Anisotropic bulk viscous string cosmological models of the Universe under a time-dependent deceleration parameter
aa r X i v : . [ phy s i c s . g e n - ph ] J un Anisotropic bulk viscous string cosmological models of theUniverse under a time-dependent deceleration parameter
ARCHANA DIXIT , RASHID ZIA , ANIRUDH PRADHAN , , Department of Mathematics, Institute of Applied Sciences & Humanities, GLAUniversity,Mathura -281 406, Uttar Pradesh, India E-mail:[email protected] E-mail:[email protected] Corresponding Author E-mail:[email protected]
Abstract
We investigate a new class of LRS Bianchi type-II cosmological models by revisitingin the paper of Mishra et al (2013) by considering a new deceleration parameter (DP)depending on the time in string cosmology for the modified gravity theory suggested byS´ a ez & Ballester (1986). We have considered the energy-momentum tensor proposedby Leteliar (1983) for bulk viscous and perfect fluid under some assumptions. To makeour models consistent with recent astronomical observations, we have used scale factor(Sharma et al et al a ( t ) = exp [ β √ βt + k ], where β and k arepositive constants and it provides a time-varying DP. By using the recent constraints( H = 73 .
8, and q = − .
54) from SN Ia data in combination with BAO and CMBobservations (Giostri et al , arXiv:1203.3213v2[astro-ph.CO]), we affirm β = 0 . k = 0 . et al (2013) and observed that the results in this paper are muchbetter, stable under perturbation and in good agreement with cosmological reflections. Keywords : String cosmology; S´ a ez-Ballester theory; Bulk viscosity; Transit Universe. Pacs No.:
The current astronomical reflexions, modern experimental data from SNe Ia [1] − [6]; CMBR[7, 8]; WMAP [9] − [12] have established two main characteristics of the universe: (a) theexistence of the anisotropic universe at the early stage of the evolution, which in due courseof time attains isotropy, and (b) the current universe is not only expanding but also the rateof expansion is increasing (i.e. accelerating universe). The SNe Ia measurements indicate auniverse which undergoes through a transition from past decelerating to present acceleratingexpansion. So, it is a challenge for theoreticians to provide satisfactory theoretical supportto these observations. 1riedmann-Robertson-Walker (FRW) spacetime describes spatially homogenized and isotropicuniverses, which can be appropriate for the contemporary universe, however since they havehigher symmetries, thus it doesn’t provide a correct matter description of the early universeand presents the poor approximations for an early universe. Therefore, those models areadditional applications for the outline of the whole evolution of the universe, that have ananisotropic nature in early time and approaches to isotropy at late times. Bianchi space-times offer a decent framework for this. Out of all, Bianchi type-II (B-type-II) frame ofreference plays a very important affirm in making models for the measurements of flour-ish of the universe throughout its early phase. Moreover, B-type-II line-element yields ananisotropic spatial curvature. Recently, Asseo and Sol [13] and Roy & Banerjee [14] stressedthe importance of B-type-II and proscribed LRS cosmological model. Kumar and Akarsu [15]mentioned B-type-II universe with anisotropic dark energy and perfect fluid. Wang [16] − [18]has investigated the models of Letelier-type within the theoretical account of LRS B-type-II.In the context of massive string, Pradhan et al. [19] have analyzed LRS Bianchi type-IIspacetime. B-type-II frame of reference is employed to analyze dark energy models withinthe new role of time-dependent DP by Maurya et al [19]. Within the present study, we tendto look into LRS B-type-II string models of the universe for perfect and viscous fluid beneaththree conditions.Next, although Einstein’s general theory of gravity (GR) explains a large number of theastrophysical phenomenon, it fails to describe some, for instance, the expanding and latetime accelerated the expansion of the universe. To deal with these, many alternative theoriesare proposed, out of which, Brans & Dicke [21] and S´ a ez & Ballester [22] scalar-tensor theo-ries are of significant involvement. In the present paper, we have studied the S´ a ez-Ballestermodified theory of gravity. In this theory, Einstein’s field equations have been modified byincorporating a dimensionless scalar field φ coupled with the metric g ij in a simple manner.This modification satisfactorily describes the weak field in which an accelerated expansionregime appears. This theory also advises an answer to the question of disappeared matter ina non-flat FRW universe.In recent years, string cosmology is gaining significant interest. Cosmic strings are topo-logically stable objects, which could be shaped throughout a phase transition within the earlyuniverse. Cosmic strings make for a significant role to study in the early universe. It is as-sumed that cosmic strings bring about to density perturbations, that cause the formation ofgalaxies or cluster of galaxies [23]. One more necessary feature of the string is that the stringtension provides rise to an efficient anisotropic pressure. Also, the stress-energy of the stringcoupled with the gravitational field is also used to elucidate several alternative cosmologicalphenomena. The pioneer works in string theory were done by many authors [24, 25]. LRSB-type-II cosmological models have been discussed [26] − [29] in different context. Recently,Pradhan et al [30] have looked into string models of accelerated expansion in f ( R, T )-gravitywith a magnetic field.Also, the dissipation effect together with bulk viscosity presents another model of darkenergy. Relaxation processes related to bulk viscosity effectively reduce the pressure in anexpanding system, in comparison the worth prescribed by the equation of state p = ωρ . Theeffective pressure becomes negative for a sufficiently large viscosity that could imitate a darkenergy behavior. The idea that the bulk viscosity drives the acceleration of the universe is2entioned in [31, 32].In recent years, many researchers [33] − [38] and references therein have investigated thecosmological universes in Saez-Ballester modified gravity theory in various contexts. Underabove-discussed perspective, the S´ a ez & Ballester field equations have been solved in an LRSB-type-II space-time in the presence of a cloud of massive string and bulk viscous fluid, undersome physically and geometrically viable assumptions. In the present paper, we are revisitingthe solutions obtained by Mishra et al [39], by assuming a scale factor a ( t ) = exp [ β √ βt + k ]which resulting into a time-dependent DP having a transition from the decelerating universeto presently accelerating universe.The plan of the manuscript is the following. Section 2 contains definitions and theoreticalcalculations. Subsec. 2 . . .
1, 3 . .
3. Results and discussions are given in Sec. 4. Stability ofcorresponding solutions is analyzed in Sect. 5. Finally, conclusions are summarized in Sec.6.
We consider an LRS B-type-II space-time [39]: ds = − dt + X dx + Y dy + 2 X xdydz + ( Y x + X ) dz (1)where X = X ( t ), Y = Y ( t ).The field equation (in gravitational units 8 πG = 1) proposed by S´ a ez & Ballester [22]: G ij − ωφ r ( φ ,i φ ,j − g ij φ ,k φ ,k ) = − T ij . (2)Here G ij = R ij − Rg ij and T ij stands for the energy-momentum tensor and φ for the scalarfield satisfying the equation rφ r − φ ,k φ ,k + 2 φ r φ ,i ; i = 0 (3)Here ω and r stand for a dimensionless coupling and arbitrary constant respectively. Acomma denotes the partial derivative whereas a semi-colon denotes partial covariant differ-entiation w. r. to t . T ij , for a cloud of massive string & bulk viscous fluid, reads: T ij = pg ij − λx i x j + ( ρ + p ) v i v j , (4)where p = p − Hξ (5)3n above Eqs. (4) and (5) the different quantities have there usual meaning as alreadydescribed in [39]. The four velocity of the particles v i = (0 , , ,
1) and a unit space-likevector x i representing the direction of string satisfy g ij v i v j = − g ij x i x j = − , v i x i = 0. InLRS Bianchi type-II metric, the mean Hubble parameter H can be defined as H = ˙ aa = 13 XX + ˙ YY ! = 13 (2 H + H ) . (6)h Here H = ˙ XX and H = ˙ YY are directional Hubble parameters in the directions of x and y axes respectively. Here a = a ( t ) is average scale factor, which, for LRS B-type-II model, iswritten as a ( t ) = ( X Y ) (7)The particle density denoted by ρ ρ follows the relation ρ = ρ ρ + λ (8)For the Metric, (1), the S´ a ez-Ballester field equations (2) & (3), along with energy-momentum tensor given by (4), we obtain the following system of field equations¨ XX + ¨ YY + ˙ XX ˙ YY + 14 Y X − ωφ r ˙ φ = λ − p (9)2 ¨ XX + ˙ X X − Y X − ωφ r ˙ φ = − p (10)˙ X X + 2 ˙ XX ˙ YY − Y X − ωφ r ˙ φ = ρ (11)¨ φ + ˙ φ XX + ˙ YY ! + r φ φ = 0 (12)In the usual notation, expansion scalar θ and the shear scalar ( σ ) are defined and given as θ = v i ; j = 3 ˙ aa = 2 ˙ XX + ˙ YY (13)and σ = 12 σ ij σ ij = 12 " X X + ˙ Y Y − θ (14)where σ ij = v i ; j + ( v i ; k v k v j + v j ; k v k v i ) + θ ( g ij + v i v j )The anisotropy parameter ( A m ) is defined as A m = 6 (cid:16) σθ (cid:17) = 2 σ H (15)4 .2 Assumptions There are four equations (9)-(12) having seven unknowns
X, Y, φ, p, ρ, ξ and λ . For deter-ministic solutions of this system, we have to take three more equations, which relates theseparameters.As suggested by Thorne [40] and followed by many researchers [41, 42], we first, assume θ is proportional to σ which gives1 √ XX − ˙ YY ! = ℓ XX + ˙ YY ! (16)where ℓ is the constant of proportionality. This yields˙ XX = m ˙ YY , (17)where m = √ ℓℓ − √ . We have select m > m = 1, as thestudy presents a picture of FRW model for m = 1. Integrating Eq. (17) and we get X = c ( Y ) m , (18)where c is a constant of integration. Any loss of generality and for simplicity, c = 1 isconsidered. Hence Eq. (18) is reduced to X = ( Y ) m (19)Secondly, we consider q as linear function of Hubble parameter [43] − [46]: q = − a ¨ a ˙ a = βH + α = β ˙ aa + α. (20)Here α , and β stand for arbitrary constants. Eq. (20) renders as a ¨ a ˙ a + β ˙ aa + α = 0, which bysolving proceeds as a = exp (cid:20) − (1 + α ) β t − α ) + lβ (cid:21) , provided α = − . (21)Here l is a constant of integration.From Eq. (21), we calculate˙ a = − (cid:18) αβ (cid:19) exp (cid:20) − (cid:18) αβ (cid:19) t (cid:21) − α ) + lβ , ¨ a = (cid:18) αβ (cid:19) exp (cid:20) − (cid:18) αβ (cid:19) t − α ) + lβ (cid:21) . (22)Eqs. (20) and (22) render the value of DP as q = −
1. We also observed the same valueof DP for α = 0. 5or α = −
1, Eq. (20) is changed into the form: q = − a ¨ a ˙ a = − βH, (23)Eq. (23) reproduced the following differential equation: a ¨ a ˙ a + β ˙ aa − . (24)The solution of above equation is found to be (Sharma et al. 2019, Garg et al. 2019) a = exp (cid:20) β p βt + k (cid:21) , (25)where k is an integrating constant. Eq. (25) is recently used by different authors [43] − [46]in different contexts.For the study of cosmic decelerated-accelerated expansion of the universe, we only con-sider the case α = − q ) and Hubble parameter H is given as q = − β √ βt + k , H = 1 √ βt + k . (26)From Eq. (26), we observe that q > t < β − k β and q < t > β − k β . By using Eqs. (19), (25) and (7), we obtain: X = ( e β √ βt + k ) m m +1 , (27) Y = ( e β √ βt + k ) m +1 . (28)From Eqs. (12), (27) and (28), we evaluate scalar field ( φ ) as φ ( t ) = " r + 22 φ Z dt ( e β √ βt + k ) + φ ! r +2 , (29)where φ and φ are integrating constants.Solving Eqs. (9)-(11) by using Eqs. (27)-(29), we obtain energy density ρ , effectivepressure p and string tension density λ as ρ = " m ( m + 2)(2 m + 1) (2 βt + k ) −
14 ( e β √ βt + k ) − m m +1 + 12 ωφ ( e β √ βt + k ) , (30)6 = " − m (2 m + 1) (2 βt + k ) + 6 mβ (2 m + 1) (2 βt + k ) − +34 ( e β √ βt + k ) − m m +1 + 12 ωφ ( e β √ βt + k ) , (31) λ = " − m + 9 m + 9(2 m + 1) (2 βt + k ) − β ( m − m + 1)(2 βt + k ) +( e β √ βt + k ) − m m +1 . (32)Accordingly, the particle density ρ p is obtained as ρ p = " (27 + 9 m − m + 1) (2 βt + k ) −
54 ( e β √ βt + k ) − m m +1 +12 ωφ ( e β √ βt + k ) − β ( m − m + 1)2 βt + k ) . (33)For calculating the other parameters, we shall consider the following three cases. p = αρ Considering perfect gas equation of state as: p = αρ, (34)where α (0 ≤ α ≤
1) is a constant. For the various values of α , we will get three types ofmodels:(i) if α = 0, we tend to get matter dominant model.(ii) if α = , we tend to get radiation dominant model.if α = 1, we get ρ = p which is termed as Zel’dovich fluid or stiff fluid model [47].Therefore by Eqs. (5) and (34), we can directly calculate the following values of ( p ) and( ξ ) : p = " mα ( m + 2)(2 m + 1) (2 βt + k ) − α e β √ βt + k ) − m m +1 + α ωφ ( e β √ βt + k ) (35) ξ = " (3 m α + 6 mα + 9 m )(2 m + 1) √ βt + k − ( α + 312 )( e β √ βt + k ) − m m +1 p βt + k − mβ (2 m + 1)(2 βt + k ) + ( α −
16 ) ωφ √ βt + k ( e β √ βt + k ) (36)7 .2 Case II: Bulk Viscous Model with ξ = ξ ρ n For most of the investigations, we found that the coefficient of bulk viscosity ξ is consideredas a simple power function of energy density and it depends on time. It is assumed, ξ = ξ ρ n , (37)where ξ and n are real constants [48] − [50]. For small density and radiative fluid, n maybe equal to 1 [51, 52]. For (0 ≤ n ≤ /
2) is good assumption to obtain realistic results, asgiven by Belinskii and Khalatnikov (1975).Using Eqs. (5), (30), (31) and (37), the expressions for ξ and p are given as: ξ = ξ " m ( m + 2)(2 m + 1) (2 βt + k ) −
14 ( e β √ βt + k ) − m m +1 + 12 ωφ ( e β √ βt + k ) n (38) p = " ξ √ βt + k " m ( m + 2)(2 m + 1) (2 βt + k ) −
14 ( e β √ βt + k ) − m m +1 +12 ωφ ( e β √ βt + k ) n − m (2 m + 1) (2 βt + k ) + 6 mβ (2 m + 1) (2 βt + k ) − + 34 ( e β √ βt + k ) − m m +1 + 12 ωφ ( e β √ βt + k ) (39) ξ = 0 For perfect fluid, the coefficient of bulk viscosity is assumed to be zero. The rest of sixunknowns
X, Y, φ, p, ρ and λ can be directly calculated from the field Eqs. (9)-(12). For ξ = 0, Eq. (5) gives p = p i.e. effective pressure equals to isotropic pressure, and theexpression is given by p = " − m (2 m + 1) (2 βt + k ) + 6 mβ (2 m + 1) (2 βt + k ) − +34 ( e β √ βt + k ) − m m +1 + 12 ωφ ( e β √ βt + k ) (40) From Eq. (26), the present value of declaration parameter can be taken as q = − βH = − β √ βt + k , where H and t have their usual meaning.By using the recent constraints ( H = 73 .
8, and q = − .
54) from SN Ia data in com-bination BAO and CMB observations [54], we concentrate the values of β = 0 . k = 0 . q versus t , (b) The plot of H versus t , Here β = 0 . k = 0 . q with respect to cosmic time t inFigure 1( a ), and observed that deceleration parameter is positive at early time and negativeat present time indicating that our models are evolving from decelerating phase ( q >
0) toaccelerating phase ( q < q for k = 0 . β = 0 . t c = β − k β . Also, when t → ∞ , q → −
1. According to SNe Ia observa-tion, the universe is accelerating at present and the value of DP lies in the range − < q < b ) shows the variation of Hubble parameter H with respect to cosmic time t asper Eq. (26). We see that H is a positive, decreasing function of time, and tends to zero as t → ∞ , which totally agrees with the established theories.The average scale factor a ( t ) in terms of redshift z is given by a ( t ) = a z , where a is thepresent value of the average scale factor a ( t ).From Eq. (25), we can get a = exp h β q βH + k i . Using the values of β = 0 . , k =0 . H = 73 . a = 8 . √ βt + kβ = ln ( a ). Also from a ( t ) = a z , we have ln ( a ) = ln ( a ) − ln (1 + z ).Substituting the above in Eq. (21), we get q ( z ) = − ln ( a ) − ln (1 + z ) (41). 9a) (b)Figure 2: (a) The plot of redshift z versus cosmic time t , (b) The plot of decelerationparameter q versus redshift z ,. Figure 2( a ) shows the fluctuation of redshift z with cosmic time t for our derived models.From the figure, we see that the redshift z is a monotonic decreasing function of cosmic time t for the present value a = 8 . z starts with a small positive value 5 .
16 at t = 0 and z → − t → ∞ for our derived models. So, we can say that t → ∞ corresponds to z → − b ), we have shown the fluctuation of q concerning for redshift z as per Eq.(41). From this figure, we see that as the redshift z decreases, the DP q is changing its phasefrom positive (decelerating phase) to negative (accelerating phase) and q → − z → − f ( T ) cosmology and observationalconstraints and cosmographic bounds on the cosmological deceleration-acceleration transi-tion redshift in F ( R ) gravity respectively.It was found in its analysis that the SNe data favor current acceleration ( z < .
5) andpast deceleration ( z > . z t = 0 . ± .
13 at (1 σ ) c.l. [3] which has been further analyzed to z t = 0 . ± . σ ) c.l. [3]. According to SNLS [57], as well as the one recently compiled by [58],yield a transition redshift z t ∼ . σ ) in better agreement with the flat ΛCDM model( z t = (2Ω Λ / Ω m ) − ∼ . . ≤ z t ≤ .
18 (2 σ , joint analysis) [59].Further, the transition redshift for our derived model comes to be z t ∼ = 1 .
965 (see Fig. 2 b )which is in good agreement with the Type Ia supernovae observations, including the farthestknown supernova SNI997ff at z ≈ . q versus z obtained in our model is compatible with the results obtained in the above references.10a) (b)(c)Figure 3: (a) Plot of energy densities versus cosmic time t , (b) The plot of ρ p λ versus cosmictime t , (c) The plot of effective pressure p versus cosmic time t . Here ω = φ = 1 , m =0 . , β = 0 . k = 0 . a ), we have shown three curves of total energy density ρ , particle energydensity ρ p and string tension density λ . We see that all the energy densities are positivedecreasing functions of time showing expanding universe. All the energy densities approachto zero as t → ∞ , meaning that the universe will keep on expanding forever. Also, we seethat λ < ρ p for an early phase of the evolution i.e., particle dominates over the string, andthen λ > ρ p in due course of evolution i.e., the string dominates over the particle thereafter.The comparative behavior of particle density ρ p and string tension density λ is also stud-ied in figure 3( b ). From figure 3( b ) we see that the ratio ρ p λ > ρ p λ > ρ p > λ i.e. the particle dominatedphase. But, as the time progresses, the ratio falls below 1 indicating the string dominatedphase. These observations are supported by Krori [61] and Kibble [62].In Figure 3( c ), we have plotted the variation of the effective pressure p concerning cosmictime t as per Eq. (31). We see that p is negative at present, which may be seed for currentaccelerated expansion of the universe.In Figure 4( a ) we have plotted the behavior of isotropic pressure p for case-I when p = αρ for three scenarios α = 0 (dust filled), α = 1 / α = 1 (stiff matterfilled) universe. In all the cases we find that the isotropic pressure is a positive decreasingfunction of time.In Figure 4( b ) we consider case-II when ξ = ξ ρ n and plotted p ( t ) for three values of n = 0 , / n , the isotropic pressure p isagain a positive decreasing function of time. We also observed in both the case-I and case-II11hat p → t → ∞ .In case-III, when ξ = 0 (i.e., in the absence of viscous effect) the isotropic and effectivepressures become equal. The behavior of the effective pressure is graphed in Figure 4( c ). Wesee that in the absence of viscosity the effective pressure becomes highly negative at the earlytime then increases and tends to a small negative value at late time.(a) (b)(c)Figure 4: (a) Plot of isotropic pressure p versus t for case I, (b) Plot of isotropic pressure p versus t for case II, (c) Plot of isotropic pressure p versus t for case III. Here ω = φ =1 , m = 0 . , β = 0 . k = 0 . a ) we have plotted the of bulk viscosity coefficient ξ for case-I when p = αρ for three scenarios α = 0 , / ξ is a positive decreasingfunction of time. In the early universe, it was high and after that it reduces gradually andtends to zero as t → ∞ . So, we can say that the nature of the fluid was highly viscous atthe time of the early universe which tends to reduce and vanish in due course of time. InFigure 5( b ) we consider case-II when ξ = ξ ρ n and plotted ξ ( t ) for three values of n = 0 , /
2& 1. Here also, we observe the same behavior for the two values of n = 1 / n = 0 the viscous effect vanishes throughout the evolution of the universe.12a) (b)Figure 5: (a) Plot of viscosity parameter ξ versus t for case I, (b) Plot of viscosity parameter ξ versus t for case II. Here ω = φ = 1 , m = 0 . , β = 0 . k = 0 . θ ), Volume scalar ( V ), shear scalar σ andanisotropy parameter( A m ) and directional Hubble parameters ( H and H ) are obtained as θ = 3 H = 3 √ βt + k (42) V = A B = e β √ βt + k (43) σ = − (cid:20) m + 1)(2 m + 1) (2 βt + k ) (cid:21) − √ βt + k (44) A m = 2 m − m + 2(2 m + 1) (45) H = mH = 3 m (2 m + 1) √ βt + k (46)In Big Bang scenario all the parameters like shear scalar ( σ ), expansion scalar ( θ ), andHubble parameter ( H ) are finite. From Eq. (43) Spatial volume ( V ) is zero at t = 0. As t → ∞ , V becomes infinite whereas θ , H , and σ approach to zero. We have tested the stability of the background solution w.r.to perturbations of the metric.For the study, we adopt the notation a i for the metric potentials. (ie. a = A and a = B ).The stability analysis is performed against the perturbations of all possible fields. Thestability of the solution has been first discussed by Chen and Kao [63]. Here perturbationwill be considered for the two expansion factor a i via a i → a B i + δa i = a B i (1 + δb i ) . (47)where δa i = a B i δb i . 13ccordingly, the perturbations of the volume scalar, directional Hubble factors, the meanHubble parameter are shown as follows: V → V B + V B X i =1 δb i , H i → H B i + δ ˙ b i ,H → H B + 13 X i =1 δ ˙ b i , X i =1 H i → X i =1 H B i + 2 X i =1 H B i .δb i (48)It can be derived that metric perturbations δb i is to linear order in δb i obey the followingequations Σ δ ¨ b i + 2Σ H B i δ ˙ b i = 0 (49)Σ δ ¨ b i + 2 ˙ V B V B δ ˙ b i + Σ δ ˙ b j H B i = 0 (50)Σ δ ˙ b j = 0 (51)From the above three equations, It can easily be seen that δ ¨ b i + ˙ V B V B δ ˙ b i = 0 (52)where V B is the background volume scalar and in this model, it is given by V B = e β √ βt + k (53)We can calculate the δb i with the help of Eq. (51), then we get δb i = e − β √ βt + k (cid:18) − p βt + k − β (cid:19) + c (54)here c is constant of integration. Therefore the actual fluctuations for each expansionfactor δa i = a B i δb i are given by δa i = a B i e − β √ βt + k (cid:18) − p βt + k − β (cid:19) + c (55)From Eq. (55) and Figure (6), we observed that for positive value of β and k, δa i approachesto zero for large t i.e. t → ∞ , δa i →
0. Consequently, the background solution is stableagainst the metric perturbation.If δb i tends to zero, from Eq. (48), we see that V → V B , H i → H B i , H → H B so we cansay that our solution is stable against the perturbation of volume scale, directional Hubbleand average Hubble parameters also. 14igure 6: Plot of metric perturbation δa i versus t . Here β = 0 . , k = 0 . c = 0 . The present study contributes to the exact solutions of the scalar-tensor theory of gravita-tion described by S´ a ez & Ballester. It is worth mentioned here that the scalar field φ plays asignificant role in the expression for the physical quantities p, ρ, p, ξ and ρ p . We find a pointtype singularity in the derived models as p, ρ, λ, ρ p diverge at t → ∞ .The model shows a phase transition from an early decelerating to present the acceleratingexpansion of the universe. The phase transition took place at z = 1 . ≈
2. Recently, Hayes et al [65] and Dunlop [65] use the comparison of Lyman- α and H- α luminosity functions todeduce the range of redshift, which currently is feasible at z ≈
2. Thus, z = 1 .
965 in ourderived models is consistent with observational value [64, 65].Also, our derived models are stable under perturbations.So, we may conclude that our models are improved from earlier works and it presents abetter picture of the universe. So it deserves attention.From Figures 4( a ), ( b ) and ( c ) we observed that the isotropic pressure, in the presence ofthe bulk viscosity λ is a decreasing function of time t and approaches to zero at late time butin the absence of bulk viscosity the presence is always negative and tends to zero at presenttime. Thus, we see the role of bulk viscosity for the evolution of the universe.Lastly, we conclude that our derived models deserve attention and show a better shapeof the universe. Acknowledgments
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