Bulk viscous cosmological model in f(R,T) theory of gravity
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International Journal of Geometric Methods in Modern Physicsc (cid:13)
World Scientific Publishing Company
Bulk Viscous Cosmological model in f ( R, T ) theory of gravity
Partha Sarathi Debnath
Department of Physics, A. B. N. Seal CollegeCoochbehar, West Bengal, Pin-736101, [email protected]
Received (Day Month Year)Revised (Day Month Year)In this paper, we have presented bulk viscous cosmological model of the universe in themodified gravity theory in which the Lagragian of the gravitational action contains ageneral function f ( R, T ), where R and T denote the curvature scalar and the trace of theenergy-momentum tensor respectively, in the framework of a flat Friedmann-Robertson-Walker model with isotropic fluid. We obtain cosmological solution in f ( R, T ) theory ofgravity, specially of particular choice f ( R, T ) = R + 2 λT , where λ is a constant, in thepresence of bulk viscosity that are permitted in Eckart theory, Truncated Israel Stewarttheory and Full Israel Stewart theory. The physical and geometrical properties of themodels in Eckart, Truncated Israel Stewart theory and Full Israel Stewart theory arestudied in detail. The analysis of the variation of bulk viscous pressure, energy density,scale factor, Hubble parameter and deceleration parameter with cosmic evolution aredone in the respective theories. The models are analyzed by comparison with recentobservational data. The cosmological models are compatible with observations. Keywords : cosmology; f ( R, T ) gravity; viscosity.
Mathematics Subject Classification 2010 : 83F05, 83D05, 35D40.
1. Introduction
Recent data [1] from cosmological observations suggest that the universe might bepassing through an accelerating phase of evolution. Although the general theory ofrelativity is a very successful theory to describe most gravitational phenomena forthe evolution of the universe, however, standard cosmological models with perfectfluids of standard fluid forms, that is radiation, matter, stiff matter fail to addressproperly the late-time acceleration of the universe. The late-time acceleration [2]problem along with the dark matter problem are the most difficult challenges tomodern gravitational theory. The late-time acceleration is attributed to a nega-tive pressure fluid dubbed as dark energy [3], but what is the dark energy for themoment we have absolutely no idea. The estimate from the recent cosmologicalobservations, PLANCK Collaboration 2015 classifies dark energy which plays a sig-nificant role to drive the acceleration, as consisting of ∼ ∼ ∼ uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Partha Sarathi Debnath
Quite a few theoretical proposals [4] came up to understand the exact fundamen-tal nature of the dark energy. Phenomenological models [5,6] appear by modifyinggravitational sector and/or matter sector of Einsteins field equation to study geo-metrical and physical features of the different phases of the universe. The modifiedtheories of gravity with proper curvature correction are also considered to accountfor dark energy. Literature [7,8,9] also discussed the reasons why modified gravityapproach is extremely attractive in the applications for late-time acceleration ofthe universe and to understand the problem of dark energy. A number of modifiedgravity theories, namely, f ( R ) (where R being the Ricci scalar curvature) [10], f ( T )(where T being the trace of the stress-energy tensor) [11,12], Horava-Lifshitz [13]and Gauss-Bonnet [14] theories have been recently proposed. Recently, Harko etal. [15] have introduced another extension of general relativity (GR) called f ( R, T )modified theory of gravity, where gravitational Lagrangian is given by an arbitraryfunction of R and T . In recent times such a modification of Einstein’s theory isfound to describe some of the observed features relevant for cosmology and astro-physics [16,17]. In f ( R, T ) gravity, literature [18] takes into account an interactingcosmological fluid described by generalized Chaplygin gas with viscosity . Relativis-tic cosmological solutions and the different phases of the universe with non-causalviscous fluid are studied [19] in f ( R, T ) gravity theory. In f ( R, T ) gravity finite-time future singularities are discussed [20]. The energy condition in f ( R, T ) gravitytheory by incorporating conservation of energy-momentum tensor is studied in theliterature [21] and it analyzed that T sector can not be chosen arbitrarily but it hasspecial form. Cosmological solution in modified f ( R, T ) gravity theory with Λ( T )gravity has been studied in literature [22] also. Cosmological solutions relevant inquantum era have been studied [23] in f ( R, T ) gravity. In f ( R, T ) gravity theorythe behavior of gravitational wave is considered in the literature [24]. The station-ary scenario between dark energy and dark matter is also studied [25] in f ( R, T )theory.It has been shown that viscosity is one of the significant aspects to study evolu-tion of the universe. A number of dissipative processes [26,27] in the early universeguide to departure from perfect fluid assumptions [28,29], which permit the pres-ence of viscosity [30,31] and is, therefore, important to explore cosmological solutionwith viscosity [32,33] to study the different phases of evolution of the universe. Inthe early universe, viscosity may originate due to various processes e.g., decouplingof matter from radiation during the recombination era, particle collisions involv-ing gravitons and formation of galaxies [34]. A non-negligible bulk viscous stressis also important at late-time evolution of the universe [35,36] as predicted by ob-servations. Eckart [37] first formulated a relativistic theory of viscosity. Howeverthe theory of Eckart suffers from shortcomings, namely, causality and stability [38].Subsequently, Israel and Stewart [39] developed a fully relativistic formulation ofthe theory which is termed as transient or extended irreversible thermodynamics (in short
EIT ) which provides a satisfactory replacement of the Eckart theory. Usinguly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt
Bulk Viscous Cosmological model in f ( R, T ) theory of gravity the transport equations obtained from EIT , cosmological solutions are explored inEinstein’s gravity [40,41,42]. Thus it is important to explore cosmological solutionswith barotropic fluid and viscosity described by the transport equation obtainedfrom
EIT in f ( R, T ) theory of gravity.In this paper we study cosmological solutions in the presence of the bulk vis-cosity described by Eckart theory, Truncated Israel Stewart theory and Full IsraelStewart theory in f ( R, T ) theory of gravity. The plan of this paper is as follows: insec. 2, we give the relevant field equations in f ( R, T ) theory of gravity. In sec. 3,cosmological solutions are presented. In sec. 4, observational data fitting are con-sidered to study constraint over model parameters. Finally, in sec. 5, we summarizethe results obtained.
2. Field Equations in f ( R, T ) gravity theory
In the f ( R, T ) theory gravity formalism action is given by [15] I = Z d x √− g (cid:18) f ( R, T ) + L m (cid:19) , (1)where we consider 8 πG = 1 , c = 1. The f ( R, T ) is an arbitrary function of Ricciscalar, R , and T (= g µν T µν ) is the trace of the energy-momentum tensor T µν . L m is the matter Lagrangian density of the matter field. The energy-momentum tensorof matter [43] is expressed as T µν = − √− g ( √− gδL m ) δg µν . Assuming that the matterLagrangian density depends only on the metric tensor g µν , not on its derivative,one is led to write T µν = g µν L m − ∂L m ∂g µν . The variation of gravitational action,given by Eq. (1), with respect to metric tensor g µν yields f R R µν − f ( R, T ) g µν + [ g µν ∇ µ ∇ µ − ∇ µ ∇ ν ] f ( R, T ) = T µν − f T [ T µν + Θ µν ] , (2)where f R and f T denote the derivative of f ( R, T ) with respect to R and T respec-tively and Θ µν is defined asΘ µν ≡ g αβ δT αβ δg µν = − T µν + g µν L m − g αβ ∂ L m ∂g µν ∂g αβ . (3)To simplify highly nonlinear field equation we require a specific form of f ( R, T ). Inthis paper we consider f ( R, T ) = R + 2 f ( T ), where f ( T ) is an arbitrary functionof the trace of the energy-momentum tensor of matter. The term f ( T ) modifies thegravitational interaction between matter and curvature. For the particular choice f ( R, T ) = R + 2 f ( T ), gravitational field equation yields R µν − Rg µν = T µν + f ( T ) g µν − f † ( T ) [ T µν + Θ µν ] , (4)where f † ( T ) denotes the derivative of f ( T ) with respect to T . We consider theflat homogeneous and isotropic space-time given by Friedmann-Robertson-Walker(FRW) metric ds = − dt + a ( t ) (cid:2) dr + r ( dθ + sin θdφ ) (cid:3) , (5)uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Partha Sarathi Debnath where a ( t ) is the scale factor of the universe. In the present study we define theenergy-momentum tensor of the metric as given by T µν = ( ρ + ¯ p ) u µ u ν − ¯ pg µν , (6)where ρ is energy density of the universe, ¯ p is the effective pressure, u µ is the fourvelocity and u µ u µ = 1. The matter Lagrangian density can be taken as L m = − ¯ p and the trace of the total energy momentum tensor is given by T = ρ − p . Theexpression of Θ µν is given by Θ µν = − T µν − ¯ p g µν . (7)Using eqs. (7) and (4), the gravitational field equations are given by R µν − Rg µν = T µν + 2 f † ( T ) T µν + (cid:2) pf † ( T ) + f ( T ) (cid:3) g µν . (8)The reconstruction of arbitrary FRW cosmologies is possible by an appropriatechoice [15] of the function f ( T ). The simplest cosmological model can be obtainedby choosing the function f ( T ) so that f ( T ) = λT , where λ is a constant actingas a coupling parameter between geometry and matter. The field equations for aparticular choice of the function f ( T ) = λT , where λ is an arbitrary constant, yield3 H = (1 + 3 λ ) ρ − λ ¯ p, (9)2 ˙ H + 3 H = λρ − (1 + 3 λ )¯ p, (10)where H = ˙ aa is the Hubble parameter and an over-dot represents derivative withrespect to cosmic time ( t ). Let the effective pressure (¯ p ) contain two parts : ¯ p = p +Π,where p is the isotropic pressure of the universe and Π ( ≤
0) is the bulk viscouspressure. Here we consider linear equation of state (EoS) of the cosmological fluid,i.e., p = ωρ, (11)where ω (1 ≥ ω ≥
0) represents EoS parameter. Using Eqs. (9)-(11), we obtainrespective expressions of energy density and bulk viscous pressure, which are givenby ρ = 31 + 4 λ H − λ (1 + 2 λ )(1 + 4 λ ) ˙ H, (12)Π = − ω )1 + 4 λ H − λ − ωλ )(1 + 2 λ )(1 + 4 λ ) ˙ H. (13)It is worthy to notice that one can recover the standard cosmological models withviscosity [32,44,45] from field Eqs. (12)-(13) for λ = 0. For physically viable cosmo-logical solutions in the presence of bulk viscosity one requires Π < | Π | << ρ .In the following section, we shall study cosmological solution in the presence of bulkviscosity and matter described by the linear EoS in the f ( R, T ) theory of gravity.uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt
Bulk Viscous Cosmological model in f ( R, T ) theory of gravity Different phases of evolution of the universe can be studied by any relevant cos-mological quantity like deceleration parameter. The deceleration parameter ( q ) isrelated to Hubble parameter ( H ) as q = ddt (cid:18) H (cid:19) − . The accelerating phases of evolution of the universe are obtained for negative valuesof the deceleration parameter ( q < q >
3. Cosmological Solutions
The bulk viscous stress satisfies following transport equation [37,39]Π + τ ˙Π = − ζH − ǫ τ Π H + ˙ ττ − ˙ ζζ − ˙ TT ! , (14)where the parameter ζ ( ≥
0) is the co-efficient of bulk viscosity, the parameter τ ( ≥
0) is the relaxation time and the constant ǫ has two values, either 0 or 1. Thetransport Eq. (14) reduces to Full Israel Stewart (FIS) theory for ǫ = 1, for ǫ = 0it reduces to Truncated Israel Stewart (TIS). One can recover Eckart theory for τ = 0. The positive entropy production due to bulk viscosity is confirmed by thepositive values of the co-efficient of bulk viscosity ( ζ ). The set of Eqs. (9)-(14) areemployed to obtain cosmological solutions. The systems of equations are not closedas the number of equations is less than the number of unknowns. It is known thatthe coefficient of bulk viscosity and relaxation time are, in general, functions oftime (or of the energy density). We, therefore, consider following relation [46,47,48] ζ = β ρ s , τ = β ρ s − , (15)where β ( >
0) and s ( >
0) are constants. It could be mentioned here that recentanalysis of B. D. Normann and I. Brevik [49,50] tend to favor the choice s = . Eckart Theory ( τ = 0 ): Using Eqs. (9)-(15), for τ = 0 we get2(1 + 3 λ − ωλ )(1 + 2 λ )(1 + 4 λ ) ˙ H + 3(1 + ω )1 + 4 λ H = 3 β " H λ − λ ˙ H (1 + 2 λ )(1 + 4 λ ) s H. (16)The evolution of the universe in Eckart theory can be obtained by using Eq. (16)which is highly nonlinear to obtain a general analytic solution. However, usingEq. (16), one can obtain numerical solutions in term of cosmic time of relevantparameter, such as deceleration parameter ( q ), Hubble parameter ( H ) and scalefactor ( a ) to study different phases of the universe studying in Eckart theory. Thelate-time behavior of the models is better revealed from the plot of decelerationparameter ( q ) vs redshift parameter ( z ). The Hubble parameter of the universe isuly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Partha Sarathi Debnath related to the redshift parameter as ( H = − z dzdt ). To study late-time evolutionof the universe, we rewrite Eq. (16) in terms of deceleration parameter ( q ) andredshift parameter ( z ), which yields2( q + 1)(1 + 3 λ − ωλ )1 + 2 λ − ω )+ 3 β ( H ) s − (1 + 4 λ ) s − (cid:20) λ ( q + 1)(1 + 2 λ ) (cid:21) s (1+ z ) s − = 0 . Here cosmic time ( t ) is related to redshift parameter ( z ) by the relation t = H − (1+ z ) [51], where the constant H has unit (Gyr) − . The lower values of the redshiftparameter ( z ) indicates late-time behavior of the universe and higher values ofthe redshift parameter ( z ) indicates early-time behavior of the universe. Figure (1)shows the plot of q vs z for different values of λ for a given set of other parameters. Itindicates that the values of the deceleration parameter ( q ) are lower for higher valuesof the coupling parameter λ at given instant of redshift parameter ( z ). For highervalues of coupling parameter ( λ ) the universe enters to the late-time accelerationphase earlier. The plot of scale factor a ( z ) vs redshift parameter ( z ) in Eckarttheory for different values of λ for a given set of other parameters, as in Fig (2),indicates that the scale factor of the universe increases more rapidly for highervalues of coupling parameter ( λ ). As the above Eq. (16) is highly non-linear andthe relativistic solution cannot be expressed in known general analytic form, weconsider special cases for simplicity. __ __ __ Λ=
0_ _ _ _ Λ= Λ=- Λ= Ω= Β= = H = - - z q Fig. 1. shows the plot of q vs z for Eckart theory in f ( R, T ) theory of gravity for different valuesof λ for a given set of other parameters. Power law model :
In this particular case scale factor of the universe exhibits power law expansion a ( t ) = a t D , where a and D are constants. For power law model, the expressionof energy density and bulk viscosity stress are given by ρ = ρ t − , Π = − Π t − , (17)uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Bulk Viscous Cosmological model in f ( R, T ) theory of gravity where ρ = D +6 λD +2 λ (1+4 λ )(1+2 λ ) D and Π = h D (1+ ω )(1+2 λ ) − λ − ωλ )(1+4 λ )(1+2 λ ) i D . For physicallypermissible (i.e., Π <
0) solutions we obtain following lower boundary limit of powerlaw exponent
D > λ − ωλ )(1+ ω )(1+2 λ ) . For power law expansion Eq. (16) becomes A + A t − s = 0 , (18)where A = 3 β (1 + 4 λ ) − s h D + λD λ i s and A = 2 − λ )(1+ ω ) D λ − ωλ . For s = Eq. (18) yields A = A = 0 , i.e., D = λ − ωλ )3(1+ ω )(1+2 λ ) and λ = − or β = 0 which isnot physically acceptable. For s = , Eq. (18) yields A + A =0, which leads to3 β (1 + 2 λ ) p (1 + 4 λ )1 + 3 λ − ωλ (cid:20) D + 2 λD λ (cid:21) + 2 − λ )(1 + ω ) D λ − ωλ = 0 . (19)Using Eq. (19) a power law accelerated universe D > > | Π | > ρ . Figure (3) shows that for power law evolution the exponent ( D ) ishigher for smaller value of EoS parameter ( ω ) and higher value of bulk viscousconstant ( β ) for a given value of other parameters in f ( R, T ) theory of gravity.So power law acceleration is suitable for smaller values of EoS parameter ( ω ) andhigher values of bulk viscous constant ( β ).3.1.2. Exponential model :
The exponential expansion ( a ( t ) ∼ exp[ H t ]) of the universe with viscosity may beobtained in f ( R, T ) gravity by setting H = H = const. In this case the equation(16) yields H = h β (1+4 λ ) − s (1+ ω )3 − s i − s . So an exponential acceleration of the universe in f ( R, T ) gravity is permitted for coupling parameter λ > − . The rate of exponentialacceleration is higher for smaller values of EoS parameter ( ω ) and higher values ofbulk viscous constant ( β ) for s < . However for s > , the rate of exponentialacceleration is higher for higher values of EoS parameter ( ω ) and smaller values ofbulk viscous constant ( β ). In exponential evolution the energy density ( ρ = const. )and bulk viscous stress remain constant (Π = const. ) parameters. Truncated Israel Stewart Theory ( ǫ = 0) : Using equations (9)-(15) for Truncated Israel Stewart (TIS) theory ( ǫ = 0) in a flatuniverse, we obtain¨ H + 3 (cid:20) ω + 3 λ + 2 ωλ λ − ωλ (cid:21) H ˙ H − λ )2(1 + 3 λ − ωλ ) H +1 β (cid:20) λ )(1 + ω ) H λ − ωλ ) + ˙ H (cid:21) " λ ) H − λ ˙ H (1 + 2 λ )(1 + 4 λ ) − s = 0 . (20)uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Partha Sarathi Debnath
The evolution of the universe in TIS theory for linear EoS in f ( R, T ) theory ofgravity can be obtained by using above Eq. (20) which is highly nonlinear to obtain ageneral analytic solution of known form of the scale factor of the universe. However,one can obtain numerical solution of the relevant parameter, such as decelerationparameter ( q ), the Hubble parameter ( H ) and scale factor ( a ( t )) in term of cosmictime to study different phases of the universe in TIS theory by using Eq. (20). Thelate-time behavior of the models is better revealed from the plot of decelerationparameter ( q ) vs redshift parameter ( z ). To study late-time evolution of the universewe rewrite the above Eq. (20) in terms of the deceleration parameter ( q ) and theredshift parameter ( z ), which yields q ′ + 2( q + 1) z − ω + 3 λ + 2 ωλ )(1 + q )(1 + 3 λ − ωλ )(1 + z ) − λ )2(1 + 3 λ − ωλ )(1 + z ) + (2 H ) − s β − s × (cid:20) λ )(1 + ω )2(1 + 3 λ − ωλ − ( q + 1) (cid:21) (cid:20) λ ) + 2 λ ( q + 1)(1 + 2 λ )(1 + 4 λ ) (cid:21) − s (1 + z ) − s = 0 . Here prime ( ′ ) represents derivative with respect to the redshift parameter ( z ).Figure (4) shows the plot of deceleration parameter ( q ) vs redshift parameter ( z )for different values of coupling parameter λ for a given set of other parametersin TIS theory. Figure (4) indicates that the values of the deceleration parameter( q ) are lower for higher values of the coupling parameter λ at given instant ofredshift parameter ( z ). For higher values of coupling parameter ( λ ) the universeenters into the late-time acceleration phase earlier. The plot of scale factor a ( z )vs redshift parameter ( z ) in TIS theory for different values of λ for a given set ofother parameters (as in Fig (5)) indicates that scale factor of the universe increasesmore rapidly for higher values of coupling parameter ( λ ). As the above Eq. (20) ishighly non-linear and the relativistic solution cannot be expressed in known generalanalytic form, we consider following special cases for simplicity.3.2.1. Power-law model :
In the power-law expansion ( a ( t ) = a t D , where a and D are constants) theexpressions of energy density and bulk viscosity stress becomes ρ = ρ t − , Π = − Π t − , (21)where the constant ρ = D +6 λD +2 λ (1+4 λ )(1+2 λ ) D and the constant Π = h D (1+ ω )(1+2 λ ) − λ − ωλ )(1+4 λ )(1+2 λ ) i D . A physically viable solution is permitted for Π < D > λ − ωλ )(1+ ω )(1+2 λ ) . Forpower law expansion Eq. (20) yields, B + B t s − = 0 , (22)where the constant B = ( λ − ωλ )(1+2 λ ) − ω ) D − λD λ − D ) and the constant B = ( ω ) Dβ − λ − ωλ ) β (1+2 λ ) )( D λ + λD (1+2 λ )(1+4 λ ) ) − s . For s = Eq. (22) yieldsuly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt
Bulk Viscous Cosmological model in f ( R, T ) theory of gravity B = B = 0, i.e., D = λ − ωλ )3(1+2 λ )(1+3 γ ) and λ = − which is not physically acceptable.For s = , one obtains B + B = 0 , which leads to4(1 + 3 λ − ωλ ) − ω )(1 + 2 λ ) D − λD − λ ) D + 1 p (1 + 4 λ ) β × [3(1 + ω )(1 + 2 λ ) D − λ − ωλ )] × (cid:20) D + 2 λD (1 + 2 λ ) (cid:21) = 0 . (23)Using Eq. (23) a power law accelerated ( D >
1) universe is obtained in TIS theoryfor different values of other parameters as shown in Fig. (6). In the Fig. (6), theshadow region corresponds physically unacceptable solution for power law expan-sion as in this region Π > | Π | > ρ . Figure (6) shows that for power lawevolution the exponent ( D ) is higher for smaller values of EoS parameter ( ω ) andhigher values of bulk viscous constant ( β ) in f ( R, T ) theory of gravity for a givenvalue of other parameters. So power law acceleration is suitable for smaller valuesof EoS parameter ( ω ) and higher value of bulk viscous constant ( β ).3.2.2. Exponential Model :
An exponential evolution ( a ( t ) ∼ exp[ H t ]) of the universe in f ( R, T ) gravity withTIS theory may be obtained by setting H = H = const. For the exponentialevolution Eq. (20) yields H = (cid:20) β (1 + 4 λ ) − s (1 + ω )3 − s (cid:21) − s . (24)The exponential accelerations of the universe in f ( R, T ) gravity with viscosity,described by TIS theory, are permitted for the coupling parameter λ > − . Therate of exponential acceleration is higher for smaller values of EoS parameter ( ω )and higher values of bulk viscous constant ( β ) for s < . However for s > ,the rate of exponential acceleration is higher for higher values of EoS parameter( ω ) and smaller values of bulk viscous constant ( β ). For exponential evolution in f ( R, T ) theory of gravity, the energy density ( ρ = const. ) and bulk viscous stress(Π = const. ) remain constant parameters. Full Israel Stewart Theory ( ǫ = 1) : Using Eqs. (9)-(15) for Full Israel Stewart (FIS) theory ( ǫ = 1) in a flat universe,we obtain the field equation¨ H + 1 + ω + 3 λ + 2 ωλ λ − ωλ H ˙ H − λ ) H λ − ωλ ) +1 β (cid:20) ω )(1 + 2 λ ) H λ − ωλ ) + ˙ H (cid:21) × " H λ − λ ˙ H (1 + 2 λ )(1 + 4 λ ) − s uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Partha Sarathi Debnath + 12 (cid:20) ω )(1 + λ ) H λ − ωλ ) + ˙ H (cid:21) × " H − ω − ω λ ) H ˙ H − λ ¨ H λ ) H − λ ˙ H = 0 , (25)where we consider barotropic behavior of temperature ( T ∼ ρ ω ω ). The evolution ofthe universe in FIS theory for linear EoS in f ( R, T ) gravity can be obtained by usingEq. (25), which is highly nonlinear, to obtain a general analytic solution. However,numerical solutions of relevant parameters such as the deceleration parameter ( q ),the Hubble parameter ( H ) and the scale factor ( a ) can be obtained for a given setof other parameters to study evolutions of the universe. The late-time behavior ofthe models is better revealed from the plot of deceleration parameter ( q ) vs redshiftparameter ( z ). To study late-time evolution of the universe we rewrite the aboveEq. (25) in terms of the deceleration parameter ( q ) and the redshift parameter ( z ),which yields q ′ + 2( q + 1) z − ω + 3 λ + 2 ωλ )(1 + q )(1 + 3 λ − ωλ )(1 + z ) − λ )2(1 + 3 λ − ωλ )(1 + z ) + (2 H ) − s β − s × (cid:20) λ )(1 + ω )2(1 + 3 λ − ωλ − ( q + 1) (cid:21) (cid:20) λ ) + 2 λ ( q + 1)(1 + 2 λ )(1 + 4 λ ) (cid:21) − s (1 + z ) − s + 12(1 + z ) × (cid:20) ω )(1 + λ )2(1 + 3 λ − ωλ ) − − q (cid:21) (cid:20) ω − ω (2( q + 1) + 2 λq ′ (1 + z )3(1 + 2 λ ) + 2 λ ( q + 1) ) (cid:21) = 0 . Figure (7) shows the plot of deceleration parameter ( q ) vs redshift parameter ( z )for different values of coupling parameter λ for a given set of other parametersin FIS theory. Figure (7) indicates that the values of the deceleration parameter( q ) are lower for higher values of the coupling parameter λ at given instant ofredshift parameter ( z ). For higher values of coupling parameter ( λ ) the universeenters into the late-time acceleration phase earlier. The plot of scale factor a ( z )vs redshift parameter ( z ) in TIS theory for different values of λ for a given set ofother parameters as in Fig (8) indicates that scale factor of the universe increasesmore rapidly for higher value of coupling parameter ( λ ). As the above Eq. (25) ishighly non-linear and the relativistic general analytic solution cannot be expressedin known form, we consider following special cases for simplicity.3.3.1. Power law model :
We can obtain the power-law expansion of the scale factor ( a ( t )) of the universein f ( R, T ) gravity with FIS theory by setting a ( t ) = a t D , where a and D areconstants. In power law model the expressions for energy density and bulk viscositystress become ρ = ρ t − , Π = − Π t − (26)where the constant ρ = D +6 λD +2 λ (1+4 λ )(1+2 λ ) D and the constant Π = h D (1+ ω )(1+2 λ ) − λ − ωλ )(1+4 λ )(1+2 λ ) i D . For physically permitted solutions Π < Bulk Viscous Cosmological model in f ( R, T ) theory of gravity shows the following lower boundary limit of power law exponent D > λ − ωλ )(1+ ω )(1+2 λ ) .For power law expansion Eq. (25) yields, C + C t s − = 0 , (27)where the constant C = 2 − D ( ω +3 λ +2 ωλ λ − ωλ ) − D (1+2 λ )2(1 − λ − ωλ ) + D ( ω )(1+2 λ ) D λ − ωλ ) − (1+2 ω )(6(1+2 λD )+2 λ )(1+ ω )(3(1+2 λD )+2 λ ) ) and the constant C = ( ω )(1+2 λ ) D β (1+3 λ − ωλ ) − β )( D λ + λD (1+2 λ )(1+4 λ ) ) − s . For s = Eq. (27) yields C = C = 0, i.e., D = λ − ωλ )3(1+2 λ )(1+3 γ ) and λ = − which is not physically acceptable. For s = , one obtains C + C = 0 , which leads to2 − D (cid:20) ω + 3 λ + 2 ωλ λ − ωλ (cid:21) − D (1 + 2 λ )2(1 − λ − ωλ ) + D × (cid:20) ω )(1 + 2 λ ) D λ − ωλ ) − (cid:21) × (cid:20) ω )(6(1 + 2 λD ) + 2 λ )(1 + ω )(3(1 + 2 λD ) + 2 λ ) (cid:21) + (cid:20) ω )(1 + 2 λ ) D β (1 + 3 λ − ωλ ) − β (cid:21) × (cid:20) D λ (3 D + 2 λ (1 + 2 λ ) ) (cid:21) = 0 . (28)Using Eq. (28) we can obtain power-law type accelerated ( D >
1) evolution for agiven set of other parameters in FIS theory as shown in Fig. (9). It is shown in Fig.(9) that the shadow regions are unsuitable for power law evolution as it correspondsto Π > | Π | > ρ . Power law exponent D is higher for smaller values of EoSparameter ( ω ) and higher values of bulk viscous constant ( β ) for given values ofother parameter as shown in Fig. (9). Power law acceleration is suitable for smallervalues of EoS parameter ( ω ) and higher values of bulk viscous constant ( β ).3.3.2. Exponential Model :
The exponential evolution with viscosity described by FIS theory may be obtainedin f ( R, T ) gravity by setting H = H = const. In this special case the energydensity ( ρ = const. ) and bulk viscous stress (Π = const. ) remain constants. Forexponential expansion the Eq. (25) yields H = (cid:20) β (1 − ω )(1 + 4 λ ) − s ω )3 − s (cid:21) − s . (29)An exponential acceleration of the universe is permitted in f ( R, T ) gravity of FIStheory for coupling parameter λ > − and EoS parameter ω = 1. The rate ofexponential acceleration is higher for smaller values of EoS parameter ( ω ) andhigher values of bulk viscous constant ( β ) for s < . However for s > , the rateof exponential acceleration is higher for higher values of EoS parameter ( ω ) andsmaller values of bulk viscous constant ( β ).uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Partha Sarathi Debnath
4. Constraints on models’ parameters from observational data :
To find out constraints among the different parameters ( a , H , D, λ ), we will fitthe models with Redshift-Magnitude Observation from New, Old and CombinedSupernova Data Sets [52]. The scale factor of the universe is related to redshiftparameter ( z ) as a ( t ) a ( t ) = z , where a ( t ) is a constant which may be consideredunity as a present value. For simplicity we consider power law and exponentialmodels of the universe. Power law model:
In power law model ( a ( t ) = a t D ) the luminous modulus is given by µ ( z ) = 5 log (cid:18) a (cid:19) D (cid:18) zD − (cid:19) (cid:16) (1 + z ) D − D − (cid:17)! + 25 , (30)where D = 1. For D = 1 i.e., neither acceleration or deceleration phase of evolutionof the universe the luminous modulus is given by µ = 5 log (cid:16) za ln(1 + z ) (cid:17) + 25.Using Eq. (30) with observational data we estimate the constraints among theparameters of the models in f ( R, T ) theory of gravity. Equation (30) is used to fitwith recent observational data [52]. The plot of z vs µ ( z ) for power law evolution ofthe universe is shown in Fig. (10). For power law evolution, it is evident that thereis reasonable fit with observational data for 2 . > D > . a = 1 × − . Exponential model:
In exponential model, the scale factor of the Universe yields a ( t ) ∼ e H t , where H is a constant parameter. The expression of luminous modulus for the Universe inthis case is given by µ ( z ) = 5 log (cid:18) z (1 + z ) H (cid:19) + 25 . (31)Equation (31) will be useful now to fit with observational data [52]. The plot of z vs µ ( z ), for exponential evolution to determine constrains for relevant parameter,is shown in Fig. (11). For exponential model, it is evident that the models fit wellwith observational data for 0 . × − > H > . × − , where H has unit( Gyr ) − .
5. Conclusion
In this paper an implementation of bulk viscous universe model in the frameworkof f ( R, T ) gravity with isotropic cosmological fluid described by linear equationof state ( p = ωρ ) has been done. Bulk viscosity described by Eckart, TruncatedIsrael Stewart (FIS) and Full Israel Stewart (FIS) theories are considered here tostudy geometrical and physical features of the universe in f ( R, T ) = R + 2 λT uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Bulk Viscous Cosmological model in f ( R, T ) theory of gravity theory of gravity, where λ is a constant acting as a coupling parameter betweenmatter and geometry. To find cosmological solutions, special cases are consideredfor non-linearity of field equations. We have studied numerical solutions, powerlaw solutions and exponential solutions in the above mentioned viscous theories.Numerical solutions for the evolution of the universe are studied by consideringcosmological parameters such as deceleration parameter ( q ) and scale factor ( a ).To study late-time behavior of the universe several plots of deceleration parameter( q ) vs redshift parameter ( z ) are given as shown in Fig (1), Fig. (4) and Fig. (7)for a given set of other parameters. The figures show how deceleration parameterdepends on redshift parameter for different values of coupling parameter ( λ ). Thefigures also show late-time acceleration is permitted in Eckart, TIS and FIS theory.For larger values of λ the universe enters into late-time acceleration phase earlierin Eckart, TIS and FIS theory. To study late-time behavior of the universe severalplots of scale factor ( a ) vs redshift parameter ( z ) are given in Fig (2), Fig. (5) andFig. (8) for a given set of other parameters. The scale factor increases more rapidlyfor larger value of λ at a given redshift parameter as shown in Fig (2), Fig. (5)and Fig. (8) in Eckart, TIS and FIS theory respectively. In the power law solution( a ( t ) ∼ t D ), the value of exponent ( D ) is higher for higher values of bulk viscousconstant ( β ) and smaller values of EoS parameter ( ω ) as shown in Fig. (3), Fig (6)and Fig. (9) for Eckart, TIS and FIS theory respectively in f ( R, T ) gravity. Powerlaw acceleration is suitable for smaller values of EoS parameter ( ω ) and highervalues of bulk viscous constant ( β ). The exponential evolution of the universe in f ( R, T ) gravity with Eckart, TIS and FIS theory is obtained for coupling parameter λ > − with constant energy density and bulk viscous stress. Redshift-Magnitudeobservations from supernova data are fitted with the theoretical models to obtainbest fit value of the different parameters as shown in Figs. (10)-(11). The powermodel fits with observational data for the values of power law exponent ( D ) in theboundary range 2 . > D > .
9. On the other hand, the exponential model fits withobservational data for the values of H , is the range 0 . × − > H > . × − ,here the unit of H is ( Gyr ) − .In conclusion, it has been observed that the scale factor increases rapidly in allviscous theories i.e., Eckart, TIS and FIS theories in f ( R, T ) gravity theory ascompared to the respective all viscous theories in standard models for any positivevalues of coupling parameter λ . It is also noted that for larger values of couplingparameter λ the universe enter into the late-time acceleration phase earlier. Acknowledgement
Author would like to thank the IUCAA Reference Centre at North Bengal Univer-sity for extending necessary research facilities to initiate the work. He would alsolike to thank unknown reviewers for their constructive critical comments to improveupon the quality of the article.uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Partha Sarathi Debnath
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Bulk Viscous Cosmological model in f ( R, T ) theory of gravity __ __ __ Λ=
0_ _ _ _ Λ= Λ=- Λ= Ω= Β= = H = @ D Š @ D Š- z a Fig. 2. shows the plot of a vs z for Eckart theory in f ( R, T ) theory of gravity for different valuesof λ for a given set of other parameters. __ __ __ Β= Β= Β= Β= Λ= P> Ω D Fig. 3. shows the plot of D vs ω for power-law evaluation for different values of β in Eckart theoryof f ( R, T ) gravity with λ = 0 . __ __ __ Λ=
0_ _ _ _ Λ= Λ=- Λ= Ω= Β= = H = - - z q Fig. 4. shows the plot of q vs z for TIS theory in f ( R, T ) theory of gravity for different values of λ for a given set of other parameters. uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Partha Sarathi Debnath __ __ __ Λ=
0_ _ _ _ Λ= Λ=- Λ= Ω= Β= = H = @ D = a ' @ D = a '' @ D = z a Fig. 5. shows the plot of a vs z for TIS theory in f ( R, T ) theory of gravity for different values of λ for a given set of other parameters. __ __ __ Β= Β= Β= Β= Λ= P> Ω D Fig. 6. shows the plot of D vs ω for Power-law evaluation in TIS theory for different values of β in f ( R, T ) gravity with λ = 0 . __ __ __ Λ=
0_ _ _ _ Λ= Λ= Λ= Ω= Β= = H = - - z q Fig. 7. shows the plot of q vs z for FIS theory in f ( R, T ) theory of gravity for different values of λ for a given set of other parameters. uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Bulk Viscous Cosmological model in f ( R, T ) theory of gravity _ _ _ _ Λ= Λ= Λ= Ω= Β= = H = @ D = a ' @ D = a '' @ D = z a Fig. 8. shows the plot of a vs z for FIS theory in f ( R, T ) theory of gravity for different values of λ for a given set of other parameters. __ __ __ Β= Β= Β= Β= Λ= P> Ω D Fig. 9. shows the plot of D vs ω for Power-law evaluation in FIS theory for different values of β in f ( R, T ) gravity with λ = 0 . > D > z Μ Fig. 10. shows the plot of µ vs z for supernova data and power law evolution of the universe for D = 2 . D = 1 . a = 1 × − . uly 5, 2019 0:33 WSPC/INSTRUCTION FILE vis-rt Partha Sarathi Debnath ´ - > H > ´ - z Μ Fig. 11. shows the plot of µ vs z for supernova data and exponential evolution of the Universewith H = 0 . × − (solid line) and H = 0 . × −3