(2+1) dimensional cosmological models in f(R; T) gravity with Λ (R; T)
aa r X i v : . [ phy s i c s . g e n - ph ] M a r (2+1) dimensional cosmological models in f ( R, T ) gravity with Λ( R, T ) Safiqul Islam a , Praveen Kumar b , G.S. Khadekar c and Tapas K Das da,d Harish-Chandra research Institute, HBNI, Chhatnag Road, Jhunsi,Allahabad-211019, India b,c
Department of Mathematics, Rashtrasant Tukadoji MaharajNagpur University, Nagpur 440033, India a safi[email protected] b [email protected] c [email protected] d [email protected] Abstract.
We intend to study a new class of cosmological models in f ( R, T ) modified theoriesof gravity, hence define the cosmological constant Λ as a function of the trace of the stressenergy-momentum-tensor T and the Ricci scalar R , and name such a model Λ( R, T ) gravitywhere we have specified a certain form of Λ(
R, T ). Λ(
R, T ) is also defined in the perfect fluidand dust case. Some physical and geometric properties of the model are also discussed. Thepressure, density and energy conditions are studied both when Λ is a positive constant andwhen Λ = Λ( t ), i.e a function of cosmological time, t. We study behavior of some cosmologicalquantities such as Hubble and deceleration parameters. The model is innovative in the sensethat it has been described in terms of both R and T and display better understanding of thecosmological observations.
1. Introduction
It is known that General Relativity is the standard theory of gravity. However alternativeequations for the gravitational field are resorted to, when the spacetime dimension is reducedto 2 + 1, due to the difficulty in defining a proper Newtonian limit [1]. The 2 + 1 dimensionalanalogue is generated by any circularly symmetric matter distribution. Circularly symmetricdistributions of charged matter in 2 + 1 dimensions are detailed in [2], where the authors havenoticed that hydroelectrostatic equilibrium for a charged fluid with equation of state p = p ( ρ )with p > θ to be proportional to the shear scalar σ [5]. Bengochea andFerraro proposed to replace the TEGR that is the torsion scalar Lagrangian T with a function f ( T ) of the torsion scalar, and studied its cosmological consequences [6]. This type of modifiedgravity is nowadays called as f ( T ) gravity theory.The authors in [7] have studied two particular models of f ( R, T ) gravity namely, f ( R ) + λT and R + 2 f ( T ) and derived their power-law solutions in homogeneous and isotropic f ( R, T )cosmology.The cosmological constant was introduced by Einstein who later rejected it after expansionnature of the universe was discovered by Hubble. The results of type Ia supernova [[8] ,[9]]show that the universe is accelerating rather than decelerating. These results suggest that ouruniverse can have a non-zero cosmological constant. In GR however it is the dark energy whichbehaves like a cosmological constant at early time and supports the accelerated expansion ofthe universe. We can remove this consideration of dark energy in our modified f ( R, T ) gravitymodel.It is observed that the violation of energy conditions in modified gravity indicates theattractive nature of gravity whereas repulsive gravity may occur for ordinary matter that satisfiesall the energy conditions [10].The organization of this paper is envisaged as follows: In section II. we generate new solutionsunder f ( R, T ) gravity and deduce the cosmological constant as a function of the Ricci scalar R , the trace of the energy momentum tensor T and another constant λ . The field equationsand solutions of the generalized metric in (2+1)-Dimensional Spacetime are shown in sectionIII. Here we study the energy conditions and various consequences when Λ is treated both as aconstant and as a function of cosmic time t, i.e Λ = Λ( t ). We find the scale factor, the Hubbleand deceleration parameters and observe their changes with time, t. The study ends with aconcluding remark.
2. Generating new solutions under f ( R, T ) models: We consider the modified gravity action as, S = 116 π Z ( √− gf ( R, T ) + √− g L m ) d x, (1)where f(R,T) is an arbitrary function of the Ricci scalar, R, T is the trace of the stress-energymomentum tensor, T µν given by T = g µν T µν and L m stands for the matter Lagrangian density. T µν is defined as, T µν = − √− g δ √− g L m δg µν , (2)Several models of f(R,T) gravity have been studied by various authors, depending on thematter source. They may be illustrated as follows,( i ) f ( R, T ) = R + 2 f ( T )( ii ) f ( R, T ) = f ( R ) + f ( T )( iii ) f ( R, T ) = f ( R ) + f ( R ) f ( T )( iv ) f ( R, T φ ) (3)where φ is a scalar field.he cosmological intricacy in (i) where f ( R, T ) = R + 2 f ( T ) has been discussed by authors[11], [12]. The cosmological model in (ii) is developed in [13] to effectively discuss the transitionfrom the matter dominated phase to an accelerated phase. With the model as (ii ), a new classof solutions pertaining to some cosmic scale functions have been investigated in [14], where theauthors study the cosmological constant Λ as a function of T .We consider the f(R,T) model given by (iii). The gravitational field equations with perfectfluid matter source are given by [15],[ f ′ ( R ) + f ′ ( R ) f ( T )] R µν − f ( R ) g µν + ( g µν - ∇ µ ∇ ν )[ f ′ ( R ) + f ′ ( R ) f ( T )]= 2 πT µν + f ( R ) f ′ ( T ) T µν + f ( R )[ f ′ ( T ) p + 12 f ( T )] g µν (4)Here T µν is the energy-momentum tensor of matter fields in the space-time, = ∇ i ∇ i and theprime denotes differentiation with respect to the argument. We consider f ( R ) = λ R, f ( R ) = λ R, f ( T ) = λ T with λ = λ = λ = λ .Considering ( g µν - ∇ µ ∇ ν ) = 0, eqn.(4) reduce significantly as, λ [ R µν − Rg µν ] = 2 πT µν + ( λ p + 12 λ T ) Rg µν (5)or as, G µν − λ ( p + 12 T ) Rg µν = 2 πλ T µν (6)Equating eqn.(4) with G µν − Λ g µν = − πT µν , (7)we get Λ( R, T ) = λ ( p + 12 T ) R (8)We have chosen a sufficiently small negative value of λ to keep in parity the same sign on theRHS of eqns.(6) and (7). The value of λ is considered to remain unchanged throughout. Theabove eqn. in the dust case p = 0, reduce asΛ = Λ( R, T ) = 12 λRT (9)In the case of perfect fluid [14] T = ρ − p when eqn.(8) reduce asΛ = 12 ρλR (10)In the case of dust universe [15] T = ρ when eqn.(9) reduce asΛ = 12 ρλR (11)In fact we observe that the cosmological constant, if treated as a function of cosmic time ’t’remains invariant both for perfect fluid as well as in dust case.We may think that the expansion of the universe is accelerating due to the positive sign ofcosmological constant. Also if one solves the Einsteins general theory of relativity equation witha positive cosmological constant, one obtains a solution for spacetime with positive Gaussiancurvature. So, we may consider to be in a de-Sitter universe. Here for a positive scalar curvature, R > ρ > − acordingto present observational data. . Field equations and solutions of the generalized metric in (2+1)-DimensionalSpacetime: In (2 + 1) dimensional gravity, we consider the metric, ds = dt − e A ( t ) dx − e B ( t ) dy , (12)where A(t) and B(t) are functions of t alone.The energy momentum tensor for perfect fluids is given by, T µν = ( ρ + p ) u µ u ν − pg µν (13)where the velocity vector u µ = (0 , ,
1) satisfying u µ u ν = 1 and u µ ∇ ν u µ = 0 , ρ and p aredefined as the energy density and pressure of the fluid respectively.Now the cosmological eqn.(6) for the energy momentum tensor defined in eqn.(13) and themetric in eqn.(12), give rise to the field equations as,˙ A ˙ B = − πλ ρ − Λ˙ B + ¨ B = 2 πλ p − Λ˙ A + ¨ A = 2 πλ p − Λ R = −
2( ˙ A + ¨ A + ˙ B + ¨ B + ˙ A ˙ B ) (14)Here a ‘.’ denotes differentiation with respect to the cosmic time t and R is the Ricci scalar.We consider some cases for generating new solutions: Case (i)(a) : Let B ( t ) = k t n + k t n ,where k and k are positive constants and n is a real number. The following relations thenfollow from eqn.(13) A ( t ) = k t n + k t n ¨ B = ¨ A = k n ( n − t n − + k n ( n + 1) t − n − (15)We derive the pressure and density as, p ( t ) = λ π [Λ + k n ( n − t n − + k n ( n + 1) × t − n − + ( k nt n − − k nt − n − ) ] ρ ( t ) = − λ π [Λ + ( k nt n − − k nt − n − ) ] (16)The Ricci scalar as a function of t from eqn.(13) is given as, R ( t ) = 4 π ( ρ − p ) λ (1 − ρλ ) (17)We observe from the Figures.(1-2) that the pressure is negative throughout and p → t → ∞ whereas the energy density is a positive decreasing function and ρ → t → ∞ . Thecosmological constant is taken as positive and Λ = 10 − . As it has very meagre value, the igure 1. The pressure is plotted against t for the values of constants, λ = − . k = − k =1, n = 0 . − (taking the cosmological constant as having constant value). Figure 2.
The density is plotted against t for the values of constants, λ = − . k = − k =1, n = 0 . − (taking the cosmological constant as having constant value). Figure 3.
The energy conditions are shown against t for the values of constants, λ = − . k = − k = 1, n = 0 . − (taking the cosmological constant as having constant value).gures does not affect any change if it is considered as having negative value as Λ = − − oreven when Λ = 0. (i)(b) : Now we thrive to search for the energy conditions which are given as,(i) Null energy condition (N.E.C) : ρ + p ≥ ρ + p ≥ ρ ≥ ρ ≥ | p | (iv) Strong energy condition (S.E.C): ρ + p ≥ ρ + 2 p ≥ (i)(c) : If however, the cosmological constant is treated as a function of t , i.e Λ = Λ( t ) thenwe find it from eqns.(9) and (16) as,Λ( t ) = − [ λ n t − n − ( k t n − k ) ( − k t n k nt n + 3 k nt n + 2 k t n + 2 k nt n − k k nt n + 3 k n ) × [ − πt n +2 − λ k nt n + 2 k λ n t n + 3 k λ n t n + 2 k λ nt n +2 k λ n t n − k k λ n t n + 3 k λ n ] − (18)It is shown graphically below, We observe that the cosmological constant is positive and Figure 4.
The cosmological constant is shown against t for the values of constants, λ = − . k = − k = 1, n = 0 . → t → ∞ . We observe the disparity between the expectedand observed values of Λ. This is the cosmological constant problem which requires furtherinvestigation to explain why it constant assumes such a meagre value today. The correspondingpressure and density terms are, p ( t ) = λ π [ k n ( n − t n − + k n ( n + 1) × t − n − + ( k nt n − − k nt − n − ) −{ λ n t − n − ( k t n − k ) ( − k t n k nt n + 3 k nt n + 2 k t n + 2 k nt n − k k nt n + 3 k n ) × ( − πt n +2 − λ k nt n + 2 k λ n t n + 3 k λ n t n + 2 k λ nt n +2 k λ n t n − k k λ n t n + 3 k λ n ) − } ] ρ ( t ) = − λ π [( k nt n − − k nt − n − ) − { λ n t − n − ( k t n − k ) ( − k t n k nt n + 3 k nt n + 2 k t n + 2 k nt n − k k nt n + 3 k n ) × ( − πt n +2 − λ k nt n +2 k λ n t n + 3 k λ n t n + 2 k λ nt n + 2 k λ n t n − k k λ n t n + 3 k λ n ) − } ] (19)or Λ = Λ( t ) the corresponding pressure, density and energy conditions are shown as below, Figure 5.
The pressure is plotted against t for the values of constants, λ = − . k = − k =1, n = 0 . Figure 6.
The density is plotted against t for the values of constants, λ = − . k = − k =1, n = 0 . Figure 7.
The energy conditions are shown against t for the values of constants, λ = − . k = − k = 1, n = 0 . i)(d) : Scale factor a(t), Hubble parameter H(t) and deceleration parameter q(t) withΛ = Λ( t ):The average scale factor a(t), Hubble parameter H(t) and deceleration parameter q(t) usingeqn.(14) are derived as [16]. a ( t ) = e A ( t )+ B ( t )2 H ( t ) = ˙ aa = ˙ B = k nt n − − k nt − n − q ( t ) = − (1 + ˙ HH ) = −{ k n ( n − t n − + k n ( n + 1) t − n − ( k nt n − − k nt − n − ) } (20) Figure 8.
The scale factor is shown against t for the values of constants, λ = − . k = − k =1, n = 0 . Figure 9.
The Hubble parameter is shown against t for the values of constants, λ = − . k = − k = 1, n = 0 .
4. Final Remarks
We have thus developed a feasible cosmological model under f ( R, T ) gravity. Various physicalaspects of the model are elucidated. Present observational results suggest that our universe can igure 10.
The deceleration parameter is shown against t for the values of constants, λ = − . k = − k = 1, n = 0 . p = ωρ , where − < ω < ω = −
1. Such form of energy-a generalization of the notion of a cosmological constant is known as dark energy. On the otherhand if ω = − , we get a quintessence field which is a hypothetical form of dark energy, moreprecisely a scalar field and postulated as an explanation of the observation of an acceleratingrate of expansion of the universe.
5. Acknowledgments
SI is thankful to S Datta, Md. A Shaikh and P. Tarafdar for providing some useful insightsin the paper.
References [1] Romero C. and Dahia F., Int. J. Theor. Phys., , 2019 (1994).[2] Cataldo M. and Cruz N., Phys. Rev. D, , 104026 (2006).[3] Barrow J. D. et al., Class. Quantum Grav. , 2845 (2014).[5] Shamir M. F., Astrophys. Space Sci., , 183-189 (2010).[6] Bengochea G. R. and Ferraro R., Phys. Rev. D, , 124019 (2009).[7] Sharif M. and Zubair M., J. Phys. Soc. Jpn. , 014002 (2013)[8] Perlmutter, S., et al., Nature,
51 (1998).[9] Reiss, A.G, et al., Astron. J. , 280-283 (2014).[11] Pawar D. D. et al., Aryabhatta Journal of Mathematics and Informatics, , 17 (2015).[12] Houndjo, M.J.S. et al., Can. J. Phys., , 548 (2013).[13] Houndjo, M.J.S., Int. J. Mod. Phys. D, , 1250003 (2012).[14] Ahmed N. et al., NRIAG Journal of Astronomy and Geophysics, , 35-47 (2016).[15] Harko T., Phys. Rev. D, , 024020 (2011).[16] Yadav A. K. et al., Int. J. Theor. Phys.,50