter $\omega$ to be constant and solve the
equation (10), the Hubble parameter obtain as a function in terms of the bulk
viscosity coefficient

$H(t)=\frac{e^{((\frac{3}{2})\int\xi(t)dt)}}{c+(\frac{3}{2})(p+\rho)\int
e^{(\frac{3}{2})\int\xi(t)dt}dt}$ (11)

where $c$ is an integral constant [42]. According to $H=\frac{{\dot{a}}}{a}$,
If we integrate from equation (11), the scale factor obtain as follows:

$a(t)=A[(\frac{3}{2})(1+\frac{p}{\rho})\int
e^{(\frac{3}{2})\int\xi(t)dt}dt+c]^{\frac{2}{3(1+\frac{p}{\rho})}}$ (12)

where $A$ is another integral constant. Note that this relation is conditional
$p\neq-\rho$, if $p=-\rho$, the scale factor is not defined in the equations.
If we consider $p=-\rho$ then we have according to equation (8):

$\dot{\rho}=9H^{2}\xi$ (13)

On the other hand, in this case $(p=-\rho)$, we will have according to
relation (10) $\omega=-1$. As a result, the bulk viscosity coefficient of the
fluid will be equal to:

$\xi=\frac{2}{3}\frac{\dot{H}}{H}$ (14)

According to relation (13) and (14) we have:

$\dot{\rho}=6H\dot{H}$ (15)

Now, by using equation (15), we will have:

$\rho=3H^{2}+o$ (16)

where $o$ is an integral constant, and if we assume its value to be
negligible, in other words, $H^{2}=\frac{\rho}{3}$, we arrive at the Friedman
equation, where the energy density of the fluid is called the density of the
dark energy. According to equations (1) and (16) the equivalent fluid pressure
will be equal to:

$p=3H^{2}\omega$ (17)

So, if $p=-\rho$ is assumed, we obtain three equations $\xi(t)$, $\rho(t)$,
$p(t)$ for the cosmic fluid with bulk viscosity, and $a(t)$ is not defined in
this case. If $p\neq-\rho$, we obtain the cosmic fluid as a cold dark matter
in the standard model of cosmology $(\Lambda CDM)$. If the parameter of the
equation of state is unequal $-1$ $(\omega\neq-1)$ and $\xi=0$, according to
equation (12), the scale factor is rewritten:

$a(t)=a_{0}(1+\frac{3}{2}H_{0}(p+\rho)t)^{\frac{2}{3(1+\frac{p}{\rho})}}$ (18)

and the energy density is equal to:

$\rho=\frac{3(H_{0})^{2}}{(1+\frac{3}{2}H_{0}(1+\frac{p}{\rho})t)^{2}}$ (19)

where $H_{0}>0$. If $\xi(t)=\xi_{0}=const$ is assumed, the Hubble parameter is
equal to:

$H(t)=\frac{H_{0}\xi_{0}e^{(\frac{3}{2})\xi_{0}t}}{H_{0}(1+\frac{p}{\rho})(e^{(\frac{3}{2})\xi_{0}t}-1)+\xi_{0}}.$
(20)

Now if we integrate from relation (20) (with condition $a_{0}=1$ ) then we
have:

$a(t)=a_{0}[1+\frac{H_{0}}{\xi_{0}}(1+\frac{p}{\rho})(e^{(\frac{3}{2})\xi_{0}t}-1)]^{\frac{2}{3(1+\frac{p}{\rho})}}$
(21)

In this case, the energy density will change according to cosmic time as
follows:

$\rho(t)=\frac{3H_{0}^{2}e^{3\xi_{0}t}}{[(1+\frac{H_{0}}{\xi_{0}})(1+\frac{p}{\rho})(e^{(\frac{3}{2})\xi_{0}t}-1)]^{2}}$
(22)

Of course, energy density can also be obtained in terms of scale factor, which
in case we will have [42]:

$\rho(a)=3H_{0}^{2}[\frac{\xi_{0}}{H_{0}(1+\frac{p}{\rho})}+(1-\frac{\xi_{0}}{H_{0}(1+\frac{p}{\rho})})a^{\frac{-2}{3(1+\frac{p}{\rho})}}]^{2}.$
(23)

Now if $\xi(t)=\xi(\rho(t))$, assuming different values for the bulk viscosity
coefficient, different results are obtained in the equations, one of the
hypotheses for the value of $\xi$ is $\xi=\alpha\rho^{s}$ ($\alpha>0$ and $s$
is a constant), which is a dependent viscosity coefficient to the power-law of
energy density [39]. In this form, the viscosity coefficient in a particular
case, for example : if $\xi(\rho)=\rho^{\frac{1}{2}}$, creates a big rip
singularity at the late of the universe that applies only in barotropic fluid
with bulk viscosity [42]. Now if $s=\frac{1}{2}$ is assumed and placed in
relation (10) we will have:

$2\dot{H}+3H^{2}(1+\frac{p}{\rho}-\sqrt{3}a\rho)=0$ (24)

if we integrate from Equation (24), the Hubble function will be equal to:

$H(t)=H_{0}[\frac{3}{2}H_{0}(1+\frac{p}{\rho}-\sqrt{3}a\rho)(t-t_{0})]^{-1}$
(25)

and if we integrate equation (25) again, the scale factor is obtained:

$a(t)=[1+\frac{3}{2}H_{0}(1+\frac{p}{\rho}-\sqrt{3}a\rho)(t-t_{0})]^{\frac{2}{3(1+\frac{p}{\rho}-\sqrt{3}a\rho)}}.$
(26)

With these relations (25) and (26) the energy density of the fluid in the case
of the variable value of the viscosity coefficient will be equal to:

$\rho(a)=3H_{0}^{2}(\frac{a}{a_{0}})^{-3(1+\frac{p}{\rho}-\sqrt{3}a\rho)}$
(27)

with Assuming $a_{0}=1$, the last relation becomes
$3H_{0}^{2}a^{-3(1+\frac{p}{\rho}-\sqrt{3}a\rho)}$. Because in this case we
assumed the viscosity coefficient to be variable, so with the use of equation
(14) we will have:

$\xi(t)=\sqrt{3}aH_{0}[1+\frac{3(1+\frac{p}{\rho}-\sqrt{3}a\rho)H_{0}(t-t_{0})}{2}]^{-1}$
(28)

In relation (27) in order for the energy density in terms of the scale factor
to be able to determine the expansion of the universe, the value of the state
parameter must be unequal $-1$ ($\omega\neq-1$)[42]. In other words,
$p\neq-\rho$, that in case the power of relation (27) remains negative and the
energy density and also the temperature decrease and with decrease energy
density , the universe is expanded. Observations show that the value of
$p+\rho$ is close to zero. so, to reduce the energy density, the condition
$p+\rho>0$ and $\sqrt{3}a\ll 1$ is necessary. In a model of dark energy is
called the phantom $\sqrt{3}a>p+\rho$, this means an increase in energy
density, which is completely opposite to the dark energy model of cosmic
expansion [7].

###  2.2 Thermodynamics of Israel-Stewart theory

The Israel-Stewart theory provides a better explanation than Eckart’s theory
and solves the non-causal and instability problems of Eckart’s theory. In this
theory, we have a causal evolution equation for the bulk viscosity pressure in
the framework of Friedman equations, which is defined as follows:

$\tau\dot{\Pi}+\Pi=-3H\xi-\frac{1}{2}\tau\Pi[3H+\frac{\dot{\tau}}{\tau}-\frac{\dot{\xi}}{\xi}-\frac{\dot{T}}{T}]$
(29)

where $\tau$ is the relaxation time for the bulk viscosity effects and $T$ is
the temperature change due to the bulk viscosity effects. We see that the
Israel-Stewart theory is more complex than the Eckart theory. If the
temperature depends only on the energy density and the density of the number
of particles according to [27] and [37] we have:

$dT=(\frac{\partial T}{\partial\rho})_{n}d\rho+(\frac{\partial T}{\partial
n})_{\rho}dn$ (30)

On the other hand, the equations of conservation of energy density and
particle density in the Israel-Stewart theory are:

$d\rho=-3H(\rho+p+\Pi)$ (31) $dn=-3Hn$ (32)

Now if we replace relations (31) and (32) in relation (30) we will have:

$dT=-3H[(\frac{\partial T}{\partial\rho})_{n}(\rho+p+\Pi)+(\frac{\partial
T}{\partial n})_{\rho}n]$ (33)

The solution to this equation is $T=T_{0}\rho^{\frac{p}{p+\rho}}$. This
solution actually shows a perfect fluid with a bulk viscosity in Eckart’s
theory with condition that in equation (33) it is $p\neq-\rho$ ($T_{0}$ is an
approximate average value of temperature). According to [39], we consider a
power- law for the bulk viscosity coefficient which is

$\xi=\xi_{0}\rho^{s}$ (34)

where $s$ and $\xi_{0}$ are a constant and arbitrary parameter, also
$\xi_{0}>0$ . Finally, according to [42] for relaxation time we have:

$\tau=\frac{\xi}{\rho}=\xi_{0}\rho^{s-1}$ (35)

Now if relations (34) and (35) are the two main hypotheses for solving
equation (29), with consider Friedmann equations (1),(2),(7), we’ll have:

$2\dot{H}+3(1+\frac{p}{\rho})H^{2}+\Pi=0$ (36)

Therefore, equation (36) can be substituted for the three equations (33),
(34), (35) and placed in the bulk viscosity pressure transfer equation (29).
After placement, the differential equation can be equated with the energy
density conservation equation (31), which in result the following relation is
obtained as [39]:

$\ddot{H}+3H\dot{H}+\frac{3^{1-s}}{\xi_{0}}\dot{H}H^{2(1-s)}-(\frac{2p+\rho}{p+\rho})\frac{\dot{H}^{2}}{H}+\frac{9}{4}H^{3}(\frac{p}{\rho}-1)+\frac{3^{2-s}(1+\frac{p}{\rho})H^{2(2-s)}}{2\xi_{0}}=0$
(37)

Because the equation (37) is so complex, this equation can only be modeled
numerically. With giving the value of the parameter $s$, the equation can be
reduced to a simpler equation. Note that for $s\neq\frac{1}{2}$ and
$s\leq-\frac{1}{2}$ there is no solution or phantom model[42]. Israel-Stewart
theory is a non-linear and causal theory for dark energy cosmic fluid models,
and a phantom solution is needed to solve equation(37)(phantom solution
correspond to cosmic data) and also for $s\geq-\frac{1}{2}$ and
$s=\frac{1}{2}$, there is a phantom solution for the equation(37) [42,43].
Unlike the previous works done in various papers which was mostly considered
$s=\frac{1}{2}$, we decided this time to use $s=1$ for consideration some
aspects of model(means, the relaxation time is constant). With placing $s=1$
in equation $(37)$ we will have:

$\ddot{H}+3H\dot{H}+\frac{1}{\xi_{0}}\dot{H}-(\frac{2p+\rho}{p+\rho})\frac{\dot{H}^{2}}{H}+\frac{9}{4}H^{3}(\frac{p}{\rho}-1)+$
$\frac{3}{2}\frac{(1+\frac{p}{\rho})H^{2}}{2\xi_{0}}=0$ (38)

with rewriting this equation we have:

$\xi_{0}\ddot{H}(1+\frac{p}{\rho})+3\xi_{0}H\dot{H}(1+\frac{p}{\rho})+\dot{H}(1+\frac{p}{\rho})-\xi_{0}(2p+\rho)\frac{\dot{H}^{2}}{H}+$
$\frac{9}{4}\xi_{0}H^{3}(\frac{p}{\rho}-1)^{2}+\frac{3}{4}H^{2}(1+\frac{p}{\rho})^{2}=0$
(39)

If $\xi_{0}=0$ then the bulk viscosity pressure will be $\Pi=0$. In this case,
relation (39) will change as follows:

$\dot{H}+\frac{3}{4}H^{2}(1+\frac{p}{\rho})=0$ (40)

In equation (39), if $p=-\rho$ or $\omega=-1$ establishes the standard
cosmological state. Therefore, if the bulk viscosity model is presented as a
solution to equation (39), it should be $p\neq-\rho$ or $\omega\neq-1$, in
other words, we should consider the standard case as an exception. One of the
solutions of equation (38) is Ansats, which is mentioned in [27] and [40] as
follows:

$H(t)=\frac{A}{t_{\alpha}-t}$ (41)

And in the general case in the form $\frac{A}{(t_{\alpha}-t)^{p}}$ which is
$p<0$ and $t_{\alpha}$ is time limited to the future which ends in a big rip
singularity and $A$ is a positive constant to describe the expansion of the
universe[41]. If we assume the recent relation (41) as a solution to the
differential equation of relation (38) and also assume
$\xi_{0}=\xi(t)=t_{\alpha}-t$, the viscosity coefficient changes with time
until the universe reaches a future singularity time $(t_{\alpha})$. An
equation in terms of description constant of the expansion of the universe
($A$) is obtained as follows:

$\frac{9}{4}A^{2}(\frac{p}{\rho}-1)+\frac{3}{4}A(\frac{p}{\rho}+5)+\frac{p+2\rho}{p+\rho}=0$
(42)

that the solution of this equation is:

$A=\frac{-\frac{3}{4}(\frac{p+5\rho}{\rho})\pm\sqrt{{\frac{9}{16}}(\frac{p+5\rho}{\rho^{2}})-9(\frac{p-\rho}{\rho})(\frac{p+2\rho}{p+\rho})}}{\frac{9}{2}(\frac{p-\rho}{\rho})}.$
(43)

Now to calculate the scale factor of this model, it is enough to integrate the
Hubble function relation (41), which in case we will have:

$a(t)=(\frac{t_{\alpha}-t_{0}}{t_{\alpha}-t})^{A}$ (44)

where $t_{0}$ is the present time. According to equations (32), (41) and (44),
we can calculate the density of the number of cosmic fluid particles with bulk
viscosity, which we have:

$n(t)=n_{0}(\frac{t_{\alpha}-t}{t_{\alpha}-t_{0}})^{3A}$ (45)

From equation (44), it is obvious that if $t_{\alpha}=t$, the expansion of the
universe becomes infinite and the density of the number of particles tends to
zero, and thus through equation (41) the bulk viscosity pressure can be
obtained[41,42,43]:

$\Pi(t)=A(2+3A(1+\frac{p}{\rho}))(t_{\alpha}-t)^{2}$ (46)

Note that the last relation establishes with condition
$H(t)=\frac{A}{t_{\alpha}-t}$ and can have different values. In the next
section, using the thermodynamic relationships of this section, we investigate
the production of positive entropy of cosmic fluids in dissipative processes
and find that with increasing the entropy of cosmic fluids with bulk viscosity
as a model of dark energy, the universe expands.

##  3 Dissipative Effects of Perfect Cosmic Fluids

In the study of the effects of bulk viscosity on cosmic fluids, as mentioned
earlier, the non-causal theory of Eckart and the causal theory of Israel-
Stewart (IS) have been defined. Bulk viscosity models with dissipative
processes can improve the expansion of the universe calculation. In fact,
entropy is produced in dissipative processes lead to the expansion of the
universe. Also entropy is produced by isentropic particles according to IS
theory [44]. In isentropic particles, the entropy of the particles is
constant, but due to the expansion of the universe, the entropy production of
these particles increases. It is noteworthy that if we consider cosmic fluids
openly so that the number of fluid particles is not maintained, then the
cosmic fluid becomes a non-equilibrium thermodynamic system, and to create
fluid particles in this state, a bulk viscosity pressure,$\Pi$ , is defined.
In addition, dissipative processes play a basic role in the evolution of the
early universe and also the big rip singularity. In fact, they violate the
dominant energy conditions (DEC), which means that $p+p+\Pi<0$, and on the
other hand, because the bulk viscosity pressure,$\Pi<0$, so these conditions
increase the energy density of fluid and according to equation (5), the
previous bulk viscosity coefficient is considered $\xi>0$. Therefore, with the
two conditions of violation of the dominant energy condition (DEC) and
$\xi>0$, the entropy of the cosmic fluid increases and as a result, it does
not violate the second law of thermodynamics [44]. Another noteworthy point is
that in the perfect causal theory of IS, in order to obtain solutions that
lead to big rip singularity, the dark component must be considered as a
phantom [44]. This is because the dark energy of the phantom is inconsistent
with the hypothesis of a perfect cosmic fluid which the fluid is reversible
and without dissipative processes. Therefore, if the dark component is
considered as a phantom then the cosmic fluids, like real fluids, is
irreversible and have dissipative processes in resulting the production of
entropy.
From the first law of thermodynamics $(4)$ also internal energy relations,
$U=V\rho$, volume $V=V_{0}a^{3}$ and the density of cosmic fluid particles
$n=\frac{N}{V}$, the following equation can be written for cosmic fluid
entropy with bulk viscosity pressure $\Pi$:

$nTdS=-3H\Pi$ (47)

If in the above relation $\Pi<0$, the right side equation becomes positive and
as a result we conclude entropy production for the expanding universe.

###  3.1 Thermodynamics of dissipative fluids

First, before the entropy results, we check the thermodynamics of dissipated
cosmic fluids. Using the relation of the first law of thermodynamics, ie
$dE=dQ-PdV$ and relation $(4)$, we can write:

$TdS=dQ=dE+PdV$ (48)

where $Q$ is the exchanged heat and $E$ is the internal energy of the
dissipated fluid. According to equation $(48)$, if no heat is exchanged in the
cosmic fluid, ie the universe is adiabatic $(dQ=0)$, then we will not have
entropy production $(dS=0)$. Therefore, one of the conditions for the
expansion of the universe and the production of entropy is that the universe
not be adiabatic. If we consider the two variables $V$ and $T$ as the two
basic variables of dissipative fluid thermodynamics, we get the criteria for
$dS$ [45]:

$dLn(T)=dLn|1+\frac{p}{\rho}|-\frac{p}{\rho}dLn(V)$ (49)

Now if we consider equation (49) with energy conservation equation $(3)$ we
can write:

$dLn(T)=dLn|1+\frac{p}{\rho}|+dLn(\rho)+dLn(V)$ (50)

that If we integrate, we have:

$\frac{V(2\rho+p)}{T}=\frac{V_{0}(2\rho_{0}+p_{0})}{T_{0}}$ (51)

On the other hand, the internal energy of cosmic fluids is defined as $E=\rho
c^{2}V$. According to this equation and equation $(51)$, the following
equation can be obtained, which is called the modified ideal gas law:

$\frac{E}{E_{0}}=\frac{T\rho}{T_{0}\rho_{0}}(\frac{p_{0}+\rho_{0}}{p+\rho})$
(52)

The dissipated cosmic fluid can be investigated under constant pressure and
constant volume conditions and the effects of their heat capacity ($C_{p}$ and
$C_{v}$). The heat capacity of a cosmic fluid at constant pressure is defined
as follows [44]:

$C_{p}=(\frac{\partial h}{\partial T})_{p}$ (53)

that $h=E+PV$ is a fluid enthalpy. Using equation $(52)$, $C_{p}$ can be
written as follows:

$C_{p}=(1+\frac{p}{\rho})\frac{E}{T}=(1+\frac{p_{0}}{\rho_{0}})\frac{E_{0}}{T_{0}}=constant$
(54)

and also for heat capacity at constant volume $C_{v}=(\frac{\partial
E}{\partial T})_{v}$ and using relations $(49)$,$(52)$ $C_{p}$ and $C_{v}$ are
related to each other as follows:

$C_{p}=\frac{(1+\frac{p}{\rho})dLn(V)-dLn(\frac{p}{\rho})}{dLn(V)}C_{v}$ (55)

Now, if we consider $P$ and $T$ as two independent thermodynamic variables of
dissipated fluid, the change in volume (expansion and compression) of the
fluid can be written as follows:

$\frac{dV}{V}=(\lambda dT-Ad\rho)$ (56)

where $\lambda$ is the coefficient of thermal expansion at constant pressure
and $A$ is the coefficient of compression at constant temperature and are
defined as follows:

$\lambda=\frac{1}{V}(\frac{\partial V}{\partial T})_{p}$ (57)

,

$A=-\frac{1}{V}(\frac{\partial V}{\partial p})_{T}$ (58)

In addition, another coefficient is defined for the adiabatic state (constant
entropy and temperature change) which is called the compression coefficient in
the adiabatic state and we have:

$B=-\frac{1}{V}(\frac{\partial V}{\partial p})_{S}$ (59)

The noteworthy point of relations $(53)$ to $(59)$ is that the ratio of heat
capacities at constant pressure and constant volume is equal to the ratio of
compression coefficients at constant temperature and constant entropy [47]:

$\frac{C_{p}}{C_{v}}=\frac{A}{B}$ (60)

Also, in order for the thermodynamic system of the dissipated fluid to be
stable (depending on the phantom and non-phantom conditions and the value
range $\omega$), all coefficients $C_{p}$,$C_{v}$,$A$,$B$ must be positive
[44]. Therefore, according to relation $(60)$ and $(55)$, the relation between
$A$ and $B$ can be written as follows:

$A=\frac{(1+\frac{p}{\rho})dLn(V)-dLn(\frac{p}{\rho})}{dLn(V)}B$ (61)

and also we have:

$\frac{\lambda}{A}=(\frac{\partial p}{\partial T})_{v}$ (62)
$\frac{\lambda}{B}=(\frac{\partial p}{\partial T})_{v}$ (63)

Now, using the modified ideal gas equation $(54)$ and equation $(49)$, the
coefficient of thermal expansion $\lambda$ can be written as follows:

$\lambda=\frac{C_{p}}{(1+\frac{p}{\rho})V}[1+\frac{\rho
dp}{\frac{p}{\rho}[\rho dp-\frac{p}{\rho}(1+\frac{p}{\rho})dLnV]}]$ (64)

that from this relation we obtain the following equations[44]:

$\lambda=\frac{C_{p}}{V}A$ (65) $\lambda=\frac{C_{p}^{2}}{VC_{v}}B$ (66)

Here, we consider two different scenarios for cosmic fluids:
1\. The cosmic fluid system is non-adiabatic.
2\. The cosmic fluid system be considered adiabatic.
If the fluid is non-adiabatic, ie entropy production is done, then we consider
the temperature to be almost constant. In this case, because entropy
production is done, so $B=0$. Also, the fluid has a bulk viscosity with
negative pressure,$\Pi<0$, and its value is greater than the equilibrium
pressure, so the effective pressure becomes negative $(p_{eff}<0)$ and as the
pressure of the fluid system decreases, the volume of the fluid system
increases $(V>0)$( this is easily understood from equation $(58)$). On the
other hand, according to the relations $(62)$ and $(65)$ in the constant
volume process, with decreasing pressure, the temperature of the fluid
increases. Therefore, with the expansion of the cosmic fluid, the entropy and
temperature both increase, and also according to equation $(65)$, the changes
between the coefficients of expansion and compression of the cosmic fluid
change linearly, as shown in Fig.1. This scenario is a acceptable conclusion
from the standard theory of cosmology that with pass the time, the universe
expands and its entropy and temperature increase.

Figure 1: Ratio of expansion coefficient to compression coefficient in non-
adiabatic fluid. It is observed that as time passes and the volume increases,
the rate of expansion of the universe decreases relative to its compression.
The slope of the diagrams is equal to $\frac{C_{p}}{V}$ with $C_{p}$
constant($C_{p}=5$) and $V$ increasing ($V=2.5,5,7.5,10$).

If the fluid is adiabatic, ie we do not have entropy production and the
entropy is assumed to be constant and unchanged, then we consider $A=0$. In
such a cosmic fluid, the viscosity pressure of the bulk is negative and the
pressure decrease continuously in the state of adiabatic expansion, and
according to the equation $(59)$ the volume of the fluid increases, in this
type of fluid(of course in the adiabatic expansion), the temperature decrease
with decreasing pressure and increasing the volume of fluid. The difference
between this scenario and the previous scenario is that first, we do not have
any entropy production in this type of fluid, but in the previous scenario,
which is our acceptable scenario, the entropy increases. The second difference
is that the expansion rate in this type of fluid (adiabatic) is more than Non-
adiabatic fluid, meaning that the universe expands more rapidly in this
scenario and also in previous scenario the temperature was increasing but in
this scenario the temperature decrease. However, this scenario can have two
specific situations:
Case 1: If the $C_{v}$ is constant and the $C_{p}$ increases then the rate of
expansion will be faster with pass the time(Fig. 2. a).
Case 2: If $C_{v}$ decreases and $C_{p}$ is constant: then the rate of
expansion increases for a limited time and then begins to compress (Fig. 2.
b). This case expresses the dark energy of the phantom well, but is not
acceptable in terms of standard cosmological theory. As we can see, the best
model for the expansion of the universe is Scenario 1, which is consistent
with both cosmological observations and standard cosmological theory. This
means that the universe continues to expand, but its rate of expansion
increases.

Figure 2: Fig.2.a(Up): The universe expand to infinity. The slope of the
diagrams is equal to $\frac{C_{p}^{2}}{VC_{v}}$ that $C_{v}$ is
constant($C_{v}=3$) and with increasing volume($V=1,3,6,10$) $C_{p}$
increase.($C_{p}=5,6,7,8)$. Fig.2.b.(Down): The universe expands until a
certain time and then stops accelerated expansion and starts to compression.

###  3.2 Investigation of entropy of dissipative fluids

First, we study the entropy of perfect fluids and then we will investigate the
entropy relations in the framework of the two theories of Eckart and IS. For a
perfect fluid, in general terms, two expressions of the equation of state,
$p=\rho\omega$, with $\omega$ constant and the particle flow number
conservation $n^{\alpha}=nu^{\alpha}$, where $u^{\alpha}$ is 4-velocity are
considered and we have:

$\dot{n}+3Hn=\frac{\dot{N}}{N}=0$ (67)

If we place equations $(67)$ and $(1)$ in the equations of conservation of
energy density $(3)$ and the first law of thermodynamics $(4)$, respectively,
the following relations are obtained:

$\dot{\rho}=-3H(p+\rho)$ (68) $TdS=Vd\rho+(p+\rho)dV$ (69)

that equation $(69)$ is also called the Gibbs relation. Now to calculate the
entropy with constant $\omega$ we have

$TdS=(1+\omega)\rho dV+Vd\rho=U(dLn(\rho)+3(1+\frac{p}{\rho})dLn(a))=0$ (70)

This equation is obtained from equations $(68)$ and $(69)$. Considering this
relation, in this case, if $\omega$ is assumed to be constant, then the
entropy is constant, which means that the fluid system is adiabatic, as we saw
in scenario 2 in the previous subsection. Now to calculate the constant
entropy value, we use the following Eulerian relation:

$TS=U+\rho V-\mu N$ (71)

Considering that the temperature and internal energy of a fluid are
thermodynamically proportional to each other $(U\propto T)$, hence we can
write the following relation:

$S=(1+\frac{p}{\rho})\frac{U}{T}-\frac{\mu N}{T}=(1+\frac{p}{\rho})\frac{\rho
V}{T}-\frac{\mu N}{T}$ (72)

In perfect cosmic fluids with constant $\omega$, entropy $S$ and the number of
fluid particles $N$ are constant, so according to equation (72) the
$\frac{\mu}{T}$ ratio must be constant, which $\mu$ is called the chemical
potential of the fluid. And based of the chemical potential, three states
occur for the cosmic fluid:
1\. If $\mu=0$ in this case $p\geq-\rho$ which in this case the cosmic fluid
is far from the behavior of the phantom.
2\. If $\mu>0$ in this case $p>-\rho$. The entropy has the least constant
value and the cosmic fluid in this case does not show the behavior of the
phantom.
3\. If $\mu<0$ in this case $p<-\rho$. We will have a phantom dark energy or
phantom cosmic fluid[25].
In case $\omega$ is the variable $(\omega(a))$, if the chemical potential is
$\mu=0$, then the parameter of the variable state, $\omega(a)$, is a constant
parameter and is always greater than or equal to $-1$, $\omega(a)\geq-1$, and
if $\mu>0$ then $\omega(a)>-1$ and away from the standard state and if $\mu<0$
two states occur, $|\mu|<\frac{S_{0}T}{V_{0}n_{0}}$ which in this case again
$\omega(a)>-1$ and if $|\mu|>\frac{S_{0}T}{V_{0}n_{0}}$, we will have the
phantom state which in this case will be $\omega(a)<-1$ [25].
Now in the framework of Eckart theory , we investigate the entropy of
dissipated cosmic fluids. In this theory, we use equation $(47)$ to calculate
the fluid entropy, which in instead of the bulk viscosity pressure $-3H\xi$,
therefore we will have:

$dS=\frac{9H^{2}\xi(t)}{nT}$ (73)

Now we calculate the thermodynamic variables of density of particles $n$ and
temperature $T$ and place them in equation $(73)$, and after we integrate from
the obtained equation, finally we obtain the entropy $S$ . In here, we
consider two cases:
1\. $\xi(t)=\xi_{0}=constant$ and $\xi_{0}>0$ : according to the relations
$(3)$ and $(4)$ in the appendix and also the particle density
$n=\frac{n_{0}}{a^{3}}$, for $T=T_{0}\rho^{\frac{p_{eff}}{1+p_{eff}}}$ we
have:

$T(t)=T_{0}\rho_{0}^{\frac{p_{eff}}{\rho+p_{eff}}}\times$
$[\frac{\xi_{0}e^{\frac{3}{2}\xi_{0}(t-t_{0})}}{H_{0}(1+\frac{p_{eff}}{\rho})(e^{\frac{3}{2}\xi_{0}(t-t_{0})}-1)+\xi_{0}}]^{\frac{2p_{eff}}{\rho+p_{eff}}}$
(74)
$n(t)=n_{0}[1+\frac{H_{0}}{\xi_{0}}(1+\frac{p_{eff}}{\rho})(e^{\frac{3}{2}\xi_{0}(t-t_{0})}-1)]^{\frac{-2\rho}{\rho+p_{eff}}}$
(75)

Now if we place two relations $(74)$ and $(75)$ in relation $(73)$ we have the
following relation:

$\frac{dS}{dt}=\frac{3\xi_{0}\rho_{0}^{\frac{\rho}{\rho+p_{eff}}}}{n_{0}T_{0}}e^{\frac{3\xi_{0}\rho}{\rho+p_{eff}}(t-t_{0})}$
(76)

If we integrate from equation $(76)$ with assuming $p=-\rho$, the entropy is
obtained:

$S(t)=S_{0}+\frac{(1+\frac{p_{eff}}{\rho})\rho_{0}^{\frac{\rho}{\rho+p_{eff}}}}{n_{0}T_{0}}(e^{\frac{3\xi_{0}\rho}{\rho+p_{eff}}(t-t_{0})}-1)$
(77)

where $S_{0}$ is a integral constant or the entropy at the present time, now
we can see, the increase in entropy is exponential. If in the Eckart theory
$\xi$ be variable and according to the hypotheses of the previous section we
consider $\xi=\alpha\rho^{s}$ $(\alpha>0)$, similar to the previous case, we
get the density of the number of particles $(n)$ and temperature $(T)$ and
then we place in $(73)$ and so integrate from the obtained relation to obtain
the entropy for the variable $\xi$. Because in this case the coefficient the
viscosity of the bulk changes as power-law, choosing a solution to obtain
entropy will not be easy, but by giving a specific value for $s$ and $\alpha$,
a simpler solution can be expressed. In a particular case, suppose
$\alpha=\frac{\sqrt{3}}{3}\gamma$ and $s=\frac{1}{2}$,
$\xi=(\frac{\sqrt{3}}{3}\gamma\rho^{\frac{1}{2}})$. Now according to equations
$(6)$ and $(7)$ in the appendix to calculate the density of the number of
particles $n$, and temperature $T$, and
$n=n_{0}a^{-3},T=T_{0}\rho_{0}^{\frac{p_{eff}}{1+p_{eff}}}$ we have:

$n(t)=n_{0}[1+\frac{3H_{0}}{2}(1+\frac{p_{eff}}{\rho}-\gamma)(t-t_{0})]^{\frac{-2\rho}{\rho+p_{eff}-\gamma}}$
(78)

,

$T(t)=T_{0}\rho_{0}^{\frac{p_{eff}}{1+p_{eff}}}[1+\frac{3H_{0}}{2}(1+\frac{p_{eff}}{\rho}-\gamma)(t-t_{0})]^{\frac{-2p_{eff}}{1+p_{eff}}}$
(79)

If we place the relations $(78,79)$ in $(73)$, we obtain the following
differential equation:

$\frac{dS}{dt}=\frac{\gamma\rho_{0}^{\frac{2\rho+p_{eff}}{\rho+p_{eff}}}}{H_{0}n_{0}T_{0}}\times[1+\frac{3H_{0}}{2}(1+\frac{p_{eff}}{\rho}-\gamma)(t-t_{0})]^{\frac{-(\frac{p_{eff}}{\rho})^{2}+(2-\gamma)(\frac{p_{eff}}{\rho})+1}{(\frac{p_{eff}}{\rho})^{2}+(2-\gamma)(\frac{p_{eff}}{\rho})+(1-\gamma)}}$
(80)

We integrate from the above relation, therefore we have:

$S(t)=S_{0}+\frac{(1+\frac{p_{eff}}{\rho})\rho_{0}^{\frac{\rho}{\rho+p_{eff}}}}{n_{0}T_{0}}[(1+\frac{3H_{0}}{2}(1+\frac{p_{eff}}{\rho}-\gamma)(t-t_{0}))^{\frac{\gamma-1}{(\frac{p_{eff}}{\rho})^{2}+(2-\gamma)(\frac{p_{eff}}{\rho})+(1-\gamma)}}-1]$
(81)

In order for the entropy to increase in this case, the power of equation
$(81)$ must be positive.
To calculate the entropy in the Israel-Stuart (IS) theory, we have the
following differential equation that it obtain from the placement Ansatz
$(41)$ in equation $(46)$, density of the number of fluid particles $(45)$ and
the relation $(9)$ in the appendix (temperature relation) in relation $(47)$,
then we have:

$\frac{dS}{dt}=M(t_{\alpha}-t)^{N}$ (82)
$M=\frac{(3A^{2})^{\frac{\rho}{\rho+p}}[2+3A(1+\frac{p}{\rho})](t_{\alpha}-t_{0})^{3A}}{n_{0}T_{0}}$
(83) $N=\frac{2p}{\rho+p}-3(1+A)$ (84)

Now we integrate from equation $(82)$ that we have(with condition
$p\neq\rho$):

$S(t)=S_{0}+(\frac{-M}{N+1})(t_{\alpha}-t)^{1+N}$ (85)

where $S_{0}$ is an integral constant or entropy at the present time. Here we
present two scenarios:
1\. If $M>0$ and $N<-1$ then we conclude that $0<\frac{p}{\rho}<\frac{1}{2}$
that we shown in Fig. 3.a. With $\omega>0$ we conclude that the cosmic fluid
is expanding.
2\. If $M<0$ and $N>-1$. in such a universe, entropy has a constant value and
then begins to decrease rapidly which totally violates standard theory and
Hubble’s law(Fig.3.b).

Figure 3: Fig.3.a(Up): Cosmic fluid entropy behavior over time. We see that,
the entropy increases infinitely up to a asymptote(here assuming $t=10$ in
term of Gpc). Then in the same asymptote, the rate of expansion of the
universe equals with its rate of compression and after these asymptote, the
average rate of expansion of the universe be lower than the average rate of
compression until the average value of entropy production of universe arrives
a horizontal asymptote which means that the average rate of expansion of the
universe is closer to zero, that we mentioned in the first scenario(for non-
adiabatic fluid). with hypothetical
conditions[$M=10,N=-2,t_{\alpha}=10,S_{0}=1$]. Fig.3.b.(Down): In this state
that it does not correspond to the standard theory at all, the entropy is
constant in a limited time and then begins to decrease very rapidly which in
fact shows an unreal universe that contradicts Hubble’s law. With hypothetical
conditions[$M=-10,N=2,t_{\alpha}=10,S_{0}=1$].

##  4 The Effects of Energy density for Dark Energy in the IS theory

In this section, we want to investigate the effects of the energy density
$\rho$ in the framework of the Israel-Stewart theory. The main parameters that
its effects are investigated : temperature $(T)$, bulk viscosity coefficient
$(\xi)$ and relaxation time $(\tau)$ (according to equations (29) to (35)).
But each of these parameters is obtained in terms of energy density $(\rho)$,
so its effects and relationships can be obtained. In this regard, if we
rewrite equation $(30)$ as energy density, we have:

$d\rho=(\frac{\partial\rho}{\partial T})_{n}dT+(\frac{\partial\rho}{\partial
n})_{T}dn$ (86)

We place the density relation of the number of fluid particles $(32)$ and the
temperature differential equation $(33)$ in equation $(86)$, we have:

$d\rho=-3H[(p+\rho+\Pi)+n(\frac{\partial\rho}{\partial T})_{n}(\frac{\partial
T}{\partial n})_{\rho}-n(\frac{\partial\rho}{\partial n})_{T}]$ (87)

that we simplify this equation, we get the equation of conservation of energy
density in equation $(31)$:

$d\rho=-3H(p+\rho+\Pi)$

Due to the bulk viscosity pressure in the Israel-Stewart (IS) theory, we have
two states:
Mode 1: $\Pi=-2\dot{H}-3(1+\frac{p}{\rho})H^{2}$ (according to the equation
$(36)$), we place this value $\Pi$ in the energy density conservation equation
$d\rho$ we have:

$d\rho=-3H(p+\rho-2\dot{H}-3H^{2}(1+\frac{p}{\rho}))$ (88)

Now we integrate from the above relation twice, then we have:

$\rho(a)=\frac{9}{20}(1+\frac{p}{\rho})H^{5}+(-\frac{1}{2}(p+\rho)+1)H^{3}$
(89)

In following, we summarize the results according to table $(1)$. Also we
plotted the obtained results in figure $(4)$. According to figure $(4A)$ that
shows the energy density in term of scale factor in standard approach. The
condition of the universe expansion and increase of the scale factor is the
ascending increase of the value of energy density. In this approach, the
pressure value is symmetrical with energy density value and with the energy
density value increase, the value of pressure becomes more negative. In figure
$(4B)$ that shows the energy density in term of scale factor in dynamical and
non-phantom approach. In this figure pressure value is greater than the energy
density value, with the universe expansion and the scale factor increase, the
energy density value increases similar in the standard approach. But in this
case as much as the energy density value be lower, the universe expands
faster. Finally, figure $(4C)$ shows the energy density in the phantom
approach, with the universe expansion and the scale factor increase, the
energy density value increases but in this case as much as the energy density
value be more, the universe expand faster and also if the energy density value
exceeds a certain value, the universe stops accelerated expansion and begins
to slow expansion and when energy density value becomes infinite, the universe
will reach a big rip singularity in the future.

Figure 4: The effects of energy density in term of scale factor on the
universe expansion in Israel-Stewart theory(IS) Fig.4.A(up): standard approach
with condition $p=-\rho$, Fig.4.B(middle): dynamical and non-phantom approach
with condition $p>-\rho$, Fig.4.C(down): phantom approach with condition
$p<-\rho$.

Mode $2$: $\Pi(t)=A(2+3A(1+\frac{p}{\rho}))(t_{\alpha}-t)^{2}$ (according to
the equation $(46)$), we place this relation in the energy density
conservation equation $d\rho$ so we have:

$d\rho=-3H[p+\rho+A(2+3A(1+\frac{p}{\rho}))(t_{\alpha}-t)^{2}]$ (90)

And, as we know, $t_{\alpha}$ is a time that leads to a singularity in the
future, and $A$ is a constant that describes the expansion of the universe(its
value was obtained in equation $(43)$). Now if we integrate from the above
relation we have:

$\rho(t)=-3H[pt+\rho
t-\frac{1}{3}A(2+3A(1+\frac{p}{\rho}))(t_{\alpha}-t)^{3}]$ (91)

Table 1: The results of the energy density effects in terms of scale factor on the universe expansion in the framework of IS theory (Mode 1). Condition |  Approach |  Description
---|---|---
$P=-\rho$ |  Standard |  In this case, the parameter of the cosmic fluid state equation ($\omega$) is considered constant as dark energy with bulk viscosity and its value is $\omega=-1$ and according to equation (1) it becomes $p=-\rho$ and also has no dynamics. In this type of approach the pressure value of cosmic fluid is negative and energy density increases and finally, the universe continues the expansion rapidly.
$P>-\rho$ |  Dynamical and non-phantom |  In this approach, first the parameter $\omega$ is not constant and increase with the expansion of the universe, that is, as the universe expands, the pressure of the cosmic fluid with bulk viscosity tends from negative to positive value. This means that the energy density value decreases. Second, this approach conforms to the thermodynamics laws and the standard cosmological model and also shows that as the universe expands, the entropy and temperature increase and the chemical potential becomes negative [24],[25].
$P<-\rho$ |  Phantom |  The energy density of a cosmic fluid increases as power-law($\rho=\rho_{0}a^{r}$), that r is a constant and positive, and $\rho_{0}$ is the initial value of the energy density of the fluid. In this approach, with the growth of the scale factor and the expansion of the universe, the energy density increases and leads to decrease in temperature and no entropy is produced. The rate of expansion in this approach is much higher and leads to a big rip singularity. This approach also violates the dominant energy conditions (DEC) and faces with many challenges and are classically and quantum unstable.[14]

We obtain the following results according to table $(2)$, we also plotted the
obtained results and relations in figure $(5)$.
In this mode, and figure $(5A)$ the results are similar to the standard
approach in table$(1)$, but with the difference that in previous mode the
energy density was in terms of scale factor but here in terms of cosmic time
and an expansion coefficient $A$ that obtained in the second section. In
figure $(5B)$ with over time and the universe expansion, the energy density
value decreases so the pressure value increases and tends to be positive.
Finally, in figure $(5C)$ with over time, the energy density value increases
and as a result the universe is expanded faster but when it crosses the
phantom divide line (here the phantom divide line has been showed by a black
dotted line on the $\rho$ axis) and the energy density becomes infinity, the
universe begins to slow expansion until it finally reaches a big rip
singularity at the time $t_{\alpha}$. In this section we could to study the
effects of energy density in terms of scale factor and in terms of cosmic time
on the expansion of the universe under the Israel-Stewart (IS) theory.
Here, Let us compared results of our research with earlier studies of Dark
Energy from viscous or inhomogeneous fluids according references [48,49,50].
In this regard, Brevik et al considered the role of a viscous (or
inhomogeneous (imperfect) equation of state) fluid in a Little Rip cosmology.
Despite the earlier observations that viscosity basically supports the Big Rip
singularity, they have demonstrated that it is also able to give rise to a
non-singular, Little Rip cosmology, which is considered to be a viable
alternative to $\Lambda$CDM cosmology. In particular, constant bulk viscosity
and a viscosity inversely proportional to the Hubble rate have been
considered. They have shown that in those cases a Little Rip cosmology may
naturally emerge. It is remarkable that for a standard fluid, $p=\omega\rho$,
the only influence of viscous effects can naturally drive the universe to a
Little Rip type evolution[48]. Also, Brevik and Timoshkin studied a general
equation of state for the dark fluid in the presence of a bulk viscosity. They
have explored the holographic principle for cosmological models with various
values for the thermodynamic parameter $\omega(\rho,t)$ and for different
forms of the bulk viscosity $\zeta(H,t)$. For each model the infrared radius,
in the form of a particle horizon, has been calculated in order to obtain the
energy conservation law. Thus, they have shown the equivalence between viscous
models and the holographic model[49]. Finally, Nojiri et al in [50] considered
the effect of modification of general equation of state of dark energy ideal
fluid by the insertion of inhomogeneous, Hubble parameter dependent term in
the late-time universe. Several explicit examples of such term which motivated
by time-dependent bulk viscosity or deviations from general relativity
studied. The corresponding late-time FRW cosmology (mainly, in its phantom
epoch) described. Also, they found that the inhomogeneous term in equation of
state helps to realize such a transition in a more natural way. It is
interesting that in the case when universe is filled with two interacting
fluids (for instance, dark energy and dark matter) the Hubble parameter
dependent term may effectively absorb the coupling between the fluids [50].
In comparing of above results with our findings in section 4, we can point out
in summary : 1- In standard approach, with over time, the universe has an
accelerated expansion (of course, before time $t_{\alpha}$). After time
$t_{\alpha}$, we assume that over time pressure tends to $p\rightarrow-2\rho$
and the universe expand slowly. In this approach the energy density value
increases rapidly but its value is symmetrical with the value of negative
pressure. 2- In dynamical and non-phantom approach, with over time, the
pressure value is negative and will change with the energy density value, and
expansion will increase, and as a result, the energy density will decrease.
Therefore, the condition for the expansion of the universe with over time is
the decrease in the dark energy density variable value. As much as the energy
density value is lower, the universe expand faster. 3- In phantom approach,
with over time, the energy density becomes bigger and tends to be infinity. As
a result, the universe expands more rapidly, but after crossing the phantom
divide line, the value of energy density becomes infinity(the maximum value of
energy density), the universe’s expansion rate gradually decreases and finally
reaches a big rip singularity after time $t_{\alpha}$ [51].

Figure 5: The effects of energy density in term of cosmic time on the universe expansion in Israel-Stewart theory(IS). Fig.5.A(up): standard approach with condition $p=-\rho$ and $A=\frac{1}{3}\mp\frac{1}{6\sqrt{\rho}}$, Fig.5.B(middle): dynamical and non-phantom approach with condition $p>-\rho$ and $A>\frac{1}{3}\mp\frac{1}{6\sqrt{\rho}}$, Fig.5.C(down): phantom approach with condition $p<-\rho$ and $A<\frac{1}{3}\mp\frac{1}{6\sqrt{\rho}}$. In this mode we considered the value of $t_{\alpha}=10$ and value of the Hubble constant $H\simeq 71\times 10^{9}$. Table 2: The results of the energy density effects in terms of cosmic time on the universe expansion in the framework of IS theory (Mode 2). Condition |  Approach |  Description
---|---|---
$P=-\rho\newline A=\frac{1}{3}\mp\frac{1}{6\sqrt{\rho}}$ |  Standard |  In this case, it is the same as in Table (1), except that the energy density is measured in terms of scale factor, but here it is defined by cosmic time and also a constant A. With Over time, the universe has an accelerated expansion (of course, before time $t_{\alpha}$). After time $t_{\alpha}$, we assume that over time pressure tends to $p\longrightarrow-2\rho$ and the universe expand slowly. In this approach the energy density value increases rapidly but its value is symmetrical with the value of negative pressure.
$P>-\rho\newline A>\frac{1}{3}\mp\frac{1}{6\sqrt{\rho}}$ |  Dynamical and non-phantom |  In this case, with over time, the pressure value is negative and will change with the energy density value[20], and expansion will increase, and as a result, the energy density will decrease (according to condition A). Therefore, the condition for the expansion of the universe with over time is the decrease in the dark energy density variable value. As much as the energy density value is lower, the universe expand faster.
$P<-\rho\newline A<\frac{1}{3}\mp\frac{1}{6\sqrt{\rho}}$ |  Phantom |  In this case, with over time, the energy density becomes bigger and tends to be infinity. As a result, the universe expands more rapidly, but after crossing the phantom divide line [21], the value of energy density becomes infinity(the maximum value of energy density), the universe’s expansion rate gradually decreases and finally reaches a big rip singularity after time $t_{\alpha}$.

##  5 Summary

In accordance with the unknown nature of dark energy, some dark energy models
has been studied as a cosmic fluid in the framework of thermodynamic laws. In
this regard, viscosity is a feature of the universe fluid content as discussed
in the present research. Therefore, in the first part of this article, we
expressed the thermodynamics of cosmic fluids in general with a constant and
variable equation of state under the two theories of Eckart and Israel-
Stewart. Then, we examined the dissipative effects of cosmic fluids and
finally examine the effects of energy density for dark energy in the Israel-
Stewart(IS) theory. The results are organized as follows:
We first investigated the thermodynamics of cosmic fluids in the dark energy
bulk viscosity model and the general relationships. Then, we expressed the
thermodynamic relationships of Eckart’s theory. In IS theory, a differential
equation is defined in terms of temperature $(T)$, bulk viscosity coefficient
$(\xi)$ and relaxation time $(\tau)$, and each of these parameters is obtained
in terms of energy density. First time, we obtained an equation for the bulk
viscosity pressure by placing the value of $s=1$ in equation $(37)$ and the
Ansats solution.
In the third section, we investigated the dissipative effects of cosmic fluids
and we concluded the best theory for studying the effects of dissipation is IS
theory, because it develops dissipation processes with slower rate. In this
section, We presented two scenarios. In the first scenario, we considered the
cosmic fluid to be non-adiabatic. We concluded that this type of fluid
conforms to the standard model of cosmology and cosmic observations, which
during it, entropy and temperature increase and causes the universe to expand.
But in the second scenario, we considered the cosmic fluid to be adiabatic and
concluded that no entropy is produced in this scenario and temperature
decreases. Also, we defined two states in this scenario, one is the
accelerated expansion state and the other is a phantom state that leads to a
singularity at the late of the universe. In the continuation of the third
section, we investigated the effects of cosmic fluid entropy in general and
then in the framework of Eckart and IS theories. We concluded that IS theory
better represents the production of entropy in dissipated cosmic fluids. Also,
in this part we defined two different scenarios for entropy. The first
scenario represents more correct behavior from cosmic fluid entropy with over
time but the second scenario is completely opposite to the first scenario.
In the fourth section, first time we expressed the effects of energy density
on the expansion of the universe in the framework of IS theory. According to
the definition of bulk viscosity pressure in IS theory, we suggested two
modes, one is energy density in terms of scale factor and the other is energy
density in terms of cosmic time and an expansion descriptor coefficient $(A)$.
The obtained results in this section are comprehensively presented in two
tables $(1)$ and $(2)$. We also plotted the energy density behavior in three
standard, dynamical and phantom approaches and obtained its results.

Acknowledgements We would like to thank all those who helped us gather the
information needed for this research.
Funding The authors did not receive support from any organization for the
submitted work.
Data availability The data that support the findings of this study are
available on request from the corre- sponding author.
Conflict of interest The authors declare that they have no known competing
financial interests or personal relationships that could have appeared to
influence the work reported in this paper. This research did not receive any
specific grant from funding agencies in the public, commercial, or not-for-
profit sectors.
Ethics approval We have followed all the rules stated in the Ethical
Responsibilities of Authors section of the journal.
Consent for publication We consent to the publication of this manuscript with
the rules of the journal.

##  6 Appendix

In this section, we consider some equations that were involved in obtaining
the equations in the previous sections. To calculate entropy we need to
calculate the particles number density of the fluid and the temperature of the
fluid. According to equation $(33)$ and its solution
$(T=T_{0}\rho^{\frac{p}{p+\rho}})$ we can write:

$\dot{T}=(\frac{p}{p+\rho})T\frac{\dot{\rho}}{\rho}+\rho^{\frac{p}{p+\rho}}\frac{df}{dx}\dot{x}$
(1)

where $df$ is a function in term of $x=T_{0}\rho^{\frac{p}{p+\rho}}$ and also
according to equations $(3)$ and $(32)$ we can write for $\dot{x}$:

$\dot{x}=\frac{\rho^{\frac{p}{p+\rho}}}{n}[(\frac{\rho}{p+\rho})\frac{\dot{\rho}}{\rho}-\frac{\dot{n}}{n}]=0$
(2)

then appendix equation (1) becomes equation
$\frac{dT}{T}=-3\frac{p+\rho}{\rho}\frac{da}{a}=\frac{p}{p+\rho}\frac{d\rho}{\rho}$
in the bulk viscosity model of a fluid with coefficient $\xi$ that $\xi$ is
constant to calculate Hubble parameter and scale factor, we have(which
equations $(74)$ and $(75)$ are concluded from them) [18]:

$H(t)=\frac{H_{0}\xi_{0}e^{\frac{3\xi_{0}(t-t_{0})}{2}}}{H_{0}(\frac{p+\rho}{\rho})(e^{\frac{3\xi_{0}(t-t_{0})}{2}}-1)+\xi_{0}}$
(3)
$a(t)=(1+\frac{H_{0}(p+\rho)}{\xi_{0}\rho}(e^{\frac{3\xi_{0}(t-t_{0})}{2}}-1))^{\frac{2\rho}{3(p+\rho)}}$
(4)

which by integrating the Hubble parameter relation, the scale factor is
obtained with condition $a(t_{0})=1$.
Finally, the energy density of the fluid can be obtained in term of scale
factor:

$\rho(a)=\rho_{0}(\frac{\xi_{0}\rho}{H_{0}(p+\rho)}+(\frac{H_{0}(p+\rho)-\xi_{0}\rho}{H_{0}(p+\rho)})a^{\frac{-2\rho}{3(p+\rho)}})^{2}$
(5)

Now, if $\xi$ is assumed to be variable as power-law,$\xi=\alpha\rho^{s}$,we
will have [18]:

$H(t)=H_{0}(1+\frac{3H_{0}}{2}(\frac{p+\rho}{\rho}-\alpha)(t-t_{0}))^{-1}$ (6)
$a(t)=(1+\frac{3H_{0}}{2}(\frac{p+\rho}{\rho}-\alpha(t-t_{0})))^{\frac{2}{3(\frac{p+\rho}{\rho}-\alpha)}}$
(7) $\rho(a)=\rho_{0}a^{-3(\frac{p+\rho}{\rho}-\alpha)}$ (8)

To calculate the fluid temperature in the bulk viscosity model in Israel-
Stewart(IS) theory in term of scale factor according to the relations $(41)$,
$(44)$ and $T=T_{0}\rho^{\frac{p}{p+\rho}}$ we will have:[16]

$T(t)=\tilde{T}(3A^{2})^{\frac{p}{p+\rho}}(t_{\alpha}-t)^{\frac{-2p}{p+\rho}}=\tilde{T}(3A^{2}(t_{\alpha}-t)^{-2})^{\frac{p}{p+\rho}}a^{\frac{2p}{A(p+\rho)}}$
(9)

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