Accelerating models with a hybrid scale factor in extended gravity
AAccelerating models with a hybrid scale factor in extended gravity
B. Mishra ∗ , S. K. Tripathy † , Sankarsan Tarai ‡ Dynamical aspects of cosmological model in an extended gravity theory have been investigated inthe present work. We have adopted a simplified approach to obtain cosmic features, which in factrequires more involved calculations. A cosmological model is constructed using a hybrid scale factorthat simulates a cosmic transit behaviour. The deceleration parameter and energy conditions havebeen obtained for the constructed model. Scalar fields have been reconstructed from the presentmodel in the extended gravity. Different diagnostic methods have been applied to analyse theviability of the constructed model. The present model almost looks like a cosmological constant fora substantial cosmic time zone and does not show any slowing down feature in near future.
PACS number : 04.50kd.
Keywords : f ( R, T ) Cosmology, Bianchi Type
V I h , Anisotropic Universe, Cosmic String. I. INTRODUCTION
The astronomical observations revealed two important phenomena on the universe: the dark energy and the darkmatter. Dark energy yields a late time acceleration of the cosmological background whereas dark matter behaves asan invisible dust matter favouring the process of gravitational clustering. The concept of the intriguing matter suchas dark energy has emerged as an alternative to the General Relativity (GR) based description of gravity [1–5]. Thesearch for alternative theories to GR has become inevitable after its failure to explain the cosmic phenomena occurringat late times. GR can be modified in many distinct directions. Consequently there are a good number of modifiedgravity theories available in literature [3, 6–9]. Different exotic matter fields simulating a positive energy density andnegative pressure are commonly used in literature as additional terms to explain the cosmic speed up phenomenon.However, it is possible to settle the issue of late time cosmic acceleration through the suitable geometrical modificationin GR without adding any exotic source of matter. The most simple geometrical modification to GR has been proposedis the f ( R ) theory, where the geometrical action contains an arbitrary function f ( R ) of the Ricci scalar R in place ofthe simple R in GR action. Some generalizations of f ( R ) gravity have been proposed recently. One such generalizationis the f ( R, T ) gravity proposed by Harko et al. [10]. In f ( R, T ) gravity, a weak coupling between the geometry andmatter is assumed and accordingly the geometrical part of the gravitational field action is modified by consideringan arbitrary function f ( R, T ) of the Ricci scalar R and the trace of the energy momentum tensor T . It is worthy tomention here that employing the trace of the energy momentum tensor in the new theory may be associated with theexistence of exotic imperfect fluids or quantum effects such as particle production.Many researchers have studied different aspects of f ( R, T ) gravity in various physical background. Sharif and Zubair[11] have studied the cosmological reconstruction of f ( R, T ) gravity in FRW universe. Shabani and Farhoudi [12]obtained the cosmological parameters in terms of some defined dimensionless parameters that are used in constructingthe dynamical equations of motion. Dynamics of an anisotropic universe is studied by Mishra et al. [13] in f ( R, T )gravity using a rescaled functional whereas Yousaf et al. [14] have shown the causes of irregular energy density in f ( R, T ) gravity. Velten and Carames [15] have challenged the viability of f ( R, T ) as an alternative modification ofgravity. Abbas and Ahmed [16] have formulated the exact solutions of the non-static anisotropic gravitating source in f ( R, T ) gravity which may lead to expansion and collapse. Baffoul et al. [17] have investigated the late-time cosmicacceleration in mimetic f ( R, T ) gravity with the Lagrange multiplier. Carvalho et al. [18] have shown the equilibriumconfigurations of white dwarfs in a modified gravity theory. Mishra et al. [19] developed a general formalism toinvestigate Bianchi type
V I h universes in an extended theory of gravity whereas the the dynamical features of themodels have been studied in [20, 21]. The investigation of different aspects of the cosmic phenomena concerning thelate time dynamics including the isotropic and anisotropic nature in the modified gravity theory requires an involvedcalculations of the dynamical parameters. Also, the results of the calculations should be in conformity with the lotof information gathered over the years from observations. In view of this, in the present work, we have developed aformalism to investigate the dynamical cosmic features in the framework of f ( R, T ) gravity. We have considered aBianchi type anisotropic and homogeneous universe for this purpose. ∗ Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India, E-mail:[email protected] † Department of Physics, Indira Gandhi Institute of Technology, Sarang, Dhenkanal, Odisha-759146, India, E-mail:tripathysunil@rediffmail.com ‡ Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India, E-mail:[email protected] a r X i v : . [ g r- q c ] A ug The paper is organised as follow: in Section II, the basic formalism of f ( R, T ) gravity and the field equations forBianchi type
V I h space time have been derived. In Section III, anisotropic nature of the cosmological model hasbeen presented. The dynamical parameters of the model have been calculated and analysed in Section IV. A scalarfield reconstruction technique has been employed in Section V to obtain the behaviour of the reconstructed scalarfields. Keeping in view the dynamically varying nature of the deceleration parameter in passing from a deceleratedphase of expansion to an accelerated one, we have employed a hybrid scale factor (HSF) in the proposed formalism toinvestigate the late time cosmic dynamics in Section VI. The viability of the constructed models have tested throughdifferent diagnostic mechanisms in Section VII and finally the concluding remarks are given in section VIII. II. BASIC FORMALISM OF f ( R, T ) THEORY AND THE FIELD EQUATIONS FOR A
BV I h SPACE-TIME
Within the scope of an extended gravity theory as proposed by Harko et al.[10], the Einstein-Hilbert action is givenby S = (cid:90) d x √− g (cid:20) π f ( R, T ) + L m (cid:21) . (1)In the above, we have used the natural unit system so that G = c = 1, where G and c are respectively the Newtoniangravitational constant and speed of light in vacuum. L m is the matter Lagrangian. The above action is different fromthat of GR where an arbitrary function of R and T ( f ( R, T )) replaces the Ricci scalar R . This interesting couplingof matter and curvature is motivated from quantum effects and leads to a non vanishing divergence of the energy-momentum tensor T ij . Such a feature of the non-minimal matter-geometry coupling provides a strong ground forcosmic acceleration. Because of this coupling, an additional force comes into play that dispels massive particles awayfrom geodesic trajectories. Since the functional form governing the matter-curvature coupling is arbitrary, differentchoices of the functional f ( R, T ) generate different models. In general, there can be three possible functional waysof coupling namely: (i) f ( R, T ) = R + 2 f ( T ), (ii) f ( R, T ) = f ( R ) + f ( T ) and (iii) f ( R, T ) = f ( R ) + f ( R ) f ( T ),where f ( T ) , f ( R ) , f ( R ) , f ( T ) and f ( T ) are some arbitrary functions of their respective arguments. In fact, therecan be infinite number of ways to chose these functions. However, the viability of the constructed models dependson the suitable choice of these functions and their ability to pass certain geometrical and observational tests. Out ofthese three types, the first choice is more like GR and can be reduced to GR under certain condition. The secondterm of the first choice in that case can be considered as a small deviation from GR. In the present work, we considerthe functional as f ( R, T ) = R + 2Λ + 2 βT where, Λ is the usual time independent cosmological constant and β isa coupling constant. This functional form resembles the first type of coupling as mentioned with R being replacedby R + 2Λ . The GR features with a cosmological constant can be recovered from the model for a vanishing β . It isneedless to mention here that the choice of the functional of the type f ( R, T ) = R + 2 f ( T ) has been widely used inliterature to address different cosmological and astrophysical issues.The field equations for the present model can be obtained by varying the action. One can refer to the procedurefollowed to obtain the field equation in Refs [10, 13]. For an arbitrary choice f ( R, T ) = f ( R ) + f ( T ) and L m = − p ,the field equations are obtained as R µν − f ( R ) f R ( R ) g µν = 1 f R ( R ) (cid:20) ( ∇ µ ∇ ν − g µν (cid:3) ) f R ( R ) + [8 π + f T ( T )] T µν + (cid:20) f T ( T ) p + 12 f ( T ) (cid:21) g µν (cid:21) . (2)In the above, p is the pressure of cosmic fluid. The partial differentiations for the model become f R = ∂f ( R ) ∂R = 1 and f T = ∂f ( T ) ∂T = 2 β , so that we can express the field equations as G µν = κT effµν , (3)with G µν = R µν − Rg µν , (4) T effµν = T µν + Λ eff ( T )8 π + 2 β g µν . (5)Here, we have a redefined Einstein matter-geometry coupling constant κ = 8 π +2 β and Λ eff ( T ) = (2 p + T ) β +Λ . For β = 0, Λ eff ( T ) becomes the usual cosmological constant Λ and the above field equation reduces to the Einstein fieldequations with a cosmological constant. For non vanishing value of β , Λ eff ( T ) becomes a time dependent quantity.We consider the universe to be filled with a cloud of one dimensional cosmic strings with string tension density ξ aligned along the x -axis. The energy-momentum tensor for such a fluid is given by T µν = ( p + ρ ) u µ u ν − pg µν − ξx µ x ν , (6)with u µ u µ = − x µ x µ = 1 (7)and u µ x µ = 0 . (8) ρ represents the energy density and is composed of the particle energy density ρ p and the string tension density ξ , ρ = ρ p + ξ .We wish to investigate dynamical aspects of the universe in f ( R, T ) theory as described above for an anisotropicBianchi type
V I h ( BV I h ) space-time given by ds = dt − A dx − B e x dy − C e hx dz , (9)where the metric potentials are considered only to depend on cosmic time. As has already been discussed in someworks of Tripathy et al.[22] and Mishra et al.[23], the exponent h in the metric can be useful if it assumes the value h = −
1. This conclusion has been derived from the null total energy concept of the whole universe known as theTryon’s conjecture [22, 23]. Going along the same line of thought, in the present study, we consider the same value of h . Now, the field equations for BV I h space-time with h = − BB + ¨ CC + ˙ B ˙ CBC + 1 A = − α ( p − ξ ) + ρβ + Λ , (10)¨ AA + ¨ CC + ˙ A ˙ CAC − A = − αp + ( ρ + ξ ) β + Λ , (11)¨ AA + ¨ BB + ˙ A ˙ BAB − A = − αp + ( ρ + ξ ) β + Λ , (12)˙ A ˙ BAB + ˙ B ˙ CBC + ˙ C ˙ ACA − A = αρ − ( p − ξ ) β + Λ , (13)˙ BB = ˙ CC . (14)Here α = 8 π + 3 β and we denote the ordinary time derivatives as overhead dots.Some relevant quantities in the context of discussion of geometrical aspect of the model includeHubble rate: H = ˙ RR = 13 (cid:32) ˙ AA + 2 ˙ BB (cid:33) , (15)Expansion scalar: θ = u l ; l = (cid:32) ˙ AA + 2 ˙ BB (cid:33) , (16)Deceleration parameter: q = − ddt (cid:18) H (cid:19) . (17) III. ANISOTROPIC NATURE OF THE MODEL
In the present work, we have considered a spatially homogeneous and anisotropic
BV I h universe with differentexpansion rates along different spatial directions. The quantities that measure the departure from spatial isotropyare the Shear scalar σ and the average anisotropy parameter A defined respectively as σ = 12 (cid:18)(cid:88) H i − θ (cid:19) = 13 (cid:32) ˙ AA − ˙ BB (cid:33) , (18) A = 13 (cid:88) i =1 (cid:18) (cid:52) H i H (cid:19) . (19) (cid:52) H i = H i − H , where H = ˙ AA , H = ˙ BB and H = ˙ CC are the directional Hubble rates along x, y and z-axesrespectively. In view of eq.(14), we have H = H . For isotropic models these quantities σ and A identically vanish.From observational perspectives, the anisotropic nature of a model is usually quantified through the estimation ofthe amplitude of shear σH at the present epoch. Using the data from differential microwave radiometers aboard theCosmic Background Explorer (COBE), Bunn et al. have placed an upper limit to this quantity as (cid:0) σH (cid:1) < × − [24]. For the best case with Ω = 1, they have obtained (cid:0) σH (cid:1) pl (cid:39) − − − [24]. In that work Bunn et al. haveconcluded that primordial anisotropy should have been fine tuned to be less than 10 − of its natural value in thePlanck era. Saadeh et al. used cosmic microwave background temperature and polarisation data from Planck andobtained a tighter limit to the anisotropic expansion as (cid:0) σH (cid:1) < . × − [25]. In view of these recent observationallimits on cosmic shear and anisotropic expansion rates, we have adopted a simple approach in the present work andhave assumed a proportional relationship between the amplitude of shear σ and Hubble rate. This assumption leads toan anisotropic relation among the directional Hubble rates H = kH . The parameter k takes care of the anisotropicfeature of the model. Obviously, k (cid:54) = 1 provides an anisotropic model.For the BV I h metric, we obtain the amplitude of shear expansion in the present epoch as (cid:16) σH (cid:17) = √ (cid:18) k − k + 2 (cid:19) . (20)Even though tighter constraints on cosmic anisotropy are available in literature and evidences against the departurefrom global isotropy are being gathered [26], these observational analysis need to be fine tuned as the analysis areprior dependent [27]. It is worth to mention here that, the cosmological principles assuming a homogeneous andisotropic universe may be a good approximation to the present universe. However, it has not yet been well proven inthe scales ≥ Gpc [28]. In view of this, in the present work, we wish to construct some accelerating anisotropic modelskeeping enough room for any amount of cosmic anisotropy. However, we can set some constraints on the parameter k basing upon the observationally found upper bound on (cid:0) σH (cid:1) . While the bounds of Bunn et al. constrain k as k = 1 . k = 1. However, in the presentwork, we consider k = 1 . (cid:0) σH (cid:1) = 4 . × − .Within the formalism discussed here it is easy to show that A = (cid:0) σH (cid:1) . Consequently, the average anisotropicparameter in the present epoch can be calculated as A = 4 . × − . Since the large scale structure of the universemay show a departure from isotropy, the cosmic anisotropy can be estimated from Hemispherical asymmetries inthe Hubble expansion. In a recent work, Kalus et al. estimated the Hubble anisotropy of supernova type Ia Hubblediagrams at low redshifts ( z < .
2) as (cid:52) HH < .
038 [29]. Using the value of the anisotropy parameter k at the presentepoch, we obtain the expansion asymmetry as (cid:52) HH = 0 . × − . One can note that, the predicted anisotropy fromour model is well within the observationally set up bounds [30]. IV. EQUATION OF STATE PARAMETER AND ENERGY CONDITIONS
The presumed anisotropic relation among the directional Hubble rates has a simplified structure and within thisformalism it can provide us a simple approach to study the cosmic dynamics. For a given anisotropic parameter k ,the directional Hubble rates become H = (cid:16) kk +2 (cid:17) H and H = H = (cid:16) k +2 (cid:17) H . Obviously for k = 1, the directionalHubble rates become equal to the Hubble parameter H . In our formalism, the presumption of the proportionalityrelation between the shear scalar and scalar expansion leads to the calculation of the equation of state (EoS) parameterin terms of the Hubble rate. Also, the energy conditions for the present model will depend on the Hubble rate. A. EoS parameter
The physical properties of the model such as pressure, energy density and string tension density are obtained fromthe field equations (10)-(14) as p = (cid:18) α − β (cid:19) [( S ( H ) − S ( H ) + S ( H )) β − S ( H ) α + ( α − β ) Λ ] , (21) ρ = (cid:18) α − β (cid:19) [ S ( H ) α − S ( H ) β − ( α − β ) Λ ] , (22) ξ = S ( H ) − S ( H ) α − β , (23)where S ( H ) = 1( k + 2) (cid:104) k + 2) ˙ H + 27 H + ( k + 2) R − ( kk +2 ) (cid:105) , (24) S ( H ) = 1( k + 2) (cid:104) k + 3 k + 2) ˙ H + 9( k + k + 1) H − ( k + 2) R − ( kk +2 ) (cid:105) , (25) S ( H ) = 1( k + 2) (cid:104) k + 1) H − ( k + 2) R − ( kk +2 ) (cid:105) . (26)Algebraic simplification of the above expressions yield p = − (cid:18) α − β (cid:19) (cid:104) φ ( k, β ) ˙ H + φ ( k, β ) H − ( α + β ) R − ( kk +2 ) − ( α − β )Λ (cid:105) , (27) ρ = (cid:18) α − β (cid:19) (cid:104) φ ( k, β ) ˙ H + φ ( k, β ) H − ( α + β ) R − ( kk +2 ) − ( α − β )Λ (cid:105) , (28) ξ = (cid:18) α − β (cid:19) (cid:104) φ ( k ) (cid:16) ˙ H + 3 H (cid:17) + 2 R − ( kk +2 ) (cid:105) , (29)where φ ( k, β ) = 3 k + 2 [( k + 1) α + ( k − β ] , (30) φ ( k, β ) = (cid:18) k + 2 (cid:19) [( k + k + 1) α + ( k − k − β ] , (31) φ ( k, β ) = − βk + 2 , (32) φ ( k, β ) = (cid:18) k + 2 (cid:19) [(2 k + 1) α − β ] , (33) φ ( k ) = 3(1 − k ) k + 2 . (34)It is interesting to note that for α + β = 0 i.e. for β = − π we have φ ( k, β ) = φ ( k, β ) and φ ( k, β ) = φ ( k, β ) (35)and consequently in the limit β → − π , p = − ρ. (36)In other words, within the scope of the present formalism, ΛCDM model with p = − ρ can be recovered from themodel for α + β = 0. Of course, overlapping of the present model with that of ΛCDM requires a negative couplingconstant.The equation of state parameter (EoS) ω is defined as the pressure to energy density ratio, ω = pρ . For α (cid:54) = ± β , itis straightforward to obtain ω as ω = − α + β ) S ( H ) − S ( H ) S ( H ) β − S ( H ) α + ( α − β ) Λ . (37)As is obvious from the above expression, the dynamical behaviour of the EoS parameter depends on the parametersof the Hubble rate H and the coupling constant β . For any realistic cosmological model, the Hubble parameter is adecreasing function of time and therefore at late phase of cosmic evolution, we expect that, the functionals S ( H ) and S ( H ) will behave alike thereby cancelling each other at late times. Therefore, for any value of β ( (cid:54) = − π, (cid:54) = − π ), theEoS parameter behaves as a cosmological constant ( ω = −
1) at late epoch. However, at an early epoch, the Hubblerate assumes a very high value thereby pushes ω to a larger value.In the limit β →
0, the model reduces to that of GR and the EoS parameter becomes ω = − S ( H ) − S ( H )Λ − S ( H ) (38)which becomes ω = − S ( H ) S ( H ) in the absence of a cosmological constant Λ term in the field equations. One can notethat, similar conclusion on the dynamical evolution of ω as above may be derived for β →
0. In other words, thedynamical behaviour of the EoS parameter will not be sensitive to the choice of the coupling constant at late times.All the trajectories of ω will behave alike at late phase of cosmic evolution. However, at an early epoch, the modelwill pass through different trajectories which may be β dependent.Another dynamical parameter is the effective cosmological constant Λ eff that appear in the equivalent EinsteinField equation for the extended gravity theory. Unlike the dynamical cosmological constant in GR, this effectivecosmological constant depends on the matter field content such as the pressure and energy density. We can obtainΛ eff as Λ eff = (cid:18) βα + β (cid:19) [( S ( H ) + S ( H )) − ] + Λ . (39)In terms of the Hubble parameter, we may express Λ eff asΛ eff = βα + β (cid:20) k + 2 ( ˙ H + 3 H ) − (cid:21) + Λ . (40)For α + β (cid:54) = 0, the magnitude of the effective cosmological constant decreases with the growth of cosmic time. Thesign of this quantity will depend on the sign of β . At late times, the behaviour of the effective cosmological constantdepends on the contribution coming from ˙ H + 3 H . In many models, this term either vanishes or have a negligiblecontribution. For such models, Λ eff reduces to (cid:16) α − βα + β (cid:17) Λ . Obviously as mentioned earlier, for a vanishing couplingconstant β , it reduces to the usual time independent cosmological constant Λ . B. Energy Conditions
Since energy conditions put some additional constraints on the viability of the models we wish to calculate thedifferent energy conditions for the constructed model in the modified gravity theory. In our formalism, the energyconditions are obtained as
NEC : ρ + p = [ S ( H ) − S ( H )] α − β ≥ , WEC : ρ = (cid:18) α − β (cid:19) [ S ( H ) α − S ( H ) β − ( α − β ) Λ ] ≥ , SEC : ρ + 3 p = (cid:18) α − β (cid:19) [( S ( H ) − S ( H )) α + (2 S ( H ) − S ( H ) + 3 S ( H )) β + 2( α − β )Λ ] ≥ , DEC : ρ − p = (cid:18) α − β (cid:19) [( S ( H ) + S ( H )) α − (2 S ( H ) − S ( H ) + S ( H )) β − α − β )Λ ] ≥ , where NEC, WEC, SEC and DEC respectively denote Null energy condition, Weak energy condition, Strong energycondition and Dominant energy condition. These energy conditions are expressed in terms of the Hubble parameteras NEC : ρ + p = − α + β (cid:20) k ( k − k + 2) H + k + 1( k + 2) 3 ˙ H (cid:21) , (41) WEC : ρ ≥ , (42) SEC : ρ + 3 p = − α − β (cid:20)(cid:18) k + k + 2( k + 2) H + k + 1( k + 2) 9 ˙ H − A (cid:19) α (cid:21) − α − β (cid:20)(cid:18) k − k − k + 2) H + 3 k − k + 2) 3 ˙ H − A (cid:19) β − α − β )Λ (cid:21) (43) DEC : ρ − p = 1 α − β (cid:20)(cid:18) k + 3 k + 2( k + 2) H + k + 1( k + 2) 3 ˙ H − A (cid:19) α (cid:21) + 1 α − β (cid:20)(cid:18) k − k − k + 2) H + k − k + 2) 3 ˙ H − A (cid:19) β − α − β )Λ (cid:21) . (44)We wish to compel our model in such a manner that the WEC be satisfied through out the cosmic evolution. Inorder to achieve this, one has to take a balance between the parameters of the Hubble rates and the choice of thecoupling constant β . Since at late times, our model overlaps with ΛCDM model, the NEC and DEC are satisfiedatleast at late phase of cosmic evolution. On the other hand, the SEC condition is violated at late times even thoughthere occurs some possibility that SEC be satisfied at an early epoch. In fact, a detailed analysis on these energycondition may be possible one the cosmic dynamics is fixed up from an assumed or derived Hubble rate. V. SCALAR FIELD RECONSTRUCTION
In GR, the late time cosmic acceleration phenomena is modelled usually through a scalar field φ which may eitherbe quintessence like or phantom like with the EoS parameter being ω ≥ − ω ≤ − S φ = (cid:90) d x √− g (cid:20) R π + (cid:15) ∂ µ φ ∂ µ φ − V ( φ ) (cid:21) , (45)where (cid:15) = +1 for quintessence field and (cid:15) = − V ( φ ) is the self interacting potential of the scalarfield. The scalar field dynamically rolls down the potential and thereby mediating for cosmic acceleration. In thiswork, we wish to draw a correspondence between the geometrically modified gravity theory discussed above with thatof the scalar field cosmology and also wish to reconstruct the scalar field along with the scalar potential. In a flatFriedman background, the energy density and pressure are expressed by ρ φ = (cid:15) φ + V ( φ ) , (46) p φ = (cid:15) φ − V ( φ ) . (47)A direct correspondence of our model with the scalar field yields˙ φ = (cid:15)α − β [ S ( H ) − S ( H )] , (48) V ( φ ) = (cid:18) α − β (cid:19) [ { S ( H ) + S ( H ) } α − { S ( H ) − S ( H ) + S ( H ) } β − α − β )Λ ] . (49)Since the factor S ( H ) − S ( H ) decreases with the cosmic evolution, we expect the magnitude of ˙ φ to decrease withcosmic time. It is worth to mention here that the exact behaviour of the scalar field will be model dependent and canbe investigated with some specific evolutionary behaviour of the Hubble rate. - 1 0 1 2- 2024681 01 2 q za = 0 . 6 9 5b = 0 . 0 8 5z t = 0 . 8 0 6( a ) - 1 0 1 20481 2 q’(z) z( b ) FIG. 1: (a)Deceleration parameter for HSF showing the transition redshift (b) q (cid:48) ( z ) as a function of redshift. The model doesnot favour a slowing down at late phase. VI. MODEL WITH A HYBRID SCALE FACTOR
The formalism developed in this work can be used to investigate certain aspects of cosmic dynamics. One can notethat all the dynamical properties are expressed in terms of the Hubble rate H . Therefore, for a given dynamics, ifthe Hubble rate is known, then it becomes easy to track the evolution history. In view of this, we employ a hybridscale factor (HSF) R = e at t b in the formalism. Here a and b are the model parameters and are constrained fromdifferent observational and physical basis. The reason behind the choice of such a scale factor is that it simulates atransition from a decelerated universe in recent past to an accelerated one. Moreover, the dynamical behaviour of HSFas predicted remains intermediate to that of the power law expansion and exponential expansion. Transit redshift z t is an important cosmological parameter which has been recently constrained from an analysis of type Ia Supernovaobservation and Hubble parameter measurements as z t = 0 .
806 [31, 32]. In a recent work, we have constrained theparameters of HSF as a = 0 .
695 and b = 0 .
085 so as to obtain a transition redshift z t = 0 .
806 [19]. The Hubbleparameter for the HSF is given by H = a + bt so that the directional Hubble rates become H = kk +2 (cid:0) a + ab (cid:1) and H = H = k +2 (cid:0) a + ab (cid:1) . The deceleration parameter for HSF is q = − b ( at + b ) . In Fig. 1(a), we have shownthe deceleration parameter q which displays the signature flipping behaviour at a suitable transit redshift. Thedeceleration parameter decreases from q = − b (cid:39) .
765 at an early time to q (cid:39) − q = − .
86. Recently, analysis from a host of Hubbleparameter measurements and type Ia Supernova observational data casts a doubt that, the universe has alreadyreached the peak of its acceleration and may be we are currently witnessing a possible slowing down [33, 34]. Sucha feature is investigated through the reconstruction of the slope of the deceleration parameter from observations. Inorder to check whether the HSF can predict such a feature we have plotted the function q (cid:48) ( z ) = dqdz as a functionof redshift z = R R − R is the scale factor at present epoch. The figure shows that there is noslowing down in cosmic acceleration at late phase of cosmic time. However, we find an interesting feature where q (cid:48) ( z )peaks up at around z = 1 .
5. In order to have a quantitative idea about the deceleration parameter, we have listedsome of its values at different epochs in table-I. - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5- 4- 202468 b = - 0.5 b = 0 b = 0.5 L C D M C P L B A w z FIG. 2: Equation of State parameter for three representative values of the coupling constant β . k = 1 . In Fig.2, the dynamical aspect of the model is assessed through the plot of the EoS parameter as function ofredshift. In the figure ω is shown for a fixed anisotropic parameter k = 1 . β . In general, the EoS parameter decreases from an initial positive value to behave like acosmological constant at late phase. The initial positive value depends on the choice of β . As expected, at late times,the model is insensitive to the choice of the coupling constant β . However at an early epoch, the EoS parameterevolves through different trajectories for different choices of β . Lower is the value of β , higher is the slope of the EoS TABLE I: Deceleration parameter at different epochsepoch z q Late phase − . . - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5- 2024681 0 w z k = 0 . 8 k = 1 . 0 0 0 0 8 1 4 k = 1 . 2 L C D M b = - 0.5
FIG. 3: Equation of State parameter for three representative values of the anisotropy parameter k . β = − . curve at late times. In other words, at an early epoch, the ω trajectory for low values of β remains in the top ofothers. The β dependent splitting in these trajectories is visible at around z = 1 .
5. At the present epoch, the modelpredicts an EoS of ω = − .
89 which is well within the observational constraints. In the figure, for a comparison, wehave also shown the trajectories for two well known ω parametrizations such as CPL [35, 36] and BA [37] given by ω ( z ) = ω + ω a z z and ω ( z ) = ω + ω z (1+ z )1+ z respectively. It is clear from the comparison that, in a time zone in therange − . ≤ z ≤ .
5, EoS from HSF is in close agreement with other models.The effect of the anisotropic parameter k on the EoS is investigated in Fig.3. In the figure we have shown theevolution of ω for a given coupling constant β = − . k namely k = 0 . , . .
2. We note here that, we have considered a specific shear expansion to Hubble rate ratio in the present epochwithin the observational limits and have constrained k to be 1 . k shows a very little departurefrom its isotropic value. Unlike that of the coupling constant, cosmic expansion anisotropy affects the cosmic dynamicsboth at an early epoch and at late times. The effect of cosmic anisotropy is almost symmetrical about z = 1 .
25. Atepochs z < .
25, higher the value of k , lower is the value of ω and at epochs z ≥ .
25, the EoS shows an oppositebehaviour i.e higher the value of k , higher is the value of ω . A quantitative idea on the effect of the cosmic expansionanisotropy on the EoS can be obtained from the values listed in Table.II. TABLE II: Variation of EoS parameter with anisotropy parameterepoch z k = 0 . k = 1 . k = 1 . − . . - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5- 0 . 9 5- 0 . 9 0- 0 . 8 5- 0 . 8 0- 0 . 7 5 w b L = 0 L = r L = 5 r L = k = 1 . 0 0 0 0 8 1 4 FIG. 4: Equation of State parameter as function of coupling constant β for three representative values of the cosmologicalconstant . k = 1 . ( d f /dt ) H ( z ) b = - 0.5 b = 0 b = 0.5
FIG. 5: Squared Slope of Reconstructed Scalar field as a function of Hubble rate. V( f ) H ( z ) b = - 0.5 b = 0 b = 0.5
FIG. 6: Scalar potential as a function of Hubble rate.TABLE III: EoS parameter at present epoch for different values of cosmological constant Λ Λ ω ( β = − ω ( β = − . ω ( β = 0) ω ( β = 0 . ω ( β = 1)0 -0.902 -0.897 -0.892 -0.888 -0.883 ρ -0.898 -0.892 -0.888 -0.883 -0.8795 ρ -0.872 -0.869 -0.866 -0.862 -0.85910 ρ -0.815 -0.819 -0.821 -0.822 -0.822 In order to assess a simultaneous effect of the coupling constant β and the cosmological constant Λ on the EoSparameter, we have shown the variation of ω at the present epoch with respect to β for four different values of Λ and a given cosmic anisotropy k = 1 . are considered as multiplesof the present value of energy density ρ as calculated from the present model. ω increases with the increase in thecoupling constant for a given value of Λ . Also for a given β , it increases with the increase in Λ . However, the netvariation of ω with respect to β decreases with an increase in Λ . To get a quantitative view, in table-III, the valuesof the EoS parameter at the present epoch are given for some representative values of the cosmological constant andthe coupling constant.In Fig.5, the quintessence like scalar fields are reconstructed from our model for three representative values of thecoupling constant β . The anisotropy parameter k and the time independent cosmological constant Λ are chosen tobe 1 . ρ respectively. As per expectation, the scalar field is found to decrease with the cosmic expansion.In Fig. 6, the evolution of the self interacting potential for the quintessence like scalar field is plotted. The selfinteracting potential increases with cosmic expansion. The choice of the coupling constant β does not affect thegeneral evolutionary behaviour of these two quantities. However with an increase in β value at a given epoch, thescalar field decreases and the potential increases. VII. DIAGNOSTIC APPROACH
There are two important diagnostic approaches used in literature. They are the determination of the state finderpair { j, s } in the j − s plane and the Om ( z ) diagnostics. These geometrical diagnostic approaches are useful tools todistinguish different Dark Energy models. While the state finder pair involve third derivatives of the scale factor, the Om ( z ) parameter involve only the first derivative of the scale factor appearing through the Hubble rate H ( z ).2 Statefinder pair
State finder pairs provide an useful tool to distinguish Dark Energy models since they involve the third derivativeof the scale factor. They are defined as j = ... RR H = ¨ HH − (2 + 3 q ) , (50) s = j − q − . . (51)In our formalism, the deceleration parameter is a time varying quantity and therefore the state finder pair evolvewith time. In Fig.7, the j − s trajectory in the j − s plane is shown for the HSF considered in this work. We observethat, our model evolves to overlap with the ΛCDM model. j s FIG. 7: j − s trajectory in the j − s plane. Om ( z ) diagnostic Another geometric diagnostic methods is the Om ( z ) diagnostic that involves first derivative of the scale factor andtherefore becomes easier to apply to distinguish between different Dark Energy models [38]. The Om ( z ) parameteris defined by Om ( z ) = E ( z ) − z ) − , (52)where E ( z ) = H ( z ) H is the dimensionless Hubble parameter. Here H is the Hubble rate at the present epoch. If Om ( z ) becomes a constant quantity, the DE model is considered to be a cosmological constant model with ω = − z with a positive slope, the model can be a phantom model with ω < −
1. For adecreasing Om ( z ) with negative slope, quintessence model are obtained ( ω > − Om ( z ) parameter forHSF is shown as a function of redshift. It can be observed from the figure that, the model looks like a cosmologicalconstant model for a substantial time zone in the recent past (0 ≤ z ≤ . Om(z) z FIG. 8: Om ( z ) parameter. VIII. CONCLUSION
In the present work, we have constructed a cosmological model in an extended theory of gravity by considering thefunctional f ( R, T ) = R + 2Λ + βT , where Λ is a constant. This model reduces to the usual GR equations with acosmological constant in the limit of a vanishing coupling constant β . Investigation of dynamical features of universein such an extended theory requires an involved calculation. In order to study certain dynamical cosmic aspects, wehave adopted an interesting approach in the present work and obtained the expressions in a more general manner.Although the cosmological principle assuming a homogeneous and isotropic universe is a good approximation to thepresent universe, it is yet to be proven in high energy scales. In view of this, we have considered an anisotropicuniverse which is more general than the FRW model for our purpose. The anisotropic model we have constructedcan be applicable to any amount of cosmic anisotropy. The anisotropic behaviour can be assessed through the valueof the anisotropy parameter at the present epoch which has been constrained as k = 1 . (cid:0) σH (cid:1) = 4 . × − . The expansion asymmetry from our model is obtained to be (cid:52) HH = 0 . × − which is in conformity with the observations.A dynamically changing universe with a feature of early deceleration and late time cosmic acceleration is simulatedthrough a hybrid scale factor. The parameters of the HSF are constrained from some physical basis to reproducethe transition redshift as obtained from different observational analysis. This HSF provides a good estimate of thedeceleration parameter and the Hubble rate at the present epoch. Recently there has been a belief that, we are atthe peak of the cosmic acceleration and the universe is now slowing down. We have investigated such a feature of theuniverse employing the HSF and obtained that there is no such slowing down in recent past or recent future.The dynamical behaviour of the model is assessed through the calculation of the EoS parameter employing theHSF. The EoS parameter decreases from a positive value in an early phase to a value closer to − β channels. Trajectory with low values of β lies in the top of all trajectories.Different diagnostic approaches have been adopted to analyse the viability of the present constructed model. At latephase, the model looks like a Λ CDM model for a substantial cosmic time zone. In the rest phase, it behaves as aquintessence field.4
Acknowledgement
BM and SKT thank IUCAA, Pune (India) for hospitality and support during an academic visit where a part of thiswork is accomplished. BM and ST acknowledge DST, New Delhi, India for providing facilities through DST-FISTlab, Department of Mathematics, where a part of this work was done. ST thanks University Grants Commission(UGC),New Delhi, India, for the financial support to carry out the research work.
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