# Bulk Viscous Cosmological Model in Brans-Dicke Theory with New Form of Time Varying Deceleration Parameter

Authors: G. P. Singh; Binaya K. Bishi
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

We have presented FRW cosmological model in the framework of Brans-Dicke theory. This paper deals with a new proposed form of deceleration parameter and cosmological constantΛ. The effect of bulk viscosity is also studied in the presence of modified Chaplygin gas equation of state (p=Aρ-B/ρn). Furthermore, we have discussed the physical behaviours of the models.


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## Body

## 1. Introduction

It has been well established that alternative theories of gravitation played an important role in understanding the models of the Universe. For the last few decades, researchers have shown more interest in alternative theories of gravitation especially scalar-tensor theories of gravity. The Brans-Dicke theory (BDT) of gravity is the one of the most successful alternative theories among all alternative theories of gravitation. This theory is consisting of a massless scalar fieldϕ and a dimensionless constant ω describing the strength of the coupling between ϕ and the matter [1]. In the BDT, gravitational constant G is treated as the reciprocal of a massless scalar field ϕ, where ϕ is expected to satisfy scalar wave equations and its source is all matter in the Universe.In a pioneering work, both research contributions by Mathiazhagan & Johri [2] and later La & Steinhardt [3] showed that the idea of inflationary expansion with a first-order phase transition can be made to work more satisfactorily if one considers the BDT in place of general relativity. The interesting consequence of BD scalar field is that the modified field equations would express the scale factor Rt as a power function of time and not as an exponential function, so that one attains the so-called “graceful exit” from the inflationary vacuum phase through a first-order phase transition. Hyperextend inflation [4] generalizes the results of extended inflation in BDT and solves the graceful exit problem in a natural way, without recourse to any fine-tuning as required in relativistic models. Romero & Barros [5] discussed the limit of the Brans-Dicke theory of gravity when ω→∞ and showed by examples that, in this limit, it is not always true that BDT reduces to general relativity. From the literature, it is known that the result of BDT is close to Einstein theory of general relativity for large value of the coupling parameter ω≥500 [6, 7]. A more recent bound on the Brans-Dicke parameter ω is ω>3300 [7]. A number of researchers [8–15] have discussed various aspects of expanding cosmological models in BDT.Cosmological observations [16, 17] and various related research clearly indicate that the constituent of the present Universe is dominated by dark energy, which constitutes about three-fourths of the whole matter of our Universe. There are several candidates for dark energy like quintessence, phantom, quintom, holographic dark energy, K-essence, Chaplygin gas, and cosmological constant. Among all the dark energy candidates, cosmological constant is the more favoured. It provides enough negative pressure to account for the acceleration and contributes an energy density of same order of magnitude compared to the energy density of the matter [18]. The discrepancy of observed value and theoretical value of cosmological constant is usually referred as cosmological constant problem in literature. This problem is the puzzling problem in standard cosmology. The cosmological constant bears a dynamical decaying character so that it might be large at early epoch and approaching to a small value at the present epoch.The effect of cosmological constant has been discussed in the literature in the context of general relativity and its alternative theories. T. Singh and T. Singh [19] presented a cosmological model in BDT by considering cosmological constant as a function of scalar field ϕ. Exact cosmological solutions in BDT with uniform cosmological “constant” have been studied by Pimentel [20]. A class of flat FRW cosmological models with cosmological “constant” in BDT have also been obtained by Ahmadi-Azar & Riazi [21]. The age of the Universe from a view point of the nucleosynthesis with Λ term in BDT was investigated by Etoh et al. [22]. Azad & Islam [23] extended the idea of T. Singh and T. Singh [19] to study cosmological constant in Bianchi type I modified Brans-Dicke cosmology. Qiang et al. [24] discussed cosmic acceleration in five-dimensional BDT using interacting Higgs and Brans-Dicke fields. Smolyakov [25] investigated a model which provides the necessary value of effective cosmological “constant” at the classical level. Recently, embedding general relativity with varying cosmological term in five-dimensional BDT of gravity in vacuum has been discussed by Reyes & Aguilar [26]. Singh et al. [27] have studied the dynamic cosmological constant in BDT.On the other side, it is known from the literature that for early evolution of the Universe, bulk viscosity is supposed to play a very important role. The presence of viscosity in the fluid explores many dynamics of the homogeneous cosmological models. The bulk viscosity coefficient determines the magnitude of the viscous stress relative to the expansion. Recently Saadat & Pourhassan [28] investigated the FRW bulk viscous cosmology with modified cosmic Chaplygin gas. Many researchers also have shown interest in FRW bulk viscous cosmological models in different contexts (see Saadat & Pourhassan [28] and references therein).Motivated by the above studies, here we have discussed the variable cosmological constantΛ for FRW metric in the context of BDT with a special form of deceleration parameter.


## 2. Field Equations

The field equation of Brans-Dicke theory in presence of cosmological constant may be written as(1)Gij-Λgij+ωϕ2ϕ;iϕ;j-12gijϕ;kϕ;k+1ϕϕ;i;j-gij□ϕ=8πϕTij,□ϕ=ϕ;i;i=8π2ω+3T,ii,where ϕ is the scalar field. The energy-momentum tensor Tij of the cosmic fluid in the presence of bulk viscosity may be be defined as(2)Tij=ρ+p+Πuiuj-p+Πgij.Let us consider a homogeneous and isotropic Universe represented by FRW space-time metric as(3)ds2=dt2-R2tdr21-kr2+r2dθ2+sin2⁡θdϕ2,where k(=1,0,-1) is the curvature parameter, which represents closed, flat, and open model of the Universe and Rt is the scale factor.The FRW metric (3) and energy-momentum tensor (2) along with Brans-Dicke field equations yield the following equations:(4)3R˙R2+3R˙ϕ˙Rϕ-ω2ϕ˙ϕ2+3kR2=8πϕρ+Λ2R¨R+R˙R2+ϕ¨ϕ+ω2ϕ˙ϕ2+2R˙ϕ˙Rϕ+kR2=-8πϕp+Π+Λϕ¨ϕ+3R˙ϕ˙Rϕ=8πϕρ-3p-3Π3+2ω+2Λ3+2ω.


## 3. Solution of the Field Equations

In order to find exact solutions of basic field equations (4), one must ensure that set of equations should be closed. Thus, two more physically reasonable relations are required among the variables.First we consider a well accepted power law relation between scale factorRt and scalar field ϕ of the form [27](5)ϕ=ϕ0Rα1and as it has been well established the expansion of present Universe is accelerating. In order to study a cosmological model with early deceleration and late time acceleration, we have proposed deceleration parameter of the form(6)q=α2+α3t1+tas the second physically plausible relation, where α2,α3∈R. The considered form of deceleration parameter is motivated by the bilinear form of deceleration parameter, Mishra & Chand [29]. Deceleration parameter is useful to classify the models of the Universe. From literature we know that deceleration parameter is a constant quantity or it depends on time. In the case when rate of expansion never changes and R˙ is constant, the scaling factor is proportional to time, which leads to zero deceleration. In case when H is constant, the deceleration parameter (q) is also constant (-1). In de-Sitter and steady state Universe such cases arises. Now we will classify the cosmological models on the basis of time dependence on Hubble parameter and deceleration parameter as follows, Bolotin et al. [30].(i)
H>0, q>0: expanding and decelerating;(ii)
H>0, q<0: expanding and accelerating;(iii)
H<0, q>0: contracting and decelerating;(iv)
H<0, q<0: contracting and accelerating;(v)
H>0, q=0: expanding, zero deceleration/constant expansion;(vi)
H<0, q=0: contracting, zero deceleration;(vii)
H=0, q=0: static.From the above classification, (i), (ii), and (v) are possible cases as in the present scenario our Universe is expanding. Again also we have found the following type of expansion exhibit by our Universe.(i)
q<-1: superexponential expansion;(ii)
-1≤q<0: exponential expansion (for q=-1 known as de-Sitter expansion);(iii)
q=0: expansion with constant rate;(iv)
-1<q<1: accelerating power expansion;(v)
q>0: decelerating expansion.We consider third physically plausible relation as the modified Chaplygin gas equation of state as follows [31, 32]:(7)p=Aρ-Bρn,where A>0, B>0 are constants and 0≤n≤1.The set of field equations (4) with the help of (5) may be written as(8)6+6α1-ωα122R˙R2+3kR2=8πϕ0Rα1ρ+Λ2+α1R¨R+2+2α1+2α12+ωα122R˙R2+kR2=-8πϕ0Rα1p+Π+Λα1R¨R+α1α1+2R˙R23+2ω=8πϕ0Rα1ρ-3p-3Π+2Λ.Equations (8) leads us to(9)23-ωα1R¨R+6-4ωα1-ωα12R˙R2+6kR2=2Λ.This equation is useful for obtaining the various cosmological solutions.Now our problem is to evaluate theRt, which is obtained from the relation(10)-H˙H2=1+q.With the help of (6) and integrating (10), we obtained(11)H=11+α3t+α2-α3ln⁡1+t+c1,where c1 is a constant of integration. The condition H→∞ when t→0 yields c1=0. Thus, (11) takes the form(12)H=11+α3t+α2-α3ln⁡1+t.Equation (12) is expressed as (13)H=11+α3t+α2-α3t-t2/2+t3/3-t4/4+t5/5-t6/6+t7/7-⋯=11+α2t+α2-α3-t2/2+t3/3-t4/4+t5/5-t6/6+t7/7-⋯=11+α2t1-α2-α31+α2t2-t23+t34-t45+t56-t67+⋯-1=11+α2t1+α2-α31+α2t2-t23+t34-t45+t56-t67+⋯+α2-α31+α22t2-t23+t34-t45+t56-t67+⋯2+α2-α31+α23t2-t23+t34-t45+t56-t67+⋯3+α2-α31+α24t2-t23+t34-t45+t56-t67+⋯4+⋯.Simplifying the above expression we obtained(14)H=11+α2t+k0+k1t+k2t2+k3t3+k4t4+Ot5,where (15)k0=α2-α321+α22k1=11+α2α2-α3241+α22-α2-α331+α2k2=11+α2α2-α341+α2-α2-α3231+α22+α2-α3381+α23k3=11+α2-α2-α351+α2+13α2-α32361+α22-α2-α3341+α23+α2-α34161+α24k4=11+α2α2-α361+α2-11α2-α32301+α22+17α2-α33481+α23-α2-α3461+α24.Integration of (14) leads us to(16)R=c2t1/1+α2eT1t,where T1t=k0t+k1(t2/2)+k2(t3/3)+k3(t4/4)+k4(t5/5)+Ot6. The solutions of the field equation (8) are expressed as follows: the energy density ρ is obtained as(17)ρ=k5tα1/1+α2eα1T1tρ1+ρ2t1+t1+α3t+α2-α3ln⁡1+t2,where k5=ϕ0c2α1/8π,ρ1=3+2w-wα2α1+3α2,ρ2=3+2w-wα3α1+3α3.The pressurep is given as(18)p=Ak5n+1tn+1α1/1+α2en+1α1T1tρ1+ρ2tn+1-B1+tn+11+α3t+α2-α3ln⁡1+t2n+2k5ntnα1/1+α2enα1T1tρ1+ρ2tn1+t1+α3t+α2-α3ln⁡1+t2.The bulk viscous stress Π is expressed as(19)k5n+1tn+1α1/1+α2en+1α1T1tρ1+ρ2t1+t1+α3t+α2-α3ln⁡1+t2n×Π=Π1+Π2t/1+t1+α3t+α2-α3ln⁡1+t2+2k/c22t2/1+α2e2T1t+Bk5ntnα1/1+α2enα1T1tρ1+ρ2t/1+t1+α3t+α2-α3ln⁡1+t2n,where Π1=-1+wα12-1+3Aα2+α2-21+Aw-1-3A+α2α1+2 and Π2=-1+wα12-1+3Aα3+α2-21+Aw-1-3A+α3α1+2.The cosmological constantΛ is expressed as(20)Λ=Λ1+Λ2t1+t1+α3t+α2-α3ln⁡1+t2+3kc22t2/1+α2e2T1t,where Λ1=-0.5wα12+wα2-2α1-3α2-1 and Λ2=-0.5wα12+wα3-2α1-3α3-1.Now, let us start with our proposed form of deceleration parameterq. The different form of deceleration parameter is evolved as a result of considered value of α2 and α3, which is expressed in Table 1. We know that in present scenario our Universe is accelerating. Thus serial numbers (2), (5), (8), and (9) of Table 1 exhibit accelerating model. Now we will discuss the deceleration parameter in serial numbers (2), (5), (8), and (9) of Table 1. For the choice of α2=α=α4, the deceleration parameter q in serial number (2) and (5) of Table 1 reduces to q=-αt/(1+t) and q=-α1+t/(1+t), respectively, which is discussed by Mishra and Chand [29]. They called this deceleration parameter as bilinear variable deceleration parameter. We will discuss the case where α2≠α≠α4 of serial number (5) of Table 1 and also serial numbers (8) and (9) of Table 1. According to the serial numbers (5), (8), and (9) of Table 1 we have three different models, which are discussed below.Table 1
Different forms of deceleration parameter depending on the parametersα2 and α3 and behaviour of cosmological models according to the deceleration parameter q.
S. number
Possible value ofα2 and α3
Form of deceleration parameterq
Behaviour of cosmological model
(1)
α
2
=
0
q
=
α
3
t
1
+
t
Decelerating
α
3
>
0
(2)
α
2
=
0
q
=
-
α
4
t
1
+
t
Accelerating
α
3
<
0
(
α
3
=
-
α
4
,
α
4
>
0
)
(3)
α
2
=
0
q
=
0
Expansion with constant rate
α
3
=
0
(4)
α
2
>
0
q
=
α
2
+
α
3
t
1
+
t
Decelerating
α
3
>
0
(5)
α
2
>
0
q
=
α
2
-
α
4
t
1
+
t
Phase transition from decelerating to accelerating
α
3
<
0
(6)
α
2
>
0
q
=
α
2
1
+
t
Decelerating
α
3
=
0
(7)
α
2
<
0
(
α
2
=
-
α
5
,
α
5
>
0
)
q
=
-
α
5
+
α
3
t
1
+
t
Phase transition from accelerating to decelerating
α
3
>
0
(8)
α
2
<
0
q
=
-
α
5
-
α
4
t
1
+
t
Accelerating
α
3
<
0
(9)
α
2
<
0
q
=
-
α
5
1
+
t
Accelerating
α
3
=
0


### 3.1. Model I

The deceleration parameterq in (6) for α2>0 and α3<0 takes the form(21)q=α2-α4t1+t,α3=-α4,α4>0.Here we noticed that q>0 for 0<t<α2/α4 and q<0 for t>α2/α4, which means that our Universe is decelerating and accelerating in the provided ranges, respectively. Thus our Universe undergoes a phase transition from decelerating to accelerating phase.For model I, the physical parameters are obtained as follows.The Hubble parameter in (12) takes the form(22)H=11-α4t+α2+α4ln⁡1+t.The scale factor Rt in (16) is expressed as(23)R=c2t1/1+α2eT1t,where T1t=k0t+k1(t2/2)+k2(t3/3)+k3(t4/4)+k4(t5/5)+Ot6. And (24)k0=α2+α421+α22k1=11+α2α2+α4241+α22-α2+α431+α2k2=11+α2α2+α441+α2-α2+α4231+α22+α2+α4381+α23k3=11+α2-α2+α451+α2+13α2+α42361+α22-α2+α4341+α23+α2+α44161+α24k4=11+α2α2+α461+α2-11α2+α42301+α22+17α2+α43481+α23-α2+α4461+α24.The FRW space-time metric in (3) takes the form (25)ds2=dt2-c22t2/1+α2e2T1tdr21-kr2+r2dθ2+sin2⁡θdϕ2with the above mentation ki,i=0,1,2,3,4. The energy density ρ, pressure p, bulk viscous stress Π, and cosmological constant Λ in (17), (18), (19), and (20) are expressed as(26)ρ=k5tα1/1+α2eα1T1tρ1+ρ2t1+t1-α4t+α2+α4ln⁡1+t2,where k5=ϕ0c2α1/8π,ρ1=3+2w-wα2α1+3α2,ρ2=3+2w+wα4α1-3α4.(27)p=Ak5n+1tn+1α1/1+α2en+1α1T1tρ1+ρ2tn+1-B1+tn+11-α4t+α2+α4ln⁡1+t2n+2k5ntnα1/1+α2enα1T1tρ1+ρ2tn1+t1-α4t+α2+α4ln⁡1+t2,k5n+1tn+1α1/1+α2en+1α1T1tρ1+ρ2t1+t1-α4t+α2+α4ln⁡1+t2n×Π=Π1+Π2t/1+t1-α4t+α2+α4ln⁡1+t2+2k/c22t2/1+α2e2T1t+Bk5ntnα1/1+α2enα1T1tρ1+ρ2t/1+t1-α4t+α2+α4ln⁡1+t2n,where Π1=-1+wα12-1+3Aα2+α2-21+Aw-1-3A+α2α1+2 and Π2=-1+wα12+1+3Aα4+α2-21+Aw-1-3A-α4α1+2.(28)Λ=Λ1+Λ2t1+t1-α4t+α2+α4ln⁡1+t2+3kc22t2/1+α2e2T1t,where Λ1=-0.5wα12+wα2-2α1-3α2-1 and Λ2=-0.5wα12-wα4+2α1+3α4+1.Figures1 and 2 represent the variation of deceleration parameter against time with different values of parameters as presented in the figures for model I. From these figures, we have noticed that when α2 is fixed and α4 is different and vice versa, deceleration parameter q is a decreasing function of time and it takes values from positive to negative, which shows that our Universe undergoes a phase transition from deceleration phase to acceleration phase. Here we observed that -1<q<1 for 0<α2≤1 and 0<α4≤1, which means that within the provided range of αii=2,4 our Universe undergoes an accelerating power expansion. It can be observed from Figures 1 and 2.Figure 1
Variation of deceleration parameter against time for fixedα2 and different α4.Figure 2
Variation of deceleration parameter against time for fixedα4 and different α2.The variation of Hubble parameterH and scale factor R against time is plotted in Figures 3 and 4, respectively for model I. As a representative case here we have presented the variation of H and R for fixed α2=0.5 and different α4 as in figures. It is found that Hubble parameter H is a decreasing function of time and approaching towards zero with the evolution of time. For 0<α4≤0.8 and α5=0.5, the scale factor R is an increasing function of time and the higher the value of α4, the lower the value of scale factor R. For α4≥0.9 and α2=0.5, the scale factor takes a bounce and increases with the evolution of time (see Figure 4).Figure 3
Variation of Hubble parameter against time for fixedα2=0.5 and different α4.Figure 4
Variation of scale factor against time for fixedα2=0.5 and different α4.Figures5 and 6 represent the variation of energy density ρ and pressure p against time, respectively for model I. From the Figure 5 we pointed out that, in the interval 0<α4≤0.8 & α2=0.5 with the time, energy density decreases for small interval of time and increases to a higher value with the evolution of time. This shows that our Universe is dominated by radiation. For α4≥0.9 and α2=0.5 the energy density is a decreasing function of time and approaches to zero with the evolution of time. In present scenario such type of qualitative behaviour of energy density is observed from observational data. From pressure profile (Figure 6) we observed that, in the intervals 0<α4≤0.8 and α2=0.5, the pressure is negative for small interval of time and increases with the evolution of time. In the intervals 0.9≤α4≤1.2 and α2=0.5, pressure is negative, which follows the observational data but, for α4>1.2, it is complex valued; thus we neglect it.Figure 5
Variation of energy density against time forα1=0.5, α2=0.5, ω=1, ϕ0=1, c2=0.1, and different α4.Figure 6
Variation of pressure against time forα1=0.5, α2=0.5, A=B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α4.The variation of bulk viscous stressΠ and cosmological constant Λ against time is plotted in Figures 7 and 8, respectively for model I. Figures indicate the qualitative and quantitative behaviour of both the parameters for open k=-1, flat k=0, and closed k=1 Universe. We have noticed the following points.Figure 7
Variation of bulk viscous stressΠ against time for α1=0.5, α2=0.5, A=B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α4. (b) shows the zooming of (a).
(a)
(b)Figure 8
Variation of cosmological constantΛ against time for α1=0.5, α2=0.5, ω=1, c2=0.1, and different α40.2,0.4,0.6,0.8,0.9,1.1,1.3. Lower two rows represent the magnified portion of the above two rows.Bulk Viscous Stress  Π  (see Figure7)
(i)
Bulk viscous stressΠ takes values from positive to negative and approaches to minus infinity -∞ with time in case of flat and closed Universe whereas it is negative-positive-negative valued for open Universe in the intervals 0<α4≤0.7 and α2=0.5.
(ii)
Bulk viscous stressΠ is positive valued and tends to infinity with the evolution of time for flat and closed Universe whereas it is negative-positive valued for open Universe in 0.7<α4≤0.8 and α2=0.5.
(iii)
Forα4>0.8 and α2=0.5, bulk viscous stress Π is positive valued and tends to infinity with the evolution of time for flat and closed Universe whereas it is negative valued for open Universe.
Cosmological Constant  Λ  (see Figure8)
(i)
Cosmological constantΛ is positive and negative for flat and open Universe and closed Universe, respectively. Cosmological constant Λ→0 when t→∞.
(ii)
In case of flat and open Universe cosmological constantΛ is positive valued for α4>0.8 and α2=0.5 whereas it is negative valued for open Universe.
(iii)
In case of flat Universe cosmological constantΛ→0 when t→∞ but for close and open Universe Λ→∞ when t→∞ and Λ→-∞ when t→∞, respectively.


### 3.2. Model II

The deceleration parameterq in (6) for α2<0 and α3<0 takes the form(29)q=-α5+α4t1+t,α3=-α4,α2=-α5,α4,α5>0.Here we noticed that q<0 for α4,α5>0, which means that our Universe is accelerating with the evolution of time.For model II, the physical parameters are obtained as follows:The Hubble parameter in (12) takes the form(30)H=11-α4t+α4-α5ln⁡1+t.The scale factor Rt in (16) is expressed as(31)R=c2t1/1-α5eT1t,where T1t=k0t+k1(t2/2)+k2(t3/3)+k3(t4/4)+k4(t5/5)+Ot6. And (32)k0=α4-α521-α52k1=11-α5α4-α5241-α52-α4-α531-α5k2=11-α5α4-α541-α5-α4-α5231-α52+α4-α5381-α53k3=11-α5-α4-α551-α5+13α4-α52361-α52-α4-α5341-α53+α4-α54161-α54k4=11-α5α4-α561-α5-11α4-α52301-α52+17α4-α53481-α53-α4-α5461-α54.The FRW space-time metric in (3) takes the form (33)ds2=dt2-c22t2/1-α5e2T1tdr21-kr2+r2dθ2+sin2⁡θdϕ2with the above mentation ki,i=0,1,2,3,4. The energy density ρ, pressure p, bulk viscous stress Π, and cosmological constant Λ in (17), (18), (19), and (20) takes the form(34)ρ=k5tα1/1-α5eα1T1tρ1+ρ2t1+t1-α4t+α4-α5ln⁡1+t2,where k5=ϕ0c2α1/8π,ρ1=3+2w+wα5α1-3α5,ρ2=3+2w+wα4α1-3α4.(35)p=Ak5n+1tn+1α1/1-α5en+1α1T1tρ1+ρ2tn+1-B1+tn+11-α4t+α4-α5ln⁡1+t2n+2k5ntnα1/1-α5enα1T1tρ1+ρ2tn1+t1-α4t+α4-α5ln⁡1+t2k5n+1tn+1α1/1-α5en+1α1T1tρ1+ρ2t1+t1-α4t+α4-α5ln⁡1+t2n×Π=Π1+Π2t/1+t1-α4t+α4-α5ln⁡1+t2+2k/c22t2/1-α5e2T1t+Bk5ntnα1/1-α5enα1T1tρ1+ρ2t/1+t1-α4t+α4-α5ln⁡1+t2n,where Π1=-1+wα12+1+3Aα5-α5+21+Aw+1+3A+α5α1+2 and Π2=-1+wα12+1+3Aα4-α5+21+Aw+1+3A+α4α1+2.(36)Λ=Λ1+Λ2t1+t1-α4t+α4-α5ln⁡1+t2+3kc22t2/1-α5e2T1t,where Λ1=-0.5wα12-wα5+2α1+3α5+1 and Λ2=-0.5wα12-wα4+2α1+3α4+1.Now we will discuss the physical parameters of the model II. Figures9 and 10 represent the variation of deceleration parameter against time for fixed α5 and different α4 and fixed α4 and different α5, respectively. Here we observed that deceleration parameter is negative and our model is accelerating.Figure 9
Variation of deceleration parameter against time for fixedα5 and different α4.Figure 10
Variation of deceleration parameter against time for fixedα4 and different α5.Figures11, 12, and 13 depict the variation of Hubble parameter H and scale factor R against time, respectively, for model II. The observations are as follows:(i)
Hubble parameterH is a decreasing function of time and tending to zero with the evolution of time. As a representative case, we have considered α5=0.5 and different α40<α4≤1.2 as in Figure 11.(ii)
Scale factor increases with the evolution of time. Here we pointed out that the qualitative behaviour of scale factorR is different for different interval of α4 and α5. As a representative case, we choose α5=0.3,0.5   and different α4 and all other parameters as in Figures 12 and 13. In the interval 0<α4≤0.2 & α4>0.9 and α5=0.3, scale factor R increases after taking a bounce where as in 0.3≤α4≤0.9 and α5=0.3, it increases gradually with the evolution of time (see Figure 12). Similar qualitative behaviour is noticed for α5=0.5 and different α4 (see Figure 13).Figure 11
Variation of Hubble parameter against time for fixedα5=0.5 and different α4.Figure 12
Variation of scale factor against time for fixedα5=0.3 and different α4.Figure 13
Variation of scale factor against time for fixedα5=0.5 and different α4.The variation of energy density and pressure against time is presented for model II in Figures14 and 15, respectively. As a representative case we choose α5=0.3   and different α4 and all other parameters are as in Figures 14 and 15. The observations are as follows:(i)
Energy density gradually decreases and approaches towards zero with the evolution of time for0<α4≤0.3 and α5=0.3.(ii)
Energy density is gradually decreased for small interval of time and tends towards infinity with the evolution of time for0.4≤α4≤0.9 and α5=0.3.(iii)
Forα4≥1 and α5=0.3, energy density tends towards zero with time. Here we pointed out that, with the increment of α4, the bounce of the energy density increases and gradually tends to zero (see Figure 14).(iv)
Pressure is negative in0<α4≤0.3 and 1≤α4≤1.5 with α5=0.3.(v)
Pressure is negative for a small interval of time and gradually increases with time and it takes values from positive to negative in the intervals0.4≤α4≤0.9 and α4≥1.6 with α5=0.3, respectively (see Figure 15).Figure 14
Variation of energy densityρ against time for α1=0.5, α5=0.3, ω=1, ϕ0=1, c2=0.1, and different α4. (b) shows the zooming of (a).
(a)
(b)Figure 15
Variation of pressurep against time for α1=0.5, α5=0.3, A=B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α4. (b) shows the zooming of (a).
(a)
(b)The variation of bulk viscous stressΠ and cosmological constant Λ against time for model II is presented in Figures 16 and 17, respectively. The observations are as follows.Figure 16
Variation of bulk viscous stressΠ against time for α1=0.5, α5=0.3, A=B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α4. (b) shows the zooming of (a). Blue line, Green line, and Red line represent k=-1, k=0, and k=1, respectively.
(a)
(b)Figure 17
Variation of cosmological constantΛ against time for α1=0.5, α5=0.3, ω=1, c2=0.1, and different α4. (b) shows the zooming of (a). Blue line, Green line, and Red line represent k=-1, k=0, and k=1, respectively.
(a)
(b)Bulk Viscous Stress  Π  (see Figure16)
(i)
It is positive valued for flat and closed Universe whereas it is negative valued for open Universe in0<α4≤0.2 and 1≤α4≤2 and α5=0.3.
(ii)
It is positive-negative valued for flat and closed Universe whereas it is negative-positive-negative valued for open Universe in0.3≤α4≤0.8 and α5=0.3.
(iii)
It is positive valued for flat and closed Universe whereas it is negative-positive valued for open Universe in0.8<α4≤1 and α5=0.3. Also it approaches towards infinity with the evolution of time in the specified interval of α4.
(iv)
Π is positive valued for flat and closed Universe whereas it is negative-positive-negative valued for open Universe in α4>2 and α5=0.3.
Cosmological Constant  Λ  (see Figure17)
(i)
For0<α4<0.3 and α5=0.3, Λ is positive valued for flat and closed Universe whereas it is negative value for open Universe. In case of flat Universe Λ→0 when t→∞ but for open and closed Universe Λ→∞ and Λ→-∞ with time respectively.
(ii)
For0.3<α4≤0.9 and α5=0.3, Λ→0 when t→∞ for open, flat, and closed Universe. In case of flat and closed Universe, cosmological constant is positive valued whereas in open Universe it is negative-positive valued.
(iii)
It is positive valued for flat and closed Universe but it is negative-positive-negative valued for open Universe inα4>0.9 and α5=0.3.


### 3.3. Model III

The deceleration parameterq in (6) for α2<0 and α3=0 takes the form(37)q=-α51+t,α2=-α5,α5>0.Here we noticed that q<0 for α5>0, which means that our Universe is accelerating with the evolution of time.For model III, the physical parameters are obtained as follows:The Hubble parameter in (12) takes the form(38)H=1t-α5ln⁡1+t.The scale factor Rt in (16) is expressed as(39)R=c2t1/1-α5eT1t,where T1t=k0t+k1(t2/2)+k2(t3/3)+k3(t4/4)+k4(t5/5)+Ot6. And (40)k0=-α521-α52k1=11-α5α5241-α52+α531-α5k2=-11-α5α541-α5+α5231-α52+α5381-α53k3=11-α5α551-α5+13α52361-α52+α5341-α53+α54161-α54k4=-11-α5α561-α5+11α52301-α52+17α53481-α53+α5461-α54.The FRW space-time metric in (3) takes the form (41)ds2=dt2-c22t2/1-α5e2T1tdr21-kr2+r2dθ2+sin2⁡θdϕ2with the above mentation ki,i=0,1,2,3,4. The energy density ρ, pressure p, bulk viscous stress Π, and cosmological constant Λ in (17), (18), (19), and (20) takes the form(42)ρ=k5tα1/1-α5eα1T1tρ1+ρ2t1+tt-α5ln⁡1+t2,where k5=ϕ0c2α1/8π,ρ1=3+2w+wα5α1-3α5,ρ2=3+2wα1.(43)p=Ak5n+1tn+1α1/1-α5en+1α1T1tρ1+ρ2tn+1-B1+tn+1t-α5ln⁡1+t2n+2k5ntnα1/1-α5enα1T1tρ1+ρ2tn1+tt-α5ln⁡1+t2k5n+1tn+1α1/1-α5en+1α1T1tρ1+ρ2t1+tt-α5ln⁡1+t2n×Π=Π1+Π2t/1+tt-α5ln⁡1+t2+2k/c22t2/1-α5e2T1t+Bk5ntnα1/1-α5enα1T1tρ1+ρ2t/1+tt-α5ln⁡1+t2n,where Π1=-1+wα12+1+3Aα5-α5+21+Aw+1+3A+α5α1+2 and Π2=-1+wα12-α5+21+Aw+1+3Aα1+2.(44)Λ=Λ1+Λ2t1+tt-α5ln⁡1+t2+3kc22t2/1-α5e2T1t,where Λ1=-0.5wα12+wα5+2α1-3α5+1 and Λ2=-0.5wα12-2wα1+3.The profile of deceleration parameter, Hubble parameter, and scale factor against time is plotted in the Figures18, 19, and 20, respectively, for model III. The observations are as follows:(i)
Deceleration parameterq is a negative valued function of time and approaches towards zero with the evolution of time. In other words we can say, at early time, our Universe is accelerating and follows an expansion with constant rate at late time (see Figure 18).(ii)
Hubble parameterH is a decreasing function of time and H→0 when t→∞. Also in this case, the higher the value of α5, the higher the value of Hubble parameter (see Figure 19).(iii)
Scale factorR is an increasing function of time and R→∞ when t→∞. Equation (39) indicates that R is not defined for α5=1. As a representative case, we considered 0<α5<1 (see Figure 20).Figures 21 and 22 depict the energy density and pressure profile against time. respectively. For α5≥1, energy density possesses physical unrealistic behaviour, so α5 is restricted to 0<α5<1. It is noticed that energy density is a decreasing function of time and ρ→0 when t→∞ (see Figure 21). Also pressure is a negative quantity with the evolution of time (see Figure 22).Figure 18
Variation of deceleration parameter against time for differentα5.Figure 19
Variation of Hubble parameter against time differentα5.Figure 20
Variation of scale factor against time for differentα5.Figure 21
Variation of energy densityρ against time for α1=0.5, ω=1, ϕ0=1, c2=0.1, and different α5.Figure 22
Variation of pressurep against time for α1=0.5, A=1, B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α5. (b) shows the zooming of (a).
(a)
(b)The profile of bulk viscous stress and cosmological constant against time is depicted in Figures23 and 24, respectively, for model III. Bulk viscous stress is positive valued for flat and closed Universe whereas it is negative valued for open Universe. Similar quantitative behaviour is observed for cosmological constant. In case of flat Universe, cosmological constant is a decreasing function of time and tends to zero with the evolution of time.Figure 23
Variation of bulk viscous stressΠ against time for α1=0.5, A=1, B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α5. (b) shows the zooming of (a).
(a)
(b)Figure 24
Variation of cosmological constantΛ against time for α1=0.5, ω=1, c2=0.1, and different α5. (b) shows the zooming of (a).
(a)
(b)


## 3.1. Model I

The deceleration parameterq in (6) for α2>0 and α3<0 takes the form(21)q=α2-α4t1+t,α3=-α4,α4>0.Here we noticed that q>0 for 0<t<α2/α4 and q<0 for t>α2/α4, which means that our Universe is decelerating and accelerating in the provided ranges, respectively. Thus our Universe undergoes a phase transition from decelerating to accelerating phase.For model I, the physical parameters are obtained as follows.The Hubble parameter in (12) takes the form(22)H=11-α4t+α2+α4ln⁡1+t.The scale factor Rt in (16) is expressed as(23)R=c2t1/1+α2eT1t,where T1t=k0t+k1(t2/2)+k2(t3/3)+k3(t4/4)+k4(t5/5)+Ot6. And (24)k0=α2+α421+α22k1=11+α2α2+α4241+α22-α2+α431+α2k2=11+α2α2+α441+α2-α2+α4231+α22+α2+α4381+α23k3=11+α2-α2+α451+α2+13α2+α42361+α22-α2+α4341+α23+α2+α44161+α24k4=11+α2α2+α461+α2-11α2+α42301+α22+17α2+α43481+α23-α2+α4461+α24.The FRW space-time metric in (3) takes the form (25)ds2=dt2-c22t2/1+α2e2T1tdr21-kr2+r2dθ2+sin2⁡θdϕ2with the above mentation ki,i=0,1,2,3,4. The energy density ρ, pressure p, bulk viscous stress Π, and cosmological constant Λ in (17), (18), (19), and (20) are expressed as(26)ρ=k5tα1/1+α2eα1T1tρ1+ρ2t1+t1-α4t+α2+α4ln⁡1+t2,where k5=ϕ0c2α1/8π,ρ1=3+2w-wα2α1+3α2,ρ2=3+2w+wα4α1-3α4.(27)p=Ak5n+1tn+1α1/1+α2en+1α1T1tρ1+ρ2tn+1-B1+tn+11-α4t+α2+α4ln⁡1+t2n+2k5ntnα1/1+α2enα1T1tρ1+ρ2tn1+t1-α4t+α2+α4ln⁡1+t2,k5n+1tn+1α1/1+α2en+1α1T1tρ1+ρ2t1+t1-α4t+α2+α4ln⁡1+t2n×Π=Π1+Π2t/1+t1-α4t+α2+α4ln⁡1+t2+2k/c22t2/1+α2e2T1t+Bk5ntnα1/1+α2enα1T1tρ1+ρ2t/1+t1-α4t+α2+α4ln⁡1+t2n,where Π1=-1+wα12-1+3Aα2+α2-21+Aw-1-3A+α2α1+2 and Π2=-1+wα12+1+3Aα4+α2-21+Aw-1-3A-α4α1+2.(28)Λ=Λ1+Λ2t1+t1-α4t+α2+α4ln⁡1+t2+3kc22t2/1+α2e2T1t,where Λ1=-0.5wα12+wα2-2α1-3α2-1 and Λ2=-0.5wα12-wα4+2α1+3α4+1.Figures1 and 2 represent the variation of deceleration parameter against time with different values of parameters as presented in the figures for model I. From these figures, we have noticed that when α2 is fixed and α4 is different and vice versa, deceleration parameter q is a decreasing function of time and it takes values from positive to negative, which shows that our Universe undergoes a phase transition from deceleration phase to acceleration phase. Here we observed that -1<q<1 for 0<α2≤1 and 0<α4≤1, which means that within the provided range of αii=2,4 our Universe undergoes an accelerating power expansion. It can be observed from Figures 1 and 2.Figure 1
Variation of deceleration parameter against time for fixedα2 and different α4.Figure 2
Variation of deceleration parameter against time for fixedα4 and different α2.The variation of Hubble parameterH and scale factor R against time is plotted in Figures 3 and 4, respectively for model I. As a representative case here we have presented the variation of H and R for fixed α2=0.5 and different α4 as in figures. It is found that Hubble parameter H is a decreasing function of time and approaching towards zero with the evolution of time. For 0<α4≤0.8 and α5=0.5, the scale factor R is an increasing function of time and the higher the value of α4, the lower the value of scale factor R. For α4≥0.9 and α2=0.5, the scale factor takes a bounce and increases with the evolution of time (see Figure 4).Figure 3
Variation of Hubble parameter against time for fixedα2=0.5 and different α4.Figure 4
Variation of scale factor against time for fixedα2=0.5 and different α4.Figures5 and 6 represent the variation of energy density ρ and pressure p against time, respectively for model I. From the Figure 5 we pointed out that, in the interval 0<α4≤0.8 & α2=0.5 with the time, energy density decreases for small interval of time and increases to a higher value with the evolution of time. This shows that our Universe is dominated by radiation. For α4≥0.9 and α2=0.5 the energy density is a decreasing function of time and approaches to zero with the evolution of time. In present scenario such type of qualitative behaviour of energy density is observed from observational data. From pressure profile (Figure 6) we observed that, in the intervals 0<α4≤0.8 and α2=0.5, the pressure is negative for small interval of time and increases with the evolution of time. In the intervals 0.9≤α4≤1.2 and α2=0.5, pressure is negative, which follows the observational data but, for α4>1.2, it is complex valued; thus we neglect it.Figure 5
Variation of energy density against time forα1=0.5, α2=0.5, ω=1, ϕ0=1, c2=0.1, and different α4.Figure 6
Variation of pressure against time forα1=0.5, α2=0.5, A=B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α4.The variation of bulk viscous stressΠ and cosmological constant Λ against time is plotted in Figures 7 and 8, respectively for model I. Figures indicate the qualitative and quantitative behaviour of both the parameters for open k=-1, flat k=0, and closed k=1 Universe. We have noticed the following points.Figure 7
Variation of bulk viscous stressΠ against time for α1=0.5, α2=0.5, A=B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α4. (b) shows the zooming of (a).
(a)
(b)Figure 8
Variation of cosmological constantΛ against time for α1=0.5, α2=0.5, ω=1, c2=0.1, and different α40.2,0.4,0.6,0.8,0.9,1.1,1.3. Lower two rows represent the magnified portion of the above two rows.Bulk Viscous Stress  Π  (see Figure7)
(i)
Bulk viscous stressΠ takes values from positive to negative and approaches to minus infinity -∞ with time in case of flat and closed Universe whereas it is negative-positive-negative valued for open Universe in the intervals 0<α4≤0.7 and α2=0.5.
(ii)
Bulk viscous stressΠ is positive valued and tends to infinity with the evolution of time for flat and closed Universe whereas it is negative-positive valued for open Universe in 0.7<α4≤0.8 and α2=0.5.
(iii)
Forα4>0.8 and α2=0.5, bulk viscous stress Π is positive valued and tends to infinity with the evolution of time for flat and closed Universe whereas it is negative valued for open Universe.
Cosmological Constant  Λ  (see Figure8)
(i)
Cosmological constantΛ is positive and negative for flat and open Universe and closed Universe, respectively. Cosmological constant Λ→0 when t→∞.
(ii)
In case of flat and open Universe cosmological constantΛ is positive valued for α4>0.8 and α2=0.5 whereas it is negative valued for open Universe.
(iii)
In case of flat Universe cosmological constantΛ→0 when t→∞ but for close and open Universe Λ→∞ when t→∞ and Λ→-∞ when t→∞, respectively.


## 3.2. Model II

The deceleration parameterq in (6) for α2<0 and α3<0 takes the form(29)q=-α5+α4t1+t,α3=-α4,α2=-α5,α4,α5>0.Here we noticed that q<0 for α4,α5>0, which means that our Universe is accelerating with the evolution of time.For model II, the physical parameters are obtained as follows:The Hubble parameter in (12) takes the form(30)H=11-α4t+α4-α5ln⁡1+t.The scale factor Rt in (16) is expressed as(31)R=c2t1/1-α5eT1t,where T1t=k0t+k1(t2/2)+k2(t3/3)+k3(t4/4)+k4(t5/5)+Ot6. And (32)k0=α4-α521-α52k1=11-α5α4-α5241-α52-α4-α531-α5k2=11-α5α4-α541-α5-α4-α5231-α52+α4-α5381-α53k3=11-α5-α4-α551-α5+13α4-α52361-α52-α4-α5341-α53+α4-α54161-α54k4=11-α5α4-α561-α5-11α4-α52301-α52+17α4-α53481-α53-α4-α5461-α54.The FRW space-time metric in (3) takes the form (33)ds2=dt2-c22t2/1-α5e2T1tdr21-kr2+r2dθ2+sin2⁡θdϕ2with the above mentation ki,i=0,1,2,3,4. The energy density ρ, pressure p, bulk viscous stress Π, and cosmological constant Λ in (17), (18), (19), and (20) takes the form(34)ρ=k5tα1/1-α5eα1T1tρ1+ρ2t1+t1-α4t+α4-α5ln⁡1+t2,where k5=ϕ0c2α1/8π,ρ1=3+2w+wα5α1-3α5,ρ2=3+2w+wα4α1-3α4.(35)p=Ak5n+1tn+1α1/1-α5en+1α1T1tρ1+ρ2tn+1-B1+tn+11-α4t+α4-α5ln⁡1+t2n+2k5ntnα1/1-α5enα1T1tρ1+ρ2tn1+t1-α4t+α4-α5ln⁡1+t2k5n+1tn+1α1/1-α5en+1α1T1tρ1+ρ2t1+t1-α4t+α4-α5ln⁡1+t2n×Π=Π1+Π2t/1+t1-α4t+α4-α5ln⁡1+t2+2k/c22t2/1-α5e2T1t+Bk5ntnα1/1-α5enα1T1tρ1+ρ2t/1+t1-α4t+α4-α5ln⁡1+t2n,where Π1=-1+wα12+1+3Aα5-α5+21+Aw+1+3A+α5α1+2 and Π2=-1+wα12+1+3Aα4-α5+21+Aw+1+3A+α4α1+2.(36)Λ=Λ1+Λ2t1+t1-α4t+α4-α5ln⁡1+t2+3kc22t2/1-α5e2T1t,where Λ1=-0.5wα12-wα5+2α1+3α5+1 and Λ2=-0.5wα12-wα4+2α1+3α4+1.Now we will discuss the physical parameters of the model II. Figures9 and 10 represent the variation of deceleration parameter against time for fixed α5 and different α4 and fixed α4 and different α5, respectively. Here we observed that deceleration parameter is negative and our model is accelerating.Figure 9
Variation of deceleration parameter against time for fixedα5 and different α4.Figure 10
Variation of deceleration parameter against time for fixedα4 and different α5.Figures11, 12, and 13 depict the variation of Hubble parameter H and scale factor R against time, respectively, for model II. The observations are as follows:(i)
Hubble parameterH is a decreasing function of time and tending to zero with the evolution of time. As a representative case, we have considered α5=0.5 and different α40<α4≤1.2 as in Figure 11.(ii)
Scale factor increases with the evolution of time. Here we pointed out that the qualitative behaviour of scale factorR is different for different interval of α4 and α5. As a representative case, we choose α5=0.3,0.5   and different α4 and all other parameters as in Figures 12 and 13. In the interval 0<α4≤0.2 & α4>0.9 and α5=0.3, scale factor R increases after taking a bounce where as in 0.3≤α4≤0.9 and α5=0.3, it increases gradually with the evolution of time (see Figure 12). Similar qualitative behaviour is noticed for α5=0.5 and different α4 (see Figure 13).Figure 11
Variation of Hubble parameter against time for fixedα5=0.5 and different α4.Figure 12
Variation of scale factor against time for fixedα5=0.3 and different α4.Figure 13
Variation of scale factor against time for fixedα5=0.5 and different α4.The variation of energy density and pressure against time is presented for model II in Figures14 and 15, respectively. As a representative case we choose α5=0.3   and different α4 and all other parameters are as in Figures 14 and 15. The observations are as follows:(i)
Energy density gradually decreases and approaches towards zero with the evolution of time for0<α4≤0.3 and α5=0.3.(ii)
Energy density is gradually decreased for small interval of time and tends towards infinity with the evolution of time for0.4≤α4≤0.9 and α5=0.3.(iii)
Forα4≥1 and α5=0.3, energy density tends towards zero with time. Here we pointed out that, with the increment of α4, the bounce of the energy density increases and gradually tends to zero (see Figure 14).(iv)
Pressure is negative in0<α4≤0.3 and 1≤α4≤1.5 with α5=0.3.(v)
Pressure is negative for a small interval of time and gradually increases with time and it takes values from positive to negative in the intervals0.4≤α4≤0.9 and α4≥1.6 with α5=0.3, respectively (see Figure 15).Figure 14
Variation of energy densityρ against time for α1=0.5, α5=0.3, ω=1, ϕ0=1, c2=0.1, and different α4. (b) shows the zooming of (a).
(a)
(b)Figure 15
Variation of pressurep against time for α1=0.5, α5=0.3, A=B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α4. (b) shows the zooming of (a).
(a)
(b)The variation of bulk viscous stressΠ and cosmological constant Λ against time for model II is presented in Figures 16 and 17, respectively. The observations are as follows.Figure 16
Variation of bulk viscous stressΠ against time for α1=0.5, α5=0.3, A=B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α4. (b) shows the zooming of (a). Blue line, Green line, and Red line represent k=-1, k=0, and k=1, respectively.
(a)
(b)Figure 17
Variation of cosmological constantΛ against time for α1=0.5, α5=0.3, ω=1, c2=0.1, and different α4. (b) shows the zooming of (a). Blue line, Green line, and Red line represent k=-1, k=0, and k=1, respectively.
(a)
(b)Bulk Viscous Stress  Π  (see Figure16)
(i)
It is positive valued for flat and closed Universe whereas it is negative valued for open Universe in0<α4≤0.2 and 1≤α4≤2 and α5=0.3.
(ii)
It is positive-negative valued for flat and closed Universe whereas it is negative-positive-negative valued for open Universe in0.3≤α4≤0.8 and α5=0.3.
(iii)
It is positive valued for flat and closed Universe whereas it is negative-positive valued for open Universe in0.8<α4≤1 and α5=0.3. Also it approaches towards infinity with the evolution of time in the specified interval of α4.
(iv)
Π is positive valued for flat and closed Universe whereas it is negative-positive-negative valued for open Universe in α4>2 and α5=0.3.
Cosmological Constant  Λ  (see Figure17)
(i)
For0<α4<0.3 and α5=0.3, Λ is positive valued for flat and closed Universe whereas it is negative value for open Universe. In case of flat Universe Λ→0 when t→∞ but for open and closed Universe Λ→∞ and Λ→-∞ with time respectively.
(ii)
For0.3<α4≤0.9 and α5=0.3, Λ→0 when t→∞ for open, flat, and closed Universe. In case of flat and closed Universe, cosmological constant is positive valued whereas in open Universe it is negative-positive valued.
(iii)
It is positive valued for flat and closed Universe but it is negative-positive-negative valued for open Universe inα4>0.9 and α5=0.3.


## 3.3. Model III

The deceleration parameterq in (6) for α2<0 and α3=0 takes the form(37)q=-α51+t,α2=-α5,α5>0.Here we noticed that q<0 for α5>0, which means that our Universe is accelerating with the evolution of time.For model III, the physical parameters are obtained as follows:The Hubble parameter in (12) takes the form(38)H=1t-α5ln⁡1+t.The scale factor Rt in (16) is expressed as(39)R=c2t1/1-α5eT1t,where T1t=k0t+k1(t2/2)+k2(t3/3)+k3(t4/4)+k4(t5/5)+Ot6. And (40)k0=-α521-α52k1=11-α5α5241-α52+α531-α5k2=-11-α5α541-α5+α5231-α52+α5381-α53k3=11-α5α551-α5+13α52361-α52+α5341-α53+α54161-α54k4=-11-α5α561-α5+11α52301-α52+17α53481-α53+α5461-α54.The FRW space-time metric in (3) takes the form (41)ds2=dt2-c22t2/1-α5e2T1tdr21-kr2+r2dθ2+sin2⁡θdϕ2with the above mentation ki,i=0,1,2,3,4. The energy density ρ, pressure p, bulk viscous stress Π, and cosmological constant Λ in (17), (18), (19), and (20) takes the form(42)ρ=k5tα1/1-α5eα1T1tρ1+ρ2t1+tt-α5ln⁡1+t2,where k5=ϕ0c2α1/8π,ρ1=3+2w+wα5α1-3α5,ρ2=3+2wα1.(43)p=Ak5n+1tn+1α1/1-α5en+1α1T1tρ1+ρ2tn+1-B1+tn+1t-α5ln⁡1+t2n+2k5ntnα1/1-α5enα1T1tρ1+ρ2tn1+tt-α5ln⁡1+t2k5n+1tn+1α1/1-α5en+1α1T1tρ1+ρ2t1+tt-α5ln⁡1+t2n×Π=Π1+Π2t/1+tt-α5ln⁡1+t2+2k/c22t2/1-α5e2T1t+Bk5ntnα1/1-α5enα1T1tρ1+ρ2t/1+tt-α5ln⁡1+t2n,where Π1=-1+wα12+1+3Aα5-α5+21+Aw+1+3A+α5α1+2 and Π2=-1+wα12-α5+21+Aw+1+3Aα1+2.(44)Λ=Λ1+Λ2t1+tt-α5ln⁡1+t2+3kc22t2/1-α5e2T1t,where Λ1=-0.5wα12+wα5+2α1-3α5+1 and Λ2=-0.5wα12-2wα1+3.The profile of deceleration parameter, Hubble parameter, and scale factor against time is plotted in the Figures18, 19, and 20, respectively, for model III. The observations are as follows:(i)
Deceleration parameterq is a negative valued function of time and approaches towards zero with the evolution of time. In other words we can say, at early time, our Universe is accelerating and follows an expansion with constant rate at late time (see Figure 18).(ii)
Hubble parameterH is a decreasing function of time and H→0 when t→∞. Also in this case, the higher the value of α5, the higher the value of Hubble parameter (see Figure 19).(iii)
Scale factorR is an increasing function of time and R→∞ when t→∞. Equation (39) indicates that R is not defined for α5=1. As a representative case, we considered 0<α5<1 (see Figure 20).Figures 21 and 22 depict the energy density and pressure profile against time. respectively. For α5≥1, energy density possesses physical unrealistic behaviour, so α5 is restricted to 0<α5<1. It is noticed that energy density is a decreasing function of time and ρ→0 when t→∞ (see Figure 21). Also pressure is a negative quantity with the evolution of time (see Figure 22).Figure 18
Variation of deceleration parameter against time for differentα5.Figure 19
Variation of Hubble parameter against time differentα5.Figure 20
Variation of scale factor against time for differentα5.Figure 21
Variation of energy densityρ against time for α1=0.5, ω=1, ϕ0=1, c2=0.1, and different α5.Figure 22
Variation of pressurep against time for α1=0.5, A=1, B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α5. (b) shows the zooming of (a).
(a)
(b)The profile of bulk viscous stress and cosmological constant against time is depicted in Figures23 and 24, respectively, for model III. Bulk viscous stress is positive valued for flat and closed Universe whereas it is negative valued for open Universe. Similar quantitative behaviour is observed for cosmological constant. In case of flat Universe, cosmological constant is a decreasing function of time and tends to zero with the evolution of time.Figure 23
Variation of bulk viscous stressΠ against time for α1=0.5, A=1, B=1, n=0.1, ω=1, ϕ0=1, c2=0.1, and different α5. (b) shows the zooming of (a).
(a)
(b)Figure 24
Variation of cosmological constantΛ against time for α1=0.5, ω=1, c2=0.1, and different α5. (b) shows the zooming of (a).
(a)
(b)


## 4. Final Statements

In this article, we have studied the FRW cosmological model with modified Chaplygin gas in the framework of Brans-Dicke theory. The approximated exact solution is obtained for modified Einstein’s field equation with the help of proposed form of deceleration parameter as in (6). We have presented three different cosmological models based on the choice of α2 and α3. The physical parameters involved in these three models are physically acceptable for some interval of α2 and α3, which follow the observational data. Here we would like conclude that, for physically acceptable cosmological models, the choice of α2 and α3 is crucial.



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