#### Viscous cosmology in new holographic dark energy model and the cosmic acceleration

Eur. Phys. J. C
Viscous cosmology in new holographic dark energy model and the cosmic acceleration
C. P. Singh 0
Milan Srivastava 0
0 Department of Applied Mathematics, Delhi Technological University , Bawana Road, Delhi 110 042 , India
In this work, we study a flat FriedmannRobertson-Walker universe filled with dark matter and viscous new holographic dark energy. We present four possible solutions of the model depending on the choice of the viscous term. We obtain the evolution of the cosmological quantities such as scale factor, deceleration parameter and transition redshift to observe the effect of viscosity in the evolution. We also emphasis upon the two independent geometrical diagnostics for our model, namely the statefinder and the Om diagnostics. In the first case we study new holographic dark energy model without viscous and obtain power-law expansion of the universe which gives constant deceleration parameter and statefinder parameters. In the limit of the parameter, the model approaches to C D M model. In new holographic dark energy model with viscous, the bulk viscous coefficient is assumed as ζ = ζ0 + ζ1 H , where ζ0 and ζ1 are constants, and H is the Hubble parameter. In this model, we obtain all possible solutions with viscous term and analyze the expansion history of the universe. We draw the evolution graphs of the scale factor and deceleration parameter. It is observed that the universe transits from deceleration to acceleration for small values of ζ in late time. However, it accelerates very fast from the beginning for large values of ζ . By illustrating the evolutionary trajectories in r −s and r −q planes, we find that our model behaves as an quintessence like for small values of viscous coefficient and a Chaplygin gas like for large values of bulk viscous coefficient at early stage. However, model has close resemblance to that of the C D M cosmology in late time. The Om has positive and negative curvatures for phantom and quintessence models, respectively depending on ζ . Our study shows that the bulk viscosity plays very important role in the expansion history of the universe.
1 Introduction
The astrophysical data obtained from high redshift surveys of
supernovae [
1–3
], Wilkinson Microwave Anisotropy Probe
(WMAP) [
4,5
] and the large scale structure from Slogan
Digital Sky Survey (SDSS) [
6,7
] support the existence of
dark energy (DE). The DE is considered as an exotic energy
component with negative pressure. The cosmological
analysis of these observations suggest that the universe consists
of about 70% DE, 30% dust matter (cold dark matter plus
baryons), and negligible radiation. It is the most accepted
idea that DE leads to the late-time accelerated expansion of
the universe. Nevertheless, the nature of such a DE is still
the source of debate. Many theoretical models have been
proposed to describe this late-time acceleration of the
universe. The most obvious theoretical candidate for DE is the
cosmological constant [
8
], which has the equation of state
(EoS) ω = −1. However, it suffers the so-called
cosmological constant (CC) problem (the fine-tuning problem) and
the cosmic coincidence problem [9–11]. Both of these
problems are related to the DE density.
In order to solve the cosmological constant problems,
many candidates such as quintessence [12,13], phantom [14],
tachyon field [15], quintom [16], holographic dark energy
[17,18], agegraphic dark energy [19,20] have been proposed
to explain the nature of DE phenomenon. Starobinsky [21]
and Kerner et al. [22] proposed an another way to explain
the accelerated expansion of the universe by modifying the
geometrical part of Einstein field equations which is known
as modified gravity theory.
In recent years, the considerable interest has been noticed
in the study of holographic dark energy (HDE) model to
explain the recent phase transition of the universe. The idea of
HDE is basically based on the holographic principle [23–25].
According to holographic principle a short distance
(ultraviolet) cut-off is related to the long distance (infrared) cut-off
L due to the limit set by the formation of a black hole [26]. Li
[18] argued that the total energy in a region of size L should
not exceed the mass of a black hole of same size for a system
with ultraviolet (UV) cut-off , thus L3ρ ≤ L M 2pl , where
ρ is the quantum zero-point energy density caused by UV
cot-off and M pl is the reduced Planck mass M −pl2 = 8π G.
The largest L allowed is the one saturating this inequality,
thus the HDE density is defined as ρ = 3c2 M 2pl L−2, where
c is a numerical constant. The UV cut-off is related to the
vacuum energy, and the infrared (IR) cut-off is related to the
large scale structure of the universe, i.e., Hubble horizon,
particle horizon, event horizon, Ricci scalar, etc. The HDE
model suffers the choice of IR cut-off problem. In the Ref.
[17], it has been discussed that the HDE model with
Hubble horizon or particle horizon can not drive the accelerated
expansion of the universe. However, HDE model with event
horizon can drive the accelerated expansion of the universe
[18]. The drawback with event horizon is that it is a global
concept of spacetime and existence of universe, depends on
the future evolution of the universe. The HDE with event
horizon is also not compatible with the age of some old high
redshift objects [27]. Gao et al. [28] proposed IR cut-off as
a function of Ricci scalar. So, the length L is given by the
average radius of Ricci scalar curvature.
As the origin of the HDE is still unknown, Granda and
Oliveros [29] proposed a new IR cut-off for HDE, known
as new holographic dark energy (NHDE), which besides the
square of the Hubble scale also contains the time derivative
of the Hubble scale. The advantage is that this NHDE model
depends on local quantities and avoids the causality
problem appearing with event horizon IR cut-off. The authors, in
their other paper [30], reconstructed the scalar field models
for HDE by using this new IR cut-off in flat Friedmann–
Robertson–Walker (FRW) universe with only DE content.
Karami and Fehri [31] generated the results of Ref. [29] for
non-flat FRW universe. Malekjani et al. [32] have studied
the statefinder diagnostic with new IR cut-off proposed in
[29] in a non-flat model. Sharif and Jawad [33] have
investigated interacting NHDE model in non-flat universe. Debnath
and Chattopadhyay [34] have considered flat FRW model
filled with mixture of dark matter and NHDE, and have
studied the statefinder and Om diagnostics. Wang and Xu [35]
have obtained the constraints on HDE model with new IR
cut-off via the Markov Chain Monte Carlo method with the
combined constraints of current cosmological observations.
Oliveros and Acero [36] have studied NHDE model with a
non-linear interaction between the DE and dark matter (DM)
in flat FRW universe.
A large number of models within modified theories can
explain the DE phenomenon. It is therefore important to find
the ways to discriminate among various competing
models. For this purpose, Sahni et al. [37] and Alam et al. [38]
introduced an important geometrical diagnostic, known as
statefinder pair {r, s} to remove the degeneracy of H0 and
q0 of different DE models. The statefinder diagnostic has
been extensively used in the literature to distinguish among
various models of DE and modified theories of gravity. The
various DE models have different evolutionary trajectories
in (r, s) plane.
In order to complement the statefinder [37,38], a new
diagnostic called Om was proposed by Sahni et al. [39] in 2008,
which is used to distinguish among the energy densities of
various DE models. The advantage of Om over the statefinder
parameters is that, Om involves only the first order derivative
of scale factor. For the C D M model Om diagnostic turns
out to be constant. We provide the mathematical expressions
of statefinder and Om diagnostic in the appropriate section.
Evolution of the universe involves a sequence of
dissipative processes. These processes include bulk viscosity, shear
viscosity and heat transport. The theory of dissipation was
proposed by Eckart [40] and the full causal theory was
developed by Israel and Stewart [41]. In the case of isotropic and
homogeneous model, the dissipative process is modeled as a
bulk viscosity, see Refs. [
42–50
]. Brevik et al. [
51
] discussed
the general account about viscous cosmology for early and
late time universe. Norman and Brevik [
52
] analyze
characteristic properties of two different viscous cosmology models
for the future universe. In other paper, Norman and Brevik
[
53
] derived a general formalism for bulk viscous and
estimated the bulk viscosity in the cosmic fluid. The HDE model
has been studied in some recent literatures [
54,55
] under the
influence of bulk viscosity. Feng and Li [56] have investigated
the viscous Ricci dark energy (RDE) model by assuming that
there is bulk viscosity in the linear barotropic fluid and RDE.
Singh and Kumar [
57
] have discussed the statefinder
diagnosis of the viscous HDE cosmology. The main motive of
this work is to explain the acceleration with the help of bulk
viscosity for new holographic dark energy (NHDE) in GR
which has not been studied sofar.
The bulk viscosity introduces dissipation by only
redefining the effective pressure, pe f f , according as pe f f = p −
3ζ H , where ζ is the bulk viscosity coefficient and H is
the Hubble parameter. In this paper, we are interested when
the universe is dominated by viscous HDE and dark
matter with Granda–Oliveros IR cut-off to study the influence
of bulk viscosity to the cosmic evolution. We consider the
general form of bulk viscosity ζ = ζ0 + ζ1 H , where ζ0
and ζ1 are the constants and H is the Hubble’s
parameter, see Refs. [42,43]. First, we discuss the non-viscous
NHDE model to find out the exact solution of the field
equations. In the second case, we find out the exact solutions of
the field equations with constant and varying bulk viscous
term. We find the exact forms of scale factor, deceleration
parameter and transition redshift and discuss the evolution
through the graphs. We also discuss the geometrical
diagnostics like statefinder parameter and Om diagnostic to
discriminate our model with C D M . We plot the trajectories
of these parameters and observe the effect of bulk viscous
coefficient.
The paper is organised as follows. In Sect. 2, we
discuss the non-viscous HDE model with new IR cut-off.
Section 3 presents the viscous NHDE model and is divided into
Sects. 3.1 and 3.2. In Sects. 3.1 and 3.2 we present the
solutions with constant and time varying bulk viscous term.
Section 4 presents the summary of the results.
2 Non-viscous NHDE model
We consider a spatially homogeneous and isotropic flat
Friedmann–Robertson–Walker (FRW) space-time, specified
by the line element
ds2 = dt 2 − a2(t ) dr 2 + r 2(dθ 2 + si n2θ dφ2) ,
(1)
where a(t ) is the scale factor, t is the cosmic time and (r, θ , φ)
are the comoving coordinates.
We consider that the Universe is filled with NHDE plus
pressureless dark matter (DM) (ignoring the contribution of
the baryonic matter here for simplicity). For Einstein field
equations Rμν − gμν R/2 = Tμν in the units where 8π G =
c = 1, we obtain the Friedmann equations for the metric (1)
as
3H 2 = ρm + ρd ,
2H˙ + 3H 2 = − pd ,
where ρm and ρd are the energy density of DM and NHDE,
respectively, and pd is the pressure of the NHDE. A relation
between ρd and pd is given by equation of state (EoS)
parameter ωd = pd /ρd . Here, H = a˙ /a is the Hubble parameter.
A dot denotes a derivative with respect to the cosmic time t .
As suggested by Granda and Oliveros in paper [29], the
energy density of HDE with the new IR cut-off is given by
ρd = 3(α H 2 + β H˙ ),
where α and β are the dimensionless parameters, which must
satisfy the restrictions imposed by the current observational
data.
Using (4), Eqs. (2) and (3) give
H˙ + (32(1++3βα ωωdd )) H 2 = 0.
The solution of (5) is given by 1
H = c0 + (2+3β ωd )
3(1+α ωd ) t ,
where c0 is an integration constant. Equation (6) can be
rewritten as
H =
H0
1 + 3H(20+(13+βαωωd)d ) (t − t0)
(8)
(9)
where H0 is the present value of the Hubble parameter at t =
t0, where NHDE starts to dominate. As we know H = a˙ /a,
Eq. (7) gives the solution for the scale factor which is given
by
(2+3β ωd )
a = a0 1 + 3(1 + α ωd )H0 (t − t0) 3(1+α ωd )
(2 + 3β ωd )
for α = −1/ωd , β = −2/3ωd ,
,
where a0 is the present value of the scale factor at a cosmic
time t = t0. Equation (8) shows the power-law a ∝ t m ,
where m is a constant, type expansion of the scale factor.
As we know that the universe will undergo with decelerated
expansion for m < 1, i.e., (2 + 3βωd ) < (3 + 3αωd ) in our
case whereas it accelerates for m > 1, i.e., (2 + 3βωd ) >
(3 + 3αωd ). For m = 1, i.e., (2 + 3βωd ) = (3 + 3αωd ),
the universe will show marginal inflation. In the absence of
NHDE, i.e., for α = β = 0, we get the dark matter dominated
scale factor, a = a0(1 + 23 H0(t − t0))2/3.
Let us consider the deceleration parameter (DP) which
is very useful parameter to discuss the behaviour of the
universe. The sign (positive or negative) of DP explains
whether the universe decelerates or accelerates. It is defined
aa¨ . From (8), we get
as q = − a˙2
3(1 + αωd )
q = (2 + 3βωd ) − 1,
which is a constant value throughout the evolution of the
universe. The universe will expand with decelerated rate for
q > 0, i.e., (2 + 3βωd ) < (3 + 3αωd ), accelerated rate for
q < 0, i.e., (2 + 3βωd ) > (3 + 3αωd ) and marginal inflation
for q = 0, i.e., (2 + 3βωd ) = (3 + 3αωd ). One can explicitly
observe the dependence of DP q on the model parameters
α, β and EoS parameter ωd under above constraints. Thus,
we can obtain a decelerated or accelerated expansion of the
universe depending on the suitable choices of these
parameters. In this case, the model does not show the phase
transition due to power-law expansion or constant DP. The model
shows marginal inflation, q = 0 when ωd = 1/3(β − α).
Using Markov chain Monte Carlo method on latest
observational data, Wang and Xu [35] have constrained the NHDE
model and obtained the best fit values of the parameters α =
0.8502−+00..00897854−+00..11026949 and β = 0.4817−+00..00787432−+00..01915756 with 1σ
and 2σ errors in flat model. In the best fit NHDE models, they
have obtained the EoS parameter ωd = −1.1414 ± 0.0608.
Putting these values of parameters (excluding the errors) in
Eq.(9), we get q = −0.7468, which shows that our NHDE
model is consistent with current observation data given in
[35].
In order to discriminate among the various DE models,
Sahni et al. [37] and Alam et al. [38] introduced a new
geometrical diagnostic pair for DE, which is known as statefinder
pair and is denoted as {r, s}. The statefinder probes the
expansion dynamics of the universe through higher derivatives of
the scale factor and is a geometrical diagnosis in the sense
that it depends on the scale factor and hence describes the
spacetime. The statefinder pair is defined as
...
a
r = a H 3 (10)
r − 1
and s = 3(q − 1/2) .
Substituting the required values from (8) and (9) into (10),
we get
(11)
(12)
9(1 + αωd ) 18(1 + αωd )2
r = 1 − (2 + 3βωd ) + (2 + 3βωd )2
,
and
s =
2(1 + αωd ) .
2 + 3βωd
From (11) and (12), we can observe that these statefinder
parameters are constant whose values depend on α, β and
ωd . As Sahni et al. [37] and Alam et al. [38] have observed
that Lambda cold dark matter ( CDM) model and standard
cold dark matter (SCDM) model have fixed point values of
statefinder parameter {r, s} = {1, 0} and {r, s} = {1, 1},
respectively. Putting the values of parameters [35] as
mentioned above, we observe that this set of data do not favor the
NHDE model over the CDM as well as SC D M model.
However, NHDE model behaves like SC D M model for
α = 3β/2. We can also observe that this model approaches
to {r, s} → {1, 0} in the limit of α → −1/ωd but there is
no such value of parameters which would clearly show the
CDM.
3 Viscous NHDE model
In the previous section, we have observed that the
nonviscous NHDE model gives constant DP which is unable
to represent the phase transition. However, the observations
show that the phase transition plays a vital role in describing
the evolution of the universe. Therefore, it will be
interesting to study the NHDE model with viscous to investigate
whether a viscous NHDE model with Granda-Oliveros IR
cut-off would be able to find the phase transition.
In an isotropic and homogeneous FRW universe, the
dissipative effects arise due to the presence of bulk viscosity
in cosmic fluids as shear viscosity plays no role. DE with
bulk viscosity has a peculiar property to cause accelerated
expansion of phantom type in the late time evolution of the
universe [
58–60
]. It can also alleviate the problem like age
problem and coincidence problem.
Let us assume that the effective pressure of NHDE is a sum
of pressure of NHDE and bulk viscosity, i.e., the universe is
filled with bulk viscous NHDE plus pressureless dark matter
(DM) (ignoring the contribution of the baryonic matter here
for simplicity). Then, the field equations (2) and (3) modify
to
3H 2 = ρm + ρd ,
2H˙ + 3H 2 = − p˜d ,
where p˜d = pd − 3H ζ is the effective pressure of NHDE.
This form of effective pressure was originally proposed by
Eckart [40] in the context of relativistic dissipative process
occurring in thermodynamic systems went out of local
thermal equilibrium. The term ζ is the bulk viscosity coefficient
[
61–63
]. On the thermodynamical grounds, ζ is
conventionally chosen to be a positive quantity and generically depends
on the cosmic time t , or redshift z, or the scale factor a, or
the energy density ρd , or a more complicated combination
form. Maartens [
64
] assumed the bulk viscous coefficient as
ζ ∝ ρn, where n is a constant. In the Refs. [42–44], the most
general form of bulk viscosity has been considered with
generalized equation of state. Following [
42–44,65
], we take the
bulk viscosity coefficient in the following form.
ζ = ζ0 + ζ1 H,
where ζ0 and ζ1 are positive constants. The motivation for
considering this bulk viscosity has been discussed in Refs.
[42–44].
From the dynamical equations (13) and (14), we can
formulate a first order differential equation for the Hubble
parameter by using Eqs. (4) and (15) as,
3(1 + αωd ) H 2
H˙ + (2 + 3βωd )
3ζ
− (2 + 3βωd )
It can be observed that Eq. (16) reduces to the non-viscous
equation (5) for ζ = 0 as discussed in previous section.
In the following subsections, we classify different viscous
NHDE models arises due to the constant and variable bulk
viscous coefficient. We analyze the behavior of the scale
factor, DP, statefinder parameter and Om diagnostic of these
different cases.
3.1 NHDE Model with constant bulk viscosity
The simplest case of viscous NHDE model is to be taken
with constant bulk viscous coefficient. Therefore, assuming
ζ1 = 0 in Eq. (15), the bulk viscous coefficient reduces to
ζ = ζ0 = const.
Using (17) into (16), we get
3(1 + αωd ) H 2
H˙ + (2 + 3βωd )
The solution of (18) in terms of cosmic time t can be given
by
H = e (2+33ζβ0ωtd ) c1 + (1 + αωd ) e (2+33ζβ0ωtd )
ζ0
−1
,
where c1 is the constant of integration. From (19), we get the
evolution of the scale factor as
a = c2 c1 +
ζ0
(2+3βωd )
(1 + αωd ) e (2+33ζβ0ωtd ) 3(1+αωd )
,
where c2 is an integration constant. The above scale factor
can be rewritten as
a =
1 +
H0(1 + αωd )
ζ0
for α = −1/ωd , ζ0 = 0
where t0 is the present cosmic time. Here, we get
exponential form of the scale factor which shows non-singular
solution. Equation (21) shows that in early stages of the
evolution, the scale factor can be approximated as a(t ) ∼
(2+3βωd )
1 + 3H(20+(13+βαωωd)d ) (t − t0) 3(1+αωd ) , and as (t − t0) → ∞, the
scale factor approaches to a form like that of the de Sitter
universe, i.e., a(t ) → exp (32ζ+0(3tβ−ωt0d)) . Thus, we observe that
the universe starts with a finite volume followed by an early
decelerated epoch, then making a transition into the
accelerated epoch in the late time of the evolution.
From (21), we can obtain the Hubble parameter in terms
of scale factor a as
H0
H (a) = (1 + αωd )
×
ζ0
H0
ζ0 (3+3αωd )
+ {(1 + αωd ) − H0 } a− (2+3βωd ) ,
(22)
where H0 is the present value of the Hubble parameter and
we have made the assumption that the present value of scale
factor is a0 = 1. The derivative of a˙ with respect to a can be
obtained as [
65
]
da˙ H0 ζ0 ζ0
da = (1 + αwd ) H0 − (1 + αwd ) − H0
× (1 +23+(α3−βwβd)wd ) a− (32(+1+3βαwwdd )) .
aT =
Equation (23) to zero, the transition scale factor aT can be
obtained as
(2+3βwd )
(1 + 3(α − β)wd ) {(1 + αwd )H0 − ζ0} 3(1+αwd )
(2 + 3βwd )ζ0
(23)
The corresponding transition redshift zT , where a = (1 +
z)−1, is
= 0.5
From (24) or (25), we observe that for ζ0 = {1+3(α−β)wd }H0 ,
3
the transition from decelerated phase to accelerated phase
occurs at aT = 1 or zT = 0, which corresponds to the
present time of the universe. On taking the observed values of
α = 0.8502 and β = 0.4817 [35], H0 = 1 and ωd = −0.5 in
this expression of ζ0, we get ζ0 = 0.15. Figure 1 represents
the evolution of the scale factor a(t ) with respect to time
(t − t0) for different values of ζ0 > 0. It is observed that the
transition from decelerated to accelerated phase takes place
in late time for small values of ζ0, i.e., in 0 < ζ0 < 0.15. The
transition from decelerated phase to accelerated one occurs at
aT = 1 for ζ0 = 0.15 which corresponds to the present time
of the universe. However, the transition takes place in early
stages of the evolution for large values of ζ0, i.e., ζ0 > 0.15.
Thus, as the value of ζ0 increases, the scale factor expands
more rapidly with exponential rate.
The result regarding the transition of the universe into the
accelerated epoch discussed above can be further verified by
studying the evolution of DP q. In this case, DP is given by
q =
ζ0
3 (1 + αωd ) − H0
Thus, we find a time-varying DP in the case of constant
viscous NHDE, which describes the phase transition of the
evolution of the universe. DP must change its sign at t = t0, i.e.,
the time at which the viscous NHDE begins to dominate.
This time can be achieved if [1 + 3(α − β)ωd ]H0 = 3ζ0. The
universe must decelerate for t < t0 and accelerate for t > t0
for any parametric values of α, β and ωd .
From (26), DP can be written in terms of scale factor as
(26)
190 Page 6 of 16
.
(31)
and β = 0.4817. On considering α = 0.8502, β = 0.4817
[35] and ωd = −0.5 in Eq. (29), we get ζ0 = 0.15 which
gives q0 = 0. Thus, the transition into accelerating phase
would occur at present time. If ζ0 > 0.15, q0 < 0, i.e., the
universe is in accelerating epoch and it entered this epoch at
an early stage. If ζ0 < 0.15, q0 > 0, i.e., the universe is in
decelerating epoch and it enters in an accelerating phase in
future. Thus, the larger the value ζ0 is, the earlier acceleration
occurs. The similar results for a fixed ζ0 also appear in Fig.
2b. The larger the values α and β is, the earlier q changes it
sign from q > 0 to q < 0 for a fixed ζ0. In both Fig. 2a, b,
we observe that q → −1 in late time of evolution.
Statefinder diagnostic
From above discussion we conclude that there is a transition
from decelerated phase to accelerated one in future for small
bulk viscous coefficient, 0 < ζ0 < 0.15. It takes place to
the present time for ζ = 0.15. However, the transition takes
place in past for ζ > 0.15. The behavior of scale factor and
deceleration parameter shows that the constant bulk viscous
coefficient plays the role of DE. In what follows, we will
present the statefinder diagnostic of the viscous NHDE model
. In this model, the statefinder parameters defined in (10) can
be obtained as
9 Hζ00 − (1 + αωd )
1 − (21++3αβωωdd ) e− (32ζ+0(3tβ−ωt0d))
(2 + 3βωd )
2
+
9
Hζ00 − (1 + αωd )
In terms of red shift z, the above equation becomes
q =
{3(1 + αωd ) − 3ζ0}
(2 + 3βωd )
⎤
3+3α(ω1d + αωd ) ⎥⎥ − 1.
(1 + z)− 2+3βωd − 1 ζ0 + (1 + αωd ) ⎦
× ⎢⎢
⎣
⎡
× ⎢⎢
⎣
⎡
When the bulk viscous parameter and all other parameters are
zero, the deceleration parameter q = 1/2, which corresponds
to a decelerating matter-dominated universe with null bulk
viscosity. However, when only the bulk viscous term ζ0 = 0,
the value of q is same as obtained in Eq. (9) for non-viscous
NHDE model.
The present value of q corresponds to z = 0 or a = 1 is,
This equation shows that if 3ζ0 = [1 + 3(α − β)ωd ], the
deceleration parameter q0 = 0. This implies that the
transition into the accelerating phase would occur at the present
time. The current DP q0 < 0 if 3ζ0 > [1 + 3(α − β)ωd ],
implying that the present universe is in the accelerating epoch
and it entered this epoch at an early stage. But q0 > 0 if
3ζ0 < [1 + 3(α − β)ωd ] implying that the present universe
is decelerating and it will be entering the accelerating phase
at a future time. The evolution of q with a is shown in Fig. 2
by taking fixed constant α and β (or ζ0), from which we can
see that the evolution of the universe is from deceleration to
acceleration. Figure 2a illustrates the evolutionary history of
DP for different value of ζ0 with ωd = −0.5, α = 0.8502
and
s =
2 Hζ00 −(1+αωd ) 1− (21++3αβωωdd) e− (32ζ+0(3tβ−ωt0d)) + 2 Hζ00 −(1+αωd ) 2 e− (62ζ+0(3tβ−ωt0d))
(2+3βωd ) (2+3βωd )2
2 (1+αωd )− Hζ00 e− (2+3βωd) − 1
3ζ0(t−t0)
(2+3βωd )
Here, these values of statefinder parameter are
timedependent and this is due to the bulk viscous coefficient ζ0.
In previous Sect. 2, we see that the diagnostic pair is constant
in the absence of viscous term. As we can observe from the
above two equations that in the limit of (t − t0) → ∞, the
model approaches to {r, s} → {1, 0} and for this limit we
get q → −1. We draw the trajectories of the statefinder pair
{r, s} in r − s plane for different values of constant ζ0 with
ωd = −0.5, H0 = t0 = 1, α = 0.8502 and β = 0.4817 as
shown in Fig. 3a. Here, we observe that the model approaches
to {r, s} → {1, 0} for all positive values of ζ0. In Fig. 3a, the
fixed points {r, s} = {1, 1} and {r, s} = {1, 0} are shown as
SC D M and C D M models, respectively.
It is observed from figures that the statefinder diagnostic
of our model can discriminate from other DE models. For
example, in quiessence with constant EoS parameter [37, 38]
and the Ricci dark energy (RDE) model [
66
], the
trajectory in r − s plane is a vertical segment, i.e. s is constant
during the evolution of the universe whereas the
trajectories for the Chaplygin gas (C G) [
67
] and the quintessence
(inverse power-law) models (Q) [37, 38] are similar to arcs
of a parabola (downward and upward) lying in the regions
s < 0, r > 1 and s > 0, r < 1, respectively. In modified
NHDE model [
68
], the trajectory in r − s is from left to right.
In holographic dark energy model with future event horizon
[
69, 70
] its evolution starts from the point s = 2/3, r = 1
and ends at C D M model fixed point in future.
In Fig. 3a, the plot reveals that the r − s plane can be
divided into two regions r < 1, s > 0 and r > 1, s < 0
which are showing the similar characteristics to Q and C G
models, respectively. The present model starts in both regions
r < 1, s > 0 and r > 1, s < 0, and end on the C D M point
in the r − s plane in far future. The trajectories in the right
side of the vertical line correspond to the different values of
plane for different values of ζ0 > 0 with ωd = −0.5, α = 0.8502 and
β = 0.4817. The arrows represent the directions of the time evolution
pair {r, q} with time. The curves are coinciding with each other for
smaller and larger values of ζ0
ζ0, i.e., ζ0 = 0.02, ζ0 = 0.10, ζ0 = 0.15 and ζ0 = 0.30
lying in the range 0 < ζ0 ≤ 0.57 whereas the trajectories
to the left side of the vertical line correspond to ζ0 > 0.57,
i.e., ζ0 = 0.60, ζ0 = 0.70, ζ0 = 0.80 and ζ0 = 1.00. This
reveals that smaller values of ζ0 give the model similar to
Q model and larger values correspond to the C G model.
We find that the evolutions are coinciding each other for all
different values of ζ0 in both regions.
We also study the evolutionary behaviour of constant
viscous NHDE model in r − q plane. For different values of
ζ0, as taken in {r, s}, the trajectories are shown in Fig. 3b for
wd = −0.5, H0 = t0 = 1, α = 0.8502 and β = 0.4817.
The SC D M model and steady state (S S) model corresponds
to fixed point {r, q} = {1, 0.5} and {r, q} = {1, −1},
respectively. It can be seen that there is a sign change of q from
positive to negative which explain the recent phase
transition. The trajectories show that viscous NHDE models
commence evolving from different points for different values of
ζ0 with respect to C D M which starts from SC D M fixed
point. The viscous NHDE model always converges to S S
model as C D M , Q and C G models in late-time evolution
of the universe. Thus, the constant viscous NHDE model is
compatible with Q and C G models.
The above discussion concludes the effect of viscous term
in NHDE model. Let us discuss the model in view point of
model parameters α and β. Figure 4a, b show the trajectories
in r − s and r − q planes, respectively, for the different values
of α and β with constant ωd = −0.5, H0 = t0 = 1 and
ζ0 = 0.02. The arrows in the diagram denote the evolution
directions of the statefinder trajectories and r −q trajectories.
From Fig. 4a, we observe that for this fixed value of ζ0 the
constant viscous NHDE model always correspond to the Q
Fig. 4 a The r − s trajectories
are plotted in r − s plane for
different values of α and β with
ωd = −0.5 and ζ0 = 0.02. The
arrows represent the directions
of the evolutions of statefinder
diagnostic pair with time. b The
r − q trajectories are plotted in
r − q plane for different values
of α and β with ωd = −0.5 and
ζ0 = 0.02. The arrows represent
the directions of the time
evolution pair {r, q} with time
model. It may start from the vicinity of SC D M model in
early time of evolution for some values of α and β, e.g.,
(α, β) = (0.8502, 0.55). In late-time of evolution the model
always converges to C D M model for any values of (α, β).
The panel (b) of Fig. 4 shows the time evolution of the
r − q trajectories in r − q plane. The horizontal line at
r = 1 corresponds to the time evolution of the C D M
model. The signature change from positive to negative in q
clearly explain the phase transition of the universe. The
constant viscous NHDE model may start from the vicinity of
the SC D M model ({r, q} = {1, 0.5}) for some values of α
and β (e.g., α = 0.8502, β = 0.55). However, the constant
viscous NHDE model approaches to the S S model as the
C D M and Q models in future . Thus, the viscous NHDE
model is compatible with the C D M and Q models with
variables model parameters and constant value of ζ0.
Thus, we conclude that our model corresponds to both Q
and C G models for the different values of viscous coefficient
ζ0 whereas for the different values of model parameters α
and β with respect to the fixed value of ζ0, our model only
corresponds to Q model. Hence, we can conclude that due
to the viscosity NHDE model is compatible with the Q and
C G models. By above analysis, we can say that the bulk
viscous coefficient and model parameters play the important
roles in the evolution of the universe, i.e., they both determine
the evolutionary behavior as well as the ultimate fate of the
universe.
Om diagnostic
In addition to statefinder {r, s}, another diagnostic model,
Om(z) is widely used to discriminate DE models. It is a new
geometrical diagnostic which combines Hubble parameter
H and redshift z. The Om(z) diagnostic [39] has been
proposed to differentiate C D M to other DE models. Many
authors [
71–73
] have studied the DE models based on Om(z)
diagnostic. Its constant behaviour with respect to z represents
that DE is a cosmological constant ( C D M ). The positive
slope of Om(z) with respect to z shows that the DE behaves
as phantom and negative slope shows that DE behaves like
quintessence. According to Ref. [39], Om(z) parameter for
spatially flat universe is defined as
H2(z)
H2 − 1
Om(z) = (1 +0z)3 − 1 ,
where H0 is the present value of the Hubble parameter. Since
the Om(z) involves only the first derivative of scale factor, so it
is easier to reconstruct it as compare to statefinder parameters.
It has been shown that the slope of Om(z) can distinguish
dynamical dark energy from the cosmological constant in a
robust manner, both with and without reference to the value
of the matter density.
On substituting the required value of H (z) from (22) in
(32), we get the value of Om(z) as
3(1+αωd ) 2
Hζ00 + 1 + αωd − Hζ00 (1 + z) 2+3βωd
− (1 + αωd )2
(1 + αωd )2[(1 + z)3 − 1]
(32)
.
(33)
For comparison, we plot Om(z) trajectory with respect to
z for different values of ζ0 > 0 (or α and β) with fixed α
and β (or ζ0), with H0 = 1 and ωd = −0.5 as shown in
Fig. 5. From Fig. 5a, we observe that for 0 < ζ0 ≤ 0.57,
the trajectory shows the negative slope, i.e., the DE behaves
like quintessence and for ζ0 > 0.57, the positive slope of the
Om trajectory is observed, i.e., the DE behaves as phantom.
For the late future stage of evolution when z = −1, we
ζ02
get Om(z) = 1 − H02(1+αωd )2 , which is the constant value of
Om(z). Thus for z = −1, the DE will correspond to C D M .
The Fig. 5b shows the Om(z) trajectory for different values
of model parameters α and β with fixed ζ0 = 0.02, ωd =
−0.5 and H0 = 1. This trajectory only shows the negative
curvature which imply that the DE behaves like quintessence.
Fig. 5 a The Om(z)
evolutionary diagram of viscous
NHDE for different values of
ζ0 > 0 with fixed ωd = −0.5,
α = 0.8502 and β = 0.4817. b
The Om(z) evolutionary diagram
of viscous NHDE for different
values of α and β with fixed
ζ0 = 0.02 with ωd = −0.5
From the above discussion with constant bulk viscous
coefficient, we find that the constant ζ0 ( or cosmological
parameters α and β ) play important roles in the evolution
of the universe, i.e., they both determine the evolutionary
behavior as well as the ultimate fate of the universe.
3.2 Solution for variable bulk viscosity coefficient
In this section, we consider two cases: (i) ζ0 = 0 and ζ1 = 0,
and (ii) ζ0 = 0 and ζ1 = 0.
Case (i) ζ0 = 0 and ζ1 = 0:
In this case, the bulk viscosity coefficient given in (15)
reduces to
which is a constant value. Such form of ζ gives no transition
phase. The positive or negative sign of q depends on whether
a →
The above equation is similar to the Eq.(5) obtained in the
case of non-viscous NHDE model in Sect. 2. The solution of
(35) for H in terms of t is given by
1
H = c3 + 3((12−+ζ31β+ωαdω)d ) t ,
where c3 represents the constant of integration. The scale
factor can be obtained as
(2+3βωd )
a = a0 1 + 3(1 −(2 ζ+1 +3βαωωdd))H0 (t − t0) 3(1−ζ1+αωd )
,
2
for ζ1 = (1 + αωd ), β = − 3ωd
The scale factor varies as power-law expansion. Now, the DP
is
q =
3(1 − ζ1 + αωd )
(2 + 3βωd )
3ζ1 < (1 + 3(α − β)ωd ) or 3ζ1 > (1 + 3(α − β)ωd ),
respectively.
Now, the statefinder parameter can be given as
9(1 + αωd − ζ1)
(2 + 3βωd )
+
In this case the statefinder pair is constant. In the limit of
ζ1 → ((21α−+3βα)ωωdd,),ththisemstoadteefil nbdeehravpeasira{sr,SsC}D→M,{1i.,e0.,}{arn,ds}fo=r
ζ1 = 2
{1, 1}.
Case (ii) ζ0 = 0 and ζ1 = 0:
Let us consider the more general form of the bulk viscous
coefficient, i.e., ζ = ζ0 + ζ1 H . Using (15) into (16), we get
H˙ +
3(1 − ζ1 + αωd ) H 2
(2 + 3βωd )
(38)
where H0 is the present value of the Hubble parameter and
we have made the assumption that the present value of scale
factor is a0 = 1. The solution of (42) for the scale factor a
in terms of t is given by
a =
1 +
H0(1 − ζ1 + αωd )
ζ0
for ζ0 = 0, ζ1 = (1 + αωd )
Here, we get an exponential type scale factor with the viscous
terms. As (t − t0) → 0, the scale factor behaves as
(2+3βωd )
3H0(1 − ζ1 + αωd )(t − t0) 3(1−ζ1+αωd )
(2+3βωd )
3(1−ζ1+αωd )
(39)
(40)
(42)
(43)
(44)
which shows power-law expansion in early time. On the other
hand, if ζ0 = H0(1 − ζ1 + αωd ) or (t − t0) → ∞, we obtain
a(t ) = exp
3ζ0(t − t0)
(2 + 3βωd )
This case corresponds the de Sitter universe which shows
accelerated expansion in the later time of evolution.
Now, with the help of (43), the Hubble parameter in terms
of scale factor can be written as
This equations shows that if both ζ0 and ζ1 are zero, the
−3(1+αωd )
Hubble parameter, H = H0a (2+3αωd ) , which corresponds to
non-viscous NHDE model. When ζ1 = 0, H reduces to Eq.
(22) which is the case of constant viscosity.
The derivative of a˙ with respect to a can be obtained from
(46), which is given by
da˙ H0
da = (1 − ζ1 + αwd )
(1 + 3(α − β)wd − 3ζ1)
× 2 + 3βwd
ζ0 ζ0
H0 − (1 − ζ1 + αwd ) − H0
3(1−ζ1+αwd )
a− (2+3βwd )
. (47)
Equating (47) to zero to get the transition scale factor aT as
(2+3βwd)
(1 + 3(α − β)wd − 3ζ1) {(1 − ζ1 + αwd )H0 − ζ0} 3(1−ζ1+αwd) .
(2 + 3βwd )ζ0
The corresponding transition redshift zT is
(1 + (α − β)wd − 3ζ1) {(1 − ζ1 + αwd )H0 − ζ0} − 3(1(2−+ζ31β+wαdw)d)
(2 + 3βwd )ζ0
It can be observed that for (ζ0 + ζ1 H0) = {1+3(α−β)wd }H0 ,
3
the transition from decelerated phase to accelerated phase
occurs at aT = 1 or zT = 0, which corresponds to the
present time of the universe. By considering the
observational value α = 0.8502 and β = 0.4817 along with
ωd = −0.5, H0 = 1, we get (ζ0 + ζ1) = 0.15. A plot of
the evolution of the scale factor is given in Fig. 6. Thus, for
0 < (ζ0 + ζ1) ≤ 0.15 the scale factor has earlier deceleration
phase followed by an acceleration phase in later stage of the
evolution. The transition from the decelerated to accelerated
expansion depends on the viscosity ζ0 and ζ1 as shown above.
For (ζ0 + ζ1) > 0.15, the transition takes place in past of the
universe and the scale factor increases with accelerated rate
forever.
aT =
zT =
.
(46)
(48)
× ⎢⎣⎢⎢
⎡
Fig. 6 The plot of the scale factor with respect to t − t0 for different
values of ζ0, ζ1 when ζ0 > 0 and ζ1 > 0 with ωd = −0.5, α = 0.8502
and β = 0.4817
The transition may also be discussed through the evolution
of DP. In this case, we get
q =
which is a time-dependent value of DP, which may describe
the transition phase of the universe. It can be observed that
DP must change its sign at t = t0. This time can be achieved
if 3(ζ0 + ζ1 H0) = {1 + 3(α − β)wd }H0. The sign of q is
positive for t < t0 and it is negative for t > t0. The values of
ζ0 and ζ1 can be obtained for a given values of ωd , α and β,
which may be obtained from observation, or vice-versa.
From (50), DP can be written in terms of scale factor as
(50)
q(a) =
{3(1 − ζ1 + αωd ) − 3ζ0}
(2 + 3βωd )
(1 − ζ1 + αωd )
3(1−ζ1+αωd )
(a 2+3βωd
− 1)ζ0 + (1 − ζ1 + αωd )
In terms of red shift z, the above equation becomes
q(z) =
{3(1 − ζ1 + αωd ) − 3ζ0}
(2 + 3βωd )
(1+z)−
(1−ζ1 + αωd )
3(1−ζ1+αωd )
2+3βωd
⎥⎥⎥ −1.
− 1 ζ0 + (1 − ζ1 + αωd ) ⎦
⎤
⎦ − 1.
(51)
⎤
(52)
When the bulk viscous parameter and all other parameter are
zero, the deceleration parameter q = 1/2, which corresponds
to a decelerating matter dominated universe with null bulk
viscosity. However, when only the bulk viscous term ζ0 = 0
and ζ1 = 0, the value of q is same as obtained in Eq. (38) for
case (i) of variable viscous NHDE model, and when ζ0 = 0
and ζ1 = 0 , Eq. (51) reduces to Eq. (27) of constant viscous
coefficient.
The present value of q corresponds to z = 0 or a = 1 is,
This equation shows that if 3(ζ0 + ζ1) = [1 + 3(α − β)ωd ],
the deceleration parameter q = 0. This implies that the
transition into the accelerating phase would occur at the present
time. The current DP q0 < 0 if 3(ζ0 +ζ1) > [1+3(α−β)ωd ],
implying that the present universe is in the accelerating epoch
and it entered this epoch at an early stage. But q0 > 0 if
3(ζ0 + ζ1) < [1 + 3(α − β)ωd ] implying that the present
universe is decelerating and it will be entering the
accelerating phase at a future time. For the observational value
α = 0.8502 and β = 0.4817 with ωd = −0.5 and H0 = 1,
we get (ζ0 + ζ1) = 0.15 which gives q0 = 0. Thus for this
s =
2 Hζ00 −(1−ζ1+αωd ) 1− 1(−2+ζ13+βωαωdd) e− (32ζ+0(3tβ−ωt0d)) + 2 Hζ00 −(1−ζ1+αωd )
(2+3βωd ) (2+3βωd )2
2 (1−ζ1+αωd )− Hζ00 e− (2+3βωd ) − 1
3ζ0(t−t0)
(2+3βωd )
As we have mentioned above , the scale factor and
deceleration parameter have been discussed to explain the
accelerating universe with viscous term or model parameters. So
it is necessary to distinguished these models in a
modelindependent manner. In what follows we will apply two
geometrical approaches to viscous NHDE model, i.e., the
statefinder and Om diagnostic from which we can compute
the evolutionary trajectories with ones of the C D M model
to show the difference among them.
In this case, the statefinder parameters defined in Eq. (10)
can be evaluated as
,
value set, the transition into accelerating phase would occur
at present time. If (ζ0 + ζ1) > 0.15, q0 < 0, i.e., the universe
is in accelerating phase and it entered this epoch at an early
stage. If (ζ0 + ζ1) < 0.15, q0 > 0, i.e., the universe is in
decelerating epoch and it will be entered into the accelerated
phase in future. This result is verified graphically which is
represented by Fig. 7a. The Fig. 7b shows the q − a graph
in q − a plane to discuss the evolution of the universe with
respect to model parameters α and β. Here, the signature
change in the value of DP can be seen by the figure. From
above discussion we say that both viscous coefficient and
model parameter have their own role in the evolution of the
universe. Some values of bulk viscous term gives the
accelerated phase from the beginning and continues to be accelerated
in late time.
From (54) and (55) it can be observed that the viscous
NHDE model converges to {r, s} → {1, 0} in the limit of
(t − t0) → ∞. This can also be achieved at (ζ0 + H0ζ1) =
H0(1+αωd ) but this is a very fixed point. Thus, the statefinder
diagnostic fails to discriminate between C D M and the
NHDE model. Here, we obtain time-dependent statefinder
pair which needs to study the general behavior. Let us see
the effect of viscosity coefficients ζ0, ζ1 and model
parameters α, β for the general form of variable viscous NHDE
model. Figure 8a shows the r − s trajectory in r − s plane for
different values of ζ0 and ζ1 with H0 = t0 = 1, α = 0.8502
and β = 0.4817. The model behaviors to Q models for
0 < (ζ0 + ζ1) ≤ 0.57 and C G models for (ζ0 + ζ1) > 0.57.
The trajectories in Q-model and CG-model both converge to
the C D M model in late time of evolution.
Fig. 7 a The q − a graph in
q − a plane for different values
of ζ0 > 0 and ζ1 > 0 with
ωd = −0.5, α = 0.8502 and
β = 0.4817. b The q − a graph
in q − a plane for different
values of α and β with ζ0 = 0.2,
ζ1 = 0.3 and ωd = −0.5
(a)
(b)
Fig. 8 a The r − s trajectories
are plotted in r − s plane for
different values of ζ0 > 0 and
ζ1 > 0 with ωd = −0.5,
α = 0.8502 and β = 0.4817.
The arrows represent the
directions of the evolutions of
statefinder diagnostic pair with
time. b The r − q trajectories
are plotted in r − q plane for
different values of ζ0 > 0 and
ζ1 > 0 with ωd = −0.5,
α = 0.8502 and β = 0.4817.
The arrows represent the
directions of the time evolution
pair {r, q} with time
Figure 8b shows the time evolution of {r, q} pair in r − q
plane for different combinations of the values of ζ0 and ζ1
with H0 = t0 = 1, α = 0.8502 and β = 0.4817. The fixed
points {r, q} = {1, 0.5} and {r, q} = {1, −1} represents the
SC D M and S S models, respectively. Since q changes its
sign from positive to negative with respect to time which
shows the phase transition of the universe from deceleration
to acceleration. In beginning this model behaves different
from the C D M but in future it behaves same as C D M
which converges to S S model in late-time. Hence the
variable viscous NHDE model always converges to S S model
as C D M , Q and C G models in late-time evolution of the
universe. For all the ranges of (ζ0 + ζ1) the trajectories
correspond to Q and C G models as in Fig. 8a. Thus the variable
viscous NHDE model is compatible with both Q and C G
models.
Thus, the viscosity coefficients are able to correspond to
both Q and C G models for different combinations of ζ0, ζ1
and also explain the phase transition of the universe.
Now, we are curious to know the behaviour of variable
viscous NHDE model with respect to the model parameters
α and β. Here, Fig. 9a, b represents the r − s and r − q
trajectories in r − s and r − q plane, respectively, for the
different values of α and β close to it’s observational value
with ωd = −0.5, H0 = t0 = 1, ζ0 = 0.02 and ζ1 = 0.03.
The evolutionary directions of both the trajectories are shown
in the figures by the arrows. In Fig. 9a, we analysed that for
this fixed value of ζ0 and ζ1 the (r, s) trajectories are lying in
the region corresponds to r < 1, s > 0 which shows that our
model is similar to the Q model. It also starts from the vicinity
of SC D M model in early time of evolution for some values
of α and β, e.g., (α, β) = (0.8502, 0.59). It is different from
RDE model and quiessence model as it produces the curved
trajectories for any values of (α, β) close to observational
value which approach to C D M in late-time of evolution as
the Q model tends to C D M model in late-time of evolution.
The r − q trajectories in r − q plane are shown by the
Fig. 9b. This model is also able to explain the phase
transition of the universe. It also starts from the neighbourhood
of the SC D M model for some values of α and β (e.g.,
α = 0.8502, β = 0.55) and approaches to S S model in
latetime for any value of α and β close to the observational value.
In future the variable viscous NHDE model approaches to the
S S model same as the C D M and Q models. Thus the
viscous NHDE model is compatible with the C D M and Q
models.
Thus, we observed from Figs. 8 and 9 that viscous NHDE
model is compatible to Q and C G models for different ranges
of viscosity coefficients in the presence of the fixed
observational value of model parameters whereas the model
parameter in the presence of fixed value of viscosity coefficients
approaches only to Q model.
Om diagnostic
Let us discuss the another geometrical parameter, i.e., Om(z)
diagnostic in viscous NHDE model. By substituting the
required values in Eq. (32), we get the Om(z) diagnostic for
ζ = ζ0 + ζ1 H as
3(1−ζ1+αωd ) 2
Hζ00 + (1 − ζ1 + αωd ) − H0
ζ0 (1 + z) 2+3βωd
− (1 − ζ1 + αωd )2
(1 − ζ1 + αωd )2[(1 + z)3 − 1]
Fig. 9 a The r − s trajectories
are plotted in r − s plane for
different values of α and β with
ωd = −0.5, ζ0 = 0.02 and
ζ1 = 0.03. The arrows represent
the directions of the evolutions
of statefinder diagnostic pair
with time. b The r − q
trajectories are plotted in r − q
plane for different values of α
and β with ωd = −0.5,
ζ0 = 0.02 and ζ1 = 0.03. The
arrows represent the directions
of the time evolution pair {r, q}
with time
Fig. 10 a The evolution of
Om(z) versus the redshift z for
different values of ζ0 > 0 and
ζ1 > 0 with ωd = −0.5,
α = 0.8502 and β = 0.4817. b
The evolution of Om(z) versus
the redshift z for different values
of α and β with ζ0 = 0.02,
ζ1 = 0.03 and ωd = −0.5
Figure 10a shows the Om(z) trajectory with respect to z
for different values of ζ0 > 0 and ζ1 > 0 corresponding to
α = 0.8502, β = 0.4817, H0 = 1 and ωd = −0.5. Here, the
trajectory represents the negative curvature, i.e., the viscous
NHDE behaves as quintessence for the limit 0 < (ζ0 + ζ1) ≤
0.57 and it shows the positive curvature, i.e., the viscous DE
behaves as phantom, for (ζ0 + ζ1) > 0.57 whereas for z =
−1, i.e., in future time we get Om(z) = 1 − H02(1−ζζ102+αωd )2 ,
which is the constant value of Om(z). Thus for z = −1, the
viscous NHDE will correspond to C D M .
Figure 10b plot the Om(z) versus z for different model
parameters α and β correspond to fixed ζ0 and ζ1. The graph
shows that there is always negative curvature for any values
of model parameters. This shows that the model behaviors
similar to quintessence model.
4 Conclusion
We have studied some viscous cosmological NHDE models
on the evolution of the universe, where the IR cutoff is given
by the modified Ricci scalar, proposed by Granda and
Olivers [29, 30]. It has been tried to demonstrate that the bulk
viscosity can also play the role as a possible candidate of
DE. We have performed a detailed study of both non-viscous
and viscous NHDE models. The component of this model is
DE and pressureless DM. We have obtained the solutions for
scale factor and deceleration parameter. We have also
studied these models from two independent geometrical point of
view, namely the statefinder parameter and Om diagnostic.
We have studied the different possible scenarios of viscous
NHDE and analyzed the evolution of the universe according
to the assumption of bulk viscous coefficient ζ . In the
following we summarize the results obtained in different sections
for non-viscous and viscous NHDE models.
In Sect. 2, we have investigated non-viscous NHDE in flat
FRW universe. We have calculated the relevant
cosmological parameters and their evolution. The evolution of scale
factor has been studied. We have obtained power-law form
of scale factor for which the model may decelerate or
accelerate depending on the constraint of model parameters. The
deceleration parameter is constant in this case. Therefore,
the model can not describe the transition phase of the
universe. The statefinder parameters are also constant. We have
observed that the observed set of data of model parameters
do not favor the NHDE model over the CDM as well as
SC D M model. However, NHDE model behaves like SC D M
model for α → 3β/2. It has been observed that this model
approaches to {r, s} → {1, 0} in the limit of α → −1/ωd
but there is no such value of parameters which would clearly
show the CDM.
In viscous NHDE model as discussed in Sect. 3, we have
considered that the matter consists of viscous holographic
dark energy and pressureless DM. We have assumed a most
general form ζ = ζ0 + ζ1 H to observe the effect of bulk
viscous coefficient in the evolution of the universe during early
and late time. We have studied three cases: (ζ0 = 0, ζ1 = 0);
(ζ0 = 0, ζ1 = 0) and (ζ0 = 0, ζ1 = 0).
In the first case where we have constant bulk viscous
coefficient, i.e., ζ = ζ0, an exponential form of the scale factor
is obtained. Therefore, the universe starts from a finite
volume followed by an early decelerated phase and then
transits into an accelerated phase in late time of evolution. As
(t − t0) → 0, the scale factor reduces to the power-law
form which corresponds to an early decelerated expansion.
As (t − t0) → ∞, the scale factor tends to the
exponential form which corresponds to acceleration similar to the de
Sitter phase. The scale factor and red shift corresponding to
the transition from decelerated to accelerated expansion has
been obtained. The evolution of scale factor has been shown
is Fig. 1. It is clear from Fig. 1 that if ζ0 = 0.15,
transition from decelerated phase to accelerated phase occurs at
aT = 1 and zT = 0, which corresponds to the present time
of the universe. The transition would takes place in past if
ζ0 > 0.15 and in future if 0 < ζ0 < 0.15.
The result regarding the transition of the universe into
accelerated epoch discussed above have been further
verified by studying the evolution of DP. The viscous NHDE
model gives time-dependent DP which would describe the
phase transition. We have obtained q in terms of a and z. The
variation of q with a has been shown in Fig. 2a, b with
varying ζ0 and constant model parameters, and varying model
parameters and constant ζ0, respectively. The evolution of
DP shows that the transition from decelerated to accelerated
epoch occurs at the present time, corresponding to q = 0
if ζ0 = 0.15. The transition would be in recent past,
corresponds to q < 0 at present ζ0 > 0.15 and the transition
into accelerating epoch will be in the future, corresponds to
q > 0 at present if 0 < ζ0 < 0.15.
As the model predicts the late time acceleration, we have
analyzed the model using statefinder parameter and Om
diagnostic to distinguished it from other DE models especially
from C D M model. The evolution of the viscous NHDE
model in the {r, s} plane is shown in Fig. 3a with different
values of ζ0 with constant α and β. It shows that the evolution
of {r, s} parameter is in such a way that r < 1, s > 0, a feature
of quintessence model where as r > 1, s < 0 corresponds
to the Chaplygin gas model. In both models, the
trajectories are coinciding with each other for any value of ζ0. The
viscous NHDE model behaving quintessence and Chaplygin
gas models in early time for different ζ0 untimely approaches
to C D M model in late time. We have also discussed the
evolutionary behavior of {r, q} to discriminate the viscous
NHDE model. The trajectory of {r, q} has been plotted in
Fig. 3b which shows the phase transition from decelerated
to accelerated phase. If 0 < ζ0 ≤ 0.57, the transition takes
place from quintessence region and approaches to S S model
in late time as C D M model approaches from SCDM.
However, if ζ0 > 0.57, the transition starts from Chaplygin gas
model and approaches to S S model in late time. Both the
trajectories in Q model and C G model are coinciding on each
other for any value of ζ0.
A study of Om diagnostic of viscous NHDE model has
been carried out in Fig. 5a for different values of ζ0 and
fixed α and β. The trajectory shows that if 0 < ζ0 ≤ 0.57,
the Om(z) trajectory shows the negative slope which means
viscous NHDE behaves like quintessence and if ζ0 > 0.57,
the positive slope of the Om(z) trajectory is observed, i.e., the
viscous NHDE behaves like phantom. In future as z → −1,
the Om(z) becomes constant, i.e, it may approach to C D M
model.
The above discussion shows that effect of bulk viscous
coefficient on NHDE model with different values of ζ0. We
have also discussed the viscous NHDE model with varying
model parameters α and β taking fixed ζ0. The trajectory for
q versus a as shown in Fig. 2b shows that the transition takes
place from decelerated to accelerated phase in future for any
values of α and β and approaches to q = −1 in late-time.
The trajectory for {r, s} and {r, q} have also been plotted
respectively in Fig. 4a, b for different values of α and β with
fixed value of ζ0. The {r, s} trajectory as shown in Fig. 4a
shows that the trajectory starts from the quintessence region,
even though some starts from the vicinity of SC D M and
approaches to C D M in late-time. The signature change of
q from positive to negative has been observed in r − q plane
as shown in Fig. 4b. The viscous NHDE model approaches to
S S model in late-time as C D M does. The Om trajectory has
been plotted in Fig. 5b for different values of α and β for fixed
ζ0. This trajectory only shows the negative curvature which
imply that the viscous NHDE behaves like quintessence.
From the above discussion with constant bulk viscous
coefficient, we find that the constant ζ0 (or cosmological
parameters α and β) play important roles in the evolution of
the universe i.e., they both determine the evolutionary
behavior as well as the ultimate fate of the universe.
In second viscous NHDE model we have assumed ζ =
ζ1 H . The solution of this model is similar to the non-viscous
NHDE one. We have obtained power-law form of scale factor
which gives constant values of DP and statefinder pairs.
In last case we have taken the most general form of bulk
viscous coefficient ζ = ζ0 + ζ1 H . The solution of this model
is similar to the constant bulk viscous coefficient ζ0. The
effect of both non-zero values of ζ0 and ζ1 have been
discussed. We have obtained exponential scale factor which
gives time-dependent DP and statefinder pairs. The transition
from decelerated to accelerated epoch has been discussed.
As (t − t0) → 0, the scale factor asymptotically gives the
power-law which shows that the model decelerates in early
time and accelerates in late-time. As (t −t0) → ∞, the model
corresponds to de Sitter like. The transition scale factor and
hence corresponding transition redshift has been calculated.
If (ζ0 + ζ1) = 0.15, the transition from decelerated to
accelerated phase occurs at aT = 1 or zT = 0, i.e., at present time
of the universe. Figure 6 shows that evolution of the scale
factor with (t − t0). If 0 < (ζ0 + ζ1) < 0.15, the scale factor
has earlier deceleration phase followed by an acceleration
phase in later stage of the evolution. For (ζ0 + ζ1) > 0.15,
the transition takes place in past of the universe and the scale
factor increases with accelerated rate forever.
The DP is time-dependent which shows phase transition
from decelerated to accelerated phase. The DP has been
written in terms of scale factor or redshift. We have calculated
the present value q0 which gives ζ0 + ζ1 = 0.15. This shows
that the transition into acceleration phase occurs at present
time. If (ζ0 + ζ1) > 0.15, q0 < 0, i.e., the universe is in
accelerating phase and it entered this epoch at an early stage.
If (ζ0 + ζ1) < 0.15, q0 > 0, i.e., the universe is in
decelerating epoch and it will be entered into the accelerated phase
in future. We have plotted q versus a for different values of
(ζ0, ζ1) with fixed model parameters and others as shown
in Fig. 7a. The Fig. 7b plots the q − a for different models
parameters α, β with fixed ζ0, ζ1 and others.
Figure 8a shows the r − s trajectory in r − s plane for
different values of ζ0 and ζ1 with constant model
parameters and others. The model behaviors to Q models for
0 < (ζ0 + ζ1) ≤ 0.57 and C G models for (ζ0 + ζ1) > 0.57.
The trajectories in Q-model and C G model both converge
to the C D M model in late time of evolution. Figure 8b
plots the trajectory of q − r for different values of (ζ0, ζ1)
with constant model parameters and others. The DP changes
its sign from positive to negative with respect to time which
shows the phase transition of the universe from deceleration
to acceleration. In beginning this model behaves different
from the C D M but in future it behaves same as C D M
which converges to S S model in late-time. Thus, the variable
viscous NHDE model is compatible with both Q and C G
model. Figure 9a, b plot the trajectories of r − s and r − q
for different model parameters (α, β) with fixed ζ0, ζ1 and
others. In Fig. 9a, we have analysed that for this fixed value
of ζ0 and ζ1 the {r, s} trajectories are lying in the region
corresponds to r < 1, s > 0 which shows that our model is
similar to the Q model. Figure 9b shows that this model is
also able to explain the phase transition of the universe. It
also starts from the neighbourhood of the SC D M model for
some values of α and β. In future the variable viscous NHDE
model approaches to the S S model same as the C D M and
Q models. Thus the viscous NHDE model is compatible with
the C D M and Q models.
We conclude that the trajectory of s − r and q − r suggest
a different behavior as compare to Ricci dark energy done
by Feng [16] where it was found that the s − r trajectory is
a vertical segment, i.e., s is constant during the evolution of
the universe. The trajectory in our viscous NHDE model is
mostly confined a parabolic curve and approaches to {r, s} =
{1, 0} in s − r plane and {r, q} = {1, −1} in q − r plane.
From Om diagnostic we find that the trajectory
represents the negative curvature, i.e, viscous NHDE behaves as
a quintessence for 0 < (ζ0 + ζ1) ≤ 0.57 and it shows the
positive curvature, i.e., the viscous DE behaves as phantom,
for (ζ0 + ζ1) > 0.57, which is graphically represented by
Fig. 10a. We have also concluded that as z → −1, we get
the constant value of Om, which corresponds to C D M
model. However, plot of Om as shown in Fig. 10b for
different model parameters with constant ζo and ζ1 reveal that
there is always negative curvature for any values of model
parameters. This shows that the viscous NHDE behaviors
similar to quintessence.
In concluding remarks, let us compare our work with
respect to the earlier studied in this direction. Feng and Li [
56
]
who investigated the viscous Ricci DE model by assuming
bulk viscous coefficient proportional to the velocity vector
of the fluid. Chattopadhyay [
55
] reported a study on
modified Chaplygin gas based reconstructed scheme for extended
HDE in the presence of bulk viscosity. In comparison to the
said work, the present work lies not only in its choice of
different bulk viscous coefficient but also in its different approach
to discuss the evolution of the universe. The present viscous
NHDE model successfully describes the present accelerated
epoch. The C D M model is attainable by present model.
The NHDE model behaves quintessence model and
Chaplygin gas model inn early time due to viscous effect.
However, it behaves only quintessence if we consider the model
parameters with fixed viscous coefficient. Our work implies
the theoretical basis for future observations to constraint the
viscous NHDE.
Acknowledgements The authors express their sincere thank to the
referee for his suggestions to improve the manuscript. One of the author
MS sincerely acknowledges the University Grant Commission, which
provided the Senior Research Fellowship under UGC-NET Scholarship.
Open Access This article is distributed under the terms of the Creative
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