Accelerating model of flat universe in f(R,T) gravity
11 Accelerating model of flat universe in f ( R, T ) gravity Nishant Singla a, , Mukesh Kumar Gupta b, , Anil Kumar Yadav c, a Department of Physics, Suresh Gyan Vihar University, Jaipur, India b School of Engineering & Technology, Suresh Gyan Vihar University, Jaipur, India c Department of Physics, United College of Engineering and Research, Greater Noida - 201306, India
Abstract
The f ( R, T ) theory of gravitation is an extended theory of gravitation in which the gravitational action containsboth the Ricci scalar R and the trace of energy momentum tensor T and hence the cosmological models based on f ( R, T ) gravity are eligible to describing late time acceleration of present universe. In this paper, we investigate anaccelerating model of flat universe with linearly varying deceleration parameter (LVDP). We apply the linearly timevarying law for deceleration parameters that generates a model of transitioning universe from early deceleratingphase to current accelerating phase. We carry out the state-finder and Om(z) analysis, and obtain that LVDP modelhave consistency with astrophysical observations. We also discuss profoundly the violation of energy-momentumconservation law in f ( R, T ) gravity and dynamical behavior of the model.
Kewwords:
Cosmological parameters; Modified theory of gravity; LVDP law; Energy conditions.
The recent observational data [1] on the late time ac-celeration of the universe and the existence of darkmatter have posed a fundamental theoretical challengeto gravitational theories. However the idea of modifi-cation of general relativity was not come to exist justafter the discovery of accelerating universe. Severalmodified theories of gravity such as Brans-Dike theory,scalar-tensor theory etc exist since a long time due tocombined motivation coming from cosmology and as-trophysics. After discovery of accelerating expansion ofuniverse, the attention of researchers towards modifiedgravity have been sought and the possibility that themodification of general relativity at cosmological scalescan explain dark energy and dark matter becomes anactive area of research since 2003 [2–6]. Harko et al. [7]have proposed a general non-minimal coupling betweenmatter and geometry by considering the effective gravi-tational Lagrangian consisting of an arbitrary functionof R and T where R and T denote the Ricci scalar andtrace of energy-momentum tensor. Thus, in f ( R, T )gravity, authors have justified, T as an argument forthe Lagrangian from exotic imperfect fluids. Thereforethe new matter and time dependent terms in gravita-tional field behaves like cosmological constant. Thusthe extra acceleration arises in f ( R, T ) gravity is notonly due to geometry of space-time but also from thematter content of universe. This extraordinary featuresof f ( R, T ) theory of gravitation has attracted many re-searchers to study and reconstruct this theory in variouscontexts of astrophysics and cosmology [8–17]. Some e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] interesting applications of f ( R, T ) gravity have beengiven in References [18, 19]. Also, in this conection, No-jiri et al. [20] have described inflation, bounce and latetime evolution in modified theory of gravity. Recently,Yadav et al [21] and Bhardwaj et al [22] have investi-gated the Bulk viscous embedded cosmological modelsin f ( R, T ) = f R ) + f R ) f T ) gravity.Today, it is well known that expansion of currentuniverse is in fact accelerating and it had evolved fromdecelerating expansion to accelerating expansion. How-ever, we still have no satisfactory explanation for thisfact that occur at energy scales (cid:39) − eV, wherewe supposedly know physics very well [23]. It is cus-tomary to note that the current accelerated expansionof universe, essentially requires either the presence ofan energy source in the context of General Relativity(GR) whose energy density decreases very slowly withthe expansion of universe [24] or a modification of GRfor describing gravitation at cosmological scales [25].Firstly, [30] have proposed linearly varying decelera-tion parameter (LVDP)law and later on, [31] have con-strained cosmological parameter of LVDP universe withH(z) + SN Ia data points. This study reveals that theLVDP model is only unbiased model about the futureof universe which ends with a big rip. Motivated fromstudies mentioned above, we assume that the decelera-tion parameter varies linearly with time to derived theaccelerating universe within the frame work of f ( R, T )gravity. We followed here the different mechanism withmodification in GR whereas [30] had produced LVDPcosmological model in GR. It is worth to note that withaid of T i.e. modification term of f ( R, T ) theory ofgravitation, EOS parameter, statefinder parameters areevolving within the range matches with recent observa- a r X i v : . [ phy s i c s . g e n - ph ] D ec tions. Recently, Moraes et al. [32] have given a new ap-proach for the conservation of energy momentum tensorin f ( R, T ) theory by choosing ordinary matter content.Thus there is need to explore f ( R, T ) theory with scalarfield however the theoretical and observational investi-gation of scalar field models is a challenging task in cos-mology. It is worth to note that f ( R, T ) theory is alsoapplicable to describe the effects of the modification ofEinstein gravity in the formulation of structure scalars.Some important applications and existence of strangestellar/compact objects within framework of f ( R, T )gravity are given in references [26–29]. It is important tonote that f ( R, T ) gravity is gravitationally responsiblefor mechanism of particle production [32–37]. RecentlyHarko et al [35] have showed that is a phenomenon oftransforming energy into momentum and vise-versa.In the present work, we extend the work carriedout by Akarsu and Dereli [30] by taking into account,the modification of GR. We investigate the f ( R, T ) = f ( R ) + 2 f ( T ) gravity model with linearly varying de-celeration parameter in FRW space-time. The paper isorganized as follows: in section 2, we present the basicmathematical formalism of f ( R, T ) = f ( R ) + 2 f ( T ). Insection 3, we have computed the physical and geomet-rical parameters of derived model. Section 4 deals withthe validation/violation of energy condition. In section5, we have discussed the violation of energy momen-tum conservation in f ( R, T ) = f ( R ) + 2 f ( T ) theoryof gravitation. In section 6 we have checked the viabil-ity of derived model through the analysis of statefinderparameters, Om(z) parameter, stability of derived so-lution and jerk parameter. The sum up of findings areaccumulated as conclusion in section 7. f ( R, T ) = f ( R ) + 2 f ( T ) for-malism The geometrically modified action in f ( R, T ) = f ( R ) +2 f ( T ) theory of gravitation is read as S = 12 κ (cid:90) [ f ( R )+2 f ( T )] √− gd x + (cid:90) L m √− gd x (1)where f ( R ) and f ( T ) are an arbitrary function of Ricciscalar R, and of the trace T of the stress-energy tensorof the matter T ij and κ = πGc .Here, the energy momentum tensor for perfect fluid dis-tribution, T ij = − pg ij + ( ρ + p ) u i u j is derived from thematter lagrangian L m . Following, Harko et al. [7], wechoose the matter Lagrangian as L m = − p , f ( R ) = R and f ( T ) = λT ; λ being the constant. p and ρ are theisotropic pressure and energy density respectively. u i is the four velocity of the fluid satisfying u i u i = 1 inco-moving co-ordinates. Thus the corresponding field equation is read as R ij − Rg ij = κ T ij + 2 ∂f∂T T ij + [ f ( T ) + 2 p ∂f∂T ] g ij (2)In this paper, we consider the flat FRW metric as ds = dt − a ( t )[ dx + dy + dz ] (3)where a ( t ) is the cosmic scale factor.In a coming co-ordinate system, the equation (2) and(3), read as3 H = (1 + 3 λ ) ρ − λp (4)2 ˙ H + 3 H = λρ − (1 + 3 λ ) p (5)In the above equations, H = ˙ aa is Hubble’s parameterand overhead dot denotes time derivatives. We havechosen the unit system such that κ = 1.The deceleration parameter is defined as q = − ¨ aa ˙ a = − − ˙ HH (6)Equations (5) and (6) lead to q = − λρ H + (1 + 3 λ ) p H + 12 (7)From equation (7), we observe that acceleration in uni-verse is possible when λρ H > (1+3 λ ) p + H H . Thus themodel under consideration is able to describe late timeacceleration of universe without inclusion of cosmologi-cal constant or dark energy component. Following, Berman [38] and Berman and Gomide [39]had proposed the special law for Hubble’s parametersthat yields the constant value of deceleration parame-ter. This power law cosmology corresponds to accel-erating expansion of universe. Later on, some authorshave investigated hybrid expansion law that describesthe expansion of universe from early deceleration phaseto current acceleration phase [31, 40]. In this paper, weassume a generalized linearly varying deceleration pa-rameter [30] in which the deceleration parameter is notconstant. q = − kt + m − k ≥ m ≥ a = a e m arctanh ( ktm − ) (9)where a is constant of integration.From equation (8), it is clear that if we set initialtime t i = 0 then q = m −
1. For m >
1, the uni-verse commence with decelerating expansion in initialphase but with the passage of time, the expansion ofuniverse undergoes from deceleration phase to acceler-ation phase. The dynamics of deceleration parameter( q ) and scalae factor ( a ) versus time are shown in Fig.1 & 2 respectively.The Hubble’s parameter is obtained as H = 2 t ( kt − m ) (10)The energy density and pressure are read as ρ = 12 t ( kt − m ) (1 + 4 λ ) − λ (1 + 2 λ )(1 + 4 λ ) × (cid:20) kt ( kt − m ) + 1 t ( kt − m ) (cid:21) (11) p = − t ( kt − m ) (1 + 4 λ ) − λ )(1 + 2 λ )(1 + 4 λ ) × (cid:20) kt ( kt − m ) + 1 t ( kt − m ) (cid:21) (12)From Fig. 3, it is clear that initially the energy den-sity is very high and it’s value decreases with passage oftime. The expression for pressure is given in equation(12) and it’s behavior is plotted in Figure 4.The density parameter (Ω) is given byΩ = − λ ) − λt ( kt − m ) λ )(1 + 4 λ ) × (cid:20) kt ( kt − m ) + 1 t ( kt − m ) (cid:21) (13)The dynamics of density parameter is shown in Fig. 4.It is evident that the overall density parameter behaveslike a flip-flop at t = 0 . t = m (cid:104) tanh (cid:16) m ln (cid:16) a a (1+ z ) (cid:17)(cid:17)(cid:105) k (14)The deceleration parameter q ( z ) in term of z can beobtained by using the relation a = a a z and (14); a isthe present value of scale factor. q ( z ) = q + k tanh (cid:18) k + q + 12 ln ( 11 + z ) (cid:19) (15) Here, q is the present value of deceleration parameter.Fig. 5 depicts the behavior of q ( z ) with respect to z for m = 1 .
25 and different values of k. The decelerationparameter starts from positive values and evolves upto some negative values q < − k is calculated by using some observationaloutcomes [41–43]. At present (z = 0), the universe isevolving with acceleration. In this section, we will apply the energy conditionsto our solution for the energy density and pressure.The main energy conditions in general relativity for theenergy-momentum tensor are expressed asNull energy condition ⇔ ρ − p ≥ ⇔ ρ ≥ ⇔ ρ + p ≥ ⇔ ρ + 3 p ≥ f ( R, T ) gravity. The above energy conditionshave been graphed in Fig. 6 for m = 1 .
25 and k = 0 . ρ + p = − λ ) (cid:20) kt ( kt − m ) + 1 t ( kt − m ) (cid:21) ≤ ρ − p = 24 t ( kt − m ) (1 + 4 λ ) + 4(1 + 4 λ ) × (cid:20) kt ( kt − m ) + 1 t ( kt − m ) (cid:21) ≥ ρ + 3 p = − t ( kt − m ) (1 + 4 λ ) − λ )(1 + 2 λ )(1 + 4 λ ) (cid:20) kt ( kt − m ) + 1 t ( kt − m ) (cid:21) ≤ a vs time. For f ( R, T ) = R + 2 λT theory of gravitation [46], wehave ∇ i T ij = − λ λ (cid:20) ∇ i ( pg ij ) + 12 ∇ i T (cid:21) (19)It is important to note that for λ = 0, ∇ i T ij = 0 andone should easily retrieves the case of GR. However, ingeneral, for f ( R, T ) gravity (i.e. λ (cid:54) = 0), the energy-momentum tensor is not conserved. Josset and Perez[47] have argued that the non-conservation of energymomentum may arise due to non unitary modificationsof quantum mechanics at Plank scale and shown that anon-conservation of energy momentum tensor leads toan effective cosmological constant which decrease withthe annihilation of energy during the cosmic expansionand can be reduced to a constant when matter densitydiminishes. In 2017, Shabani and Ziaie [48] have inves-tigated that the violation of energy - momentum conser- vation in f ( R, T ) theory of gravity can provide acceler-ated expansion. Later on, Shabani and Ziaie [49] haveconstructed a model of accelerating universe in whichan effective field is conserved in f ( R, T ) gravity ratherthan the usual energy-momentum tensor. In this paper,we have developed a LVDP model in f ( R, T ) theorywhich quantify the violation of energy-momentum con-servation through a deviation factor ∆, defined as∆ = ˙ ρ + 3 H ( ρ + p ) (20)Here, ∆ (cid:54) = 0.The value of ∆ may be positive or negative dependingon weather the energy flows into the matter field ei-ther in outward or inward direction respectively. In Fig.7, we have shown the non-conservation of energy mo-mentum for LVDP law. Fig. 7 depicts that the energy-momentum conservation is validated only for a very lim-ited period of time (1 . ≤ t ≤ . λ .Figure 3: ρ vs time.Figure 4: Density parameter (Ω) vs time.Table 1: The computed values of k from observational results q Source/Ref. red-shift k-0.5 Aviles et al. (2017) [JLA + Union 2.1] z = 0.15 0.75-0.7 Mukherjee and Banerjee (2017) [OHD + SNe + BAO] z = 0.29 0.95-0.9 Moresco et al. (2012) [WMAP + SN Ia] z = 0.55 1.15
Figure 5: q(z) versus z for m = 1 .
25 and different values of k.Figure 6: Validation/Violation of energy conditionsFigure 7: Non-conservation of energy-momentum.
The statefinders parameters [50] are obtained as r = ˙¨ aaH = 2 + 4 m + 6 kt + 3 k t − m (1 + kt )2( − m + kt ) × Exp (cid:18) − Arctanh (1 − kt/m ) m (cid:19) (21) s = r − q − / − + mt − kt × (cid:20) − g Exp (cid:18) − Arctanh (1 − kt/m ) m (cid:19)(cid:21) (22)where g = m +6 kt +3 k t − m (1+ kt )2( − m + kt ) .The remarkable feature of statefinders is that theseparameters depend on scale factor and its time deriva-tives and hence are geometric in nature. Fig. 8 ex-hibits the evolutionary trajectories of derived model ins-r plane. Moreover, the well known flat Λ CDM modelcorresponds to the points s = 0 and r = 1 in s-r plane.The blue dot in Fig. 8 at (s, r) = (0,1) shows the posi-tion of flat Λ CDM model.
The Om(z) parameter is read as Om ( z ) = (cid:16) H ( z ) H (cid:17) − z ) − H is the present value of Hubble’s parameter.The Om(z) parameter of derived model is given by Om ( z ) = (cid:104) m [1 − tanh [ m ln (1 / z )]] (cid:105) − z ) − m = 1 .
25 whereas theright panel explores the nature of Om(z) parameter inthe range 0 ≤ m ≤
3. In both the panel, Om(z) param-eter is negative and monotonically increasing within theinterval 0 ≤ z ≤ . In this section, we examine the stability of derived modelwith respect to the perturbation in scale factor as fol-lowing. a → a B + δa = a B (1 + δα ) (25)where δα = δaa B denotes the small deviation in per-turbed term δa and a B is the background scale factor. With reference to equation (25), the perturbations ofvolume scalar and expansion scalar are read as V → V B + V B δα, θ → θ B + θ B δα (26)where V B and θ B denote background volume scalarand expansion scalar respectively.Following, Saha et al ( [54]), δα satisfy the followingequations. δ ¨ α + ˙ V B V B δ ˙ α = 0 (27)The background volume scalar is given by V B = a = a e m arctanh ( ktm − ) (28)Integrating equation (27, we obtain δα = c − c e m arctanh ( ktm − ) t (29)where c and c are the constants of integration.Thus the actual fluctuation in the derived solutionis obtained as δa = δ e m arctanh ( ktm − ) − δ e m arctanh ( ktm − ) t (30)where δ = c a and δ = c a .The straightforward behaviour of actual fluctuation inthe derived solution is shown in Fig. 10. We observethat the value of δa was null at initial epoch i.e. t = 0and increase slowly with the evolution of universe andfinally approaches to a very small positive value. Thuswe conclude that the actual fluctuation in derived modelis very small which is not desirable to effect the physicalproperties of universe. The jerk, snap and lerk parameters [55–59] of derivedmodel are obtained as j = ˙¨ aaH = − k t m ( kt + 1) − kt − m − s = ¨¨ aaH = 3 k t − m (cid:0) k t + 6 kt + 2 (cid:1) +9 k t + m (12 kt + 11) + 6 kt − m + 1 (32) l = ˙¨¨ aaH = − k t − k t − m (cid:0) k t + 22 kt + 7 (cid:1) Figure 8: Dynamics of r & s .Figure 9: Om(z) parameter versus z for m = 1 .
25 (left panel) and 1 ≤ m ≤ − k t + 10 m (cid:0) k t + 9 k t + 6 kt + 1 (cid:1) +10 m (6 kt + 5) − kt − m − H , q and j . The gold sample of SN Iaobservational data predicts the value of jerk parame-ter as j = 2 . +0 . − . [61]. Recently, Amirhashchi &Amirhashchi [59] and Muthukrishna & Parkinson [62]have estimated the present values of j , s and l by usingdifferent observational data sets. In this paper, we have investigated an acceleratingmodel of flat universe in modified theory of gravitywithout inclusion of cosmological constant. Therefore, the cosmological constant problems are not associatedwith the model under consideration. According to theobservations and associated analysis, the universe un-dergoes an accelerated expansion in the present epoch.The universe might have transitioned from early de-celerating phase to current accelerating phase whichclearly indicate that the evolving deceleration parame-ter displays a signature flipping behavior that is why, inthis paper we have considered LVDP law to constructthe model of transitioning universe in f ( R, T ) gravity.The model straightforward consider the implicationsof Hubble’s parameter, density parameter and decel-eration parameter. Collectively, all these confirm thetransition of universe and matter-geometry approachclearly indicates the flipping of early deceleration phaseto acceleration at present epoch.At t = 0, both the scale factor and volume becomeszero which means the derived model have point typesingularity. Initially, q = m −
1, may be positive ornegative depending upon the value of m but for nonzero cosmic time (i. e. t = m/k), we obtain q = − δa versus t.Figure 11: Plot of jerk, snap and lerk parameter versus t.which shows asymptotic expansion. The plot of energydensity (Fig. 3) illustrates that ρ is very high at ini-tial time and it deceases sharply with passage of timeand ultimately approaches to zero at t → ∞ . Fig. 5exhibits the dynamics of density parameter (Ω) versustime.From Fig 5, it is analyzed that the present valueof deceleration parameter is in the range − . ≤ q ≤− .
90 for m = 1 .
25 and different value of free parame-ter k. Also we observe that for k = 0 . q = − . q matches with recent astrophysical ob-servations [63]. Hence, for all graphical analysis we havechosen k = 0 .
95 and m = 1 .
25. The weak and domi-nant energy conditions are satisfying the model becauseof the non-increasing and positive energy density whichdecreases with expansion of universe. The violation ofstrong energy condition confirms the accelerating ex-pansion of universe. We have discussed the violation of energy momentum tensor for derived model that lead toa sort of accelerated expansion of universe. The f ( R, T )theory of gravity explain an accelerating expansion ofuniverse at the cost of non conservation of energy mo-mentum tensor of matter. The state-finder analysisshows that LVDP cosmological models approaches thestandard Λ
CDM model in future. We observe thatwhen the redshift z is increasing within the interval0 ≤ z ≤ .
5, the Om(z) is monotonically increasing,which also indicates an accelerated expansion of uni-verse.
Acknowledgments
The authors are grateful to S. D. Odintsov, Z. Yousafand B. Saha for fruitful comments on the paper.0
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