Accelerating Universe in Hybrid and Logarithmic Teleparallel Gravity
Sanjay Mandal, Snehasish Bhattacharjee, S. K. J. Pacif, P.K. Sahoo
AAccelerating Universe in Hybrid and Logarithmic Teleparallel Gravity
Sanjay Mandal , Snehasish Bhattacharjee , S. K. J. Pacif , P.K. Sahoo Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India Department of Astronomy, Osmania University, Hyderabad-500007, India and Department of Mathematics, School of Advanced Sciences,Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India.
Teleparallel gravity is a modified theory of gravity for which the Ricci scalar R of the underlyinggeometry in the action is replaced by an arbitrary functional form of torsion scalar T . In doingso, cosmology in f ( T ) gravity becomes greatly simplified owing to the fact that T contains onlythe first derivatives of the vierbeins. The article exploits this appealing nature of f ( T ) gravity andpresent cosmological scenarios from hybrid and logarithmic teleparallel gravity models of the form f = e mT T n and f = D log( bT ) respectively, where m , n , D and b are free parameters constrained tosuffice the late time acceleration. We employ a well motivated parametrization of the decelerationparameter having just one degree of freedom constrained with a χ test from 57 data points of Hubbledata set in the redshift range 0 . < z < .
36, to obtain the expressions of pressure, density andEoS parameter for both the teleparallel gravity models and study their temporal evolution. We findthe deceleration parameter to experience a signature flipping for the χ value of the free parameterat z tr (cid:39) . PACS numbers: 04.50.Kd
I. INTRODUCTION
Several Observations reveal that the universe is accelerating for the very second time in its 13.7 billion year longlifetime [1]. It has now been agreed that a cosmological entity with almost three-quarters of the energy budget ofthe universe coupled with a EoS parameter ω (cid:39) − f ( R ) gravity, f ( G ) gravity, f ( R, T ) gravity, etc have widespread use in modern cosmology (For a recent review on modified gravity see [4]. Alsosee [5] for some interesting cosmological applications of modified gravity).Teleparallel gravity is a well established and well motivated modified theory of gravity inspired from f ( R ) gravity [6](See [7] for a review on teleparallel gravity). In teleparallel gravity, the Ricci scalar R of the underlying geometry inthe action is replaced by an arbitrary functional form of torsion scalar T . Thus, in teleparallel gravity, instead of usingthe torsionless Levi-Civita connection (which is usually assumed in GR), the curvatureless Weitzenb¨ock connection isemployed in which the corresponding dynamical fields are the four linearly independent verbeins, and T is related tothe antisymmetric connection following from the non-holonomic basis [3, 8].Linear f ( T ) gravity models are the teleparallel equivalent of GR (TEGR) [9]. Nonetheless, f ( T ) gravity differsignificantly from f ( R ) gravity in the fact that the field equations in f ( T ) gravity are always at second-order comparedto the usual fourth-order in f ( R ) gravity. This owes to the fact that the torsion scalar contains only the first derivativesof the vierbeins and thus makes cosmology in f ( T ) gravity much simpler. However, Despite being a second-ordertheory, very few exact solutions of the field equations have been reported in literature. Power law solutions in FLRWspacetime have been reported in [12], while for anisotropic spacetimes in [10]. Solutions for Bianchi I spacetime andstatic spherically spacetimes can be found in [3] and [11] respectively.Since cosmology in f ( T ) gravity is much simpler compared to other modified gravity theories, it has been employed tomodel inflation [13], late time acceleration [14] and big bounce [15]. The instability epochs of self-gravitating objectscoupled with anisotropic radiative matter content and the instability of cylindrical compact object in f ( T ) gravityhave been discussed in Ref. [16, 17].The manuscript is organized as follows: In Section II we present an overview of f ( T ) gravity. In Section III wedescribe the kinematic variables obtained from a parametrization of deceleration parameter used to obtain the exactsolutions of the field equations. In Section IV we present the hybrid and logarithmic teleparallel gravity models andobtain the expressions of pressure, density and EoS parameter. In Section V we present some geometric diagnostics ofthe parametrization of deceleration parameter. In Section VI we study the energy conditions for both the teleparallelgravity models. In Section VII we obtain some observational bounds on the free parameters of the parametrizationby performing a chi-square test using Hubble datasets with 57 datapoints, Supernovae datasets consisting of 580 datapoints from Union2 . a r X i v : . [ phy s i c s . g e n - ph ] A p r VIII we present our results and conclusions.
II. OVERVIEW OF f ( T ) GRAVITY
The action in teleparallel gravity is represented as S = 116 πG (cid:90) [ T + f ( T )] ed x, (1)where e = det ( e iµ ) = √− g and G is Newtonian gravitational constant. The gravitational field in this framework arisesdue to torsion defined as T γµν ≡ e γi ( ∂ µ e iν − ∂ ν e iµ ) . (2)The contracted form of torsion tensor reads T ≡ T γµν T γµν + 12 T γµν T νµγ − T γγµ T νµν . (3)varying the action S + L m , where L m represent the matter Lagrangian yields the field equations as e − ∂ µ ( ee γi S µνγ )(1 + f T ) − (1 + f T ) e λi T γµλ S νµγ + e γi S µνγ ∂ µ ( T ) f T T + 14 e νi [ T + f ( T )] = k e γi T ( M ) νγ , (4)where f T = df ( T ) /dT , f T T = d f ( T ) /dT , the “superpotential” tensor S µνγ written in terms of cotorsion K µνγ = − ( T µνγ − T νµα − T µνα ) as S µνγ = ( K µνγ + δ µγ T ανα − δ νγ T αµα ) and T ( M ) νγ represents the energy-momentum tensor to thematter Lagrangian L m . For a flat FLRW universe with the metric denoted as ds = dt − a ( t ) dx µ dx ν , (5)where a ( t ) the scale factor, gives e iµ = diag (1 , a, a, a ) . (6)Employing (5) into the field equation (4), the modified Friedman equations reads H = 8 πG ρ − f T f T , (7)˙ H = − (cid:20) πG ( ρ + p )1 + f T + 2 T f
T T (cid:21) , (8)where H ≡ ˙ a/a denote the Hubble parameter and dots represent the derivative with respect to time and ρ and p bethe energy density and pressure of the matter content and T = − H . From equations (7) and (8), we obtain theexpressions of density ρ , pressure p and EoS parameter ω respectively as ρ = 3 H + f H f T (9) p = − H (1 + f T − H f T T ) − (3 H + f H f T ) (10) ω = pρ = − − H (1 + f T − H f T T )(3 H + f + 6 H f T ) (11)where we set 8 πG = 1. Furthermore, the continuity equation reads˙ ρ + 3 H (1 + ω ) ρ = 0 , (12) III. KINEMATIC VARIABLES
The system of field equations described above has only two independent equations with four unknowns. To solvethe system completely and in order to study the temporal evolution of energy density, pressure and EoS parameter,we need two more constraint equations (extra conditions). In literature, there are several arguments to choosethese equations (see [18] for details). The method is well known as the model independent way approach to studycosmological models that generally considers a parametrizations of any kinematic variables such as Hubble parameter,deceleration parameter, jerk parameter and EoS parameter and provide the necessary supplementary equation [19].Bearing that in mind, we shall work with a parametrization of deceleration parameter proposed in [20] as q = − α − (cid:16) z (cid:17) α (13)where α shall be constrained from a chi-square test using any observational datasets (ref. section VII). The motivationuse this parametrization is driven by the fact that equation (13) allows a signature flipping for − > α > − • α = − z = 0. • α < − z < • α ≥ − H = β (cid:20) (cid:18)
11 + z (cid:19) α (cid:21) (14)where β is the integration constant. To obtain (14), we used the relation H ( z ) H = exp (cid:20)(cid:90) z q ( z (cid:48) )1 + z (cid:48) dz (cid:48) (cid:21) (15) α =- α =- α =- α =- - - - z q ( z ) FIG. 1: Plot of deceleration parameter ( q ) as a function of redshift ( z ) for different values of model parameter α showingdiverge evolutionary dynamics. Higher order derivatives of deceleration parameter such as jerk ( j ), snap ( s ) and lerk ( l ) parameters provideimportant information about the evolution of the universe. They are represented as [21] j ( z ) = (1 + z ) dqdz + q (1 + 2 q ) ,s ( z ) = − (1 + z ) djdz − j (2 + 3 q ) ,l ( z ) = − (1 + z ) dsdz − s (3 + 4 q )The jerk parameter represents the evolution of deceleration parameter. Since q can be constrained from observations,jerk parameter is used to predict the future. Additionally, the jerk parameter along with higher derivatives such assnap and lerk parameters provide useful insights into the emergence of sudden future singularities [21].From Fig. 2 and Fig. 4, the jerk and lerk parameters are observed to have decreasing behaviors. Also, as thevalue of α decreases, the parameters assumes higher values at redshift z = 0. Both of these parameters are positivewhich represents an accelerated expansion. The snap parameter is negative for all α which also denote an acceleratedexpansion. Interestingly, the jerk parameter does not attain unity at z = 0 which clearly does not coincide withΛCDM model. Interestingly, this implies that the late time acceleration can be caused due to modifications of gravity.It is therefore encouraging to study the dynamics of EoS parameter which may arise purely due to geometric effectsin the framework of modified gravity theories such as teleparallel gravity. α =- α =- α =- α =- - z j ( z ) FIG. 2: Jerk parameter as a function of redshift. α =- α =- α =- α =- - - - - z s ( z ) FIG. 3: Snap parameter as a function of redshift. α =- α =- α =- α =- - z l ( z ) FIG. 4: Lerk parameter as a function of redshift.
IV. COSMOLOGY WITH TELEPARALLEL GRAVITYA. Hybrid Teleparallel Gravity
For the first case, we presume the functional form of teleparallel gravity to be f ( T ) = e mT T n , (16)where m ≥ n constants. Interestingly, this model takes power-law and exponential forms depending on thevalues of n and m . Particularly: • For m = 0 Eq. (16) reduces to f ( T ) = T n (power law). • For and n = 0, Eq. (16) reduces to f ( T ) = e mT (exponential).Using Eq. (16) in Eq. (7) and Eq. (8), the expressions of energy density ρ , pressure p and EoS parameter ω readsrespectively as ρ = 3 K + 6 n ( − K ) n e − mK (cid:18) − n + 6 nK (cid:19) , (17) p = − (cid:18) αK − e − tαβ αβ − e − tαβ (cid:19) × (cid:26) − − K ) n e − mK (cid:20) m + 4 mn − Km − n K − n ( n − K (cid:21)(cid:27) − K − n ( − K ) n e − mK (cid:18) − n + 6 nK (cid:19) (18) ω = − − (cid:18) αK − e − tαβ αβ − e − tαβ (cid:19) × (cid:26) − − K ) n e − mK (cid:20) m + 4 mn − Km − n K − n ( n − K (cid:21)(cid:27) × (cid:26) K + 6 n ( − K ) n e − mK (cid:18) − n + 6 nK (cid:19)(cid:27) − (19)where K = e − tαβ β ( − e − tαβ ) . FIG. 5: Energy density as a function of redshift for α = − . , β = 31 . m = 0 . FIG. 6: Energy density as a function of redshift for α = − . , β = 31 . n = 5.FIG. 7: EoS parameter as a function of redshift for α = − . , β = 31 . m = 0 . α = − . , β = 31 . n = 5. B. Logarithmic Teleparallel Gravity
For the second case, we presume the functional form of teleparallel gravity to be f ( T ) = D log( bT ) , (20)where D and b < ρ , pressure p and EoS parameter ω readsrespectively as ρ = − D + 3 K + D bK ) , (21) p = D − K − (cid:18) D K (cid:19) (cid:18) αK − e − tαβ αβ − e − tαβ (cid:19) − D bK ) , (22) ω = − − (cid:18) D K (cid:19) (cid:18) αK − e − tαβ αβ − e − tαβ (cid:19) × (cid:26) − D + 3 K + D bK ) (cid:27) − (23) FIG. 9: Energy density as a function of redshift for α = − . , β = 31 . b = − α = − . , β = 31 . D = 0 . FIG. 11: EoS parameter as a function of redshift for α = − . , β = 31 . b = − α = − . , β = 31 . D = 0 . V. GEOMETRICAL DIAGNOSTICSA. Statefinder Diagnostics
Due to the fact that the number of dark energy models are quite large and increasing on a daily basis, it becomesabsolutely necessary to find a method to distinguish a particular model from the well established DE models like theΛCDM, SCDM, HDE, CG and Quintessence. With that reasoning, [22] proposed the { r, s } diagnostics where r and s are defined as r = ˙¨ aaH ,s = r − (cid:0) q − (cid:1) , (cid:18) q (cid:54) = 12 (cid:19) . Different combinations of r and s represent different dark energy models. Particularly: • For ΛCDM → ( r = 1 , s = 0). • For SCDM → ( r = 1 , s = 1). • For HDE → ( r = 1 , s = ). • For CG → ( r > , s < • For Quintessence → ( r < , s > { r, s } diagnostics tool is that different dark energy models exhibit different trajectories in the { r, s } plane. The deviation from the point { r, s } = { , } represent deviation from the well agreed ΛCDM model.Furthermore, the values of r and s could in principle be inferred from observations [23] and therefore could be veryuseful in discriminating dark energy models in the near future.The expression of r and s parameters for our model reads r = 1 + α (cid:16) z (cid:17) α (cid:110) α + (cid:16) z (cid:17) α (3 + 2 α ) (cid:111)(cid:110) (cid:16) z (cid:17) α (cid:111) (24) s = α − (cid:16) z (cid:17) α + 33 + (cid:16) z (cid:17) α (3 + 2 α ) (25)In Fig. 13, the { r, s } plane is shown for the parametrization (13) where the arrows indicate the direction of temporalevolution. The model is observed to deviate significantly from the point (0 ,
1) initially and is extremely sensitive tothe value of α . For α ≤ −
2, the model initially starts its journey from the territory of Chaplygin gas ( r > , s < α > −
1, the model at high redshifts stays in the Quintessenceregion but again approaches towards ΛCDM. Interestingly, for α = − . ∼ − . r = 1 , s = ) which is the region of HDE. However,at late times the model is observed to coincide with ( r = 1 , s = 0). Therefore, the parametrization used in this workis interesting and warrants further attention. Λ CDM SCDM α =- α =- α =- α =- - s r FIG. 13: { r, s } plane for the redshift range z [ − ,
5] for different values of α . In addition to the { r, s } plane, we construct the { r, q } plane to get additional understanding of the parametrization(13). In { r, q } plane, the solid line in the middle represents the evolution of the ΛCDM universe and also divide theplane into two sections. The upper section belong to Chaplygin gas models and the lower section to Quintessencemodels.0 SCDM α =- α =- α =- α =- Λ CDMdS CHAPLYGIN GASQUINTESSENCE - - q r FIG. 14: { r, q } plane for the redshift range z [ − ,
5] for different values of α . We observe from Fig. 14, that except for α = − .
5, all the profiles starts from r > , q > r < , − < q < r = 1 , q = −
1. However, for α < − q is always negative and therefore the profile does not start from the SCDMuniverse. B. Om Diagnostic
Another very useful diagnostic tool constructed from the Hubble parameter is the Om diagnostic which essentiallyprovide a null test of the ΛCDM model [24]. This tool easily captures the dynamical nature of dark energy modelsfrom the slope of Om ( z ). If the slope of this diagnostic tool were to be positive, it would imply a Quintessence nature( ω > −
1) whereas the opposite would prefer a Phantom nature ( ω < − ω = − Om ( z ) = (cid:16) H ( z ) H (cid:17) − z + 3 z + 3 z (26)From Fig. 15, we observe a negative slope for α > − α ≤ − Om ( z ) increases with redshift and therefore represents an Quintessence darkenergy model. Hence, the value of α dictates the nature of the underlying dark energy model represented by theparametrization (13). α =- α =- α =- α =- - z O m ( z ) FIG. 15: Om ( z ) for different values of α . VI. ENERGY CONDITIONS
Based upon the Raychaudhuri equation, the energy conditions are essential to describe the behavior of the compati-bility of timelike, lightlike or spacelike curves [25] and often used to understand the dreadful singularities [26]. Energyconditions in teleparallel gravity have been studied in [27]. Energy conditions also provide the corners in parameterspaces since they violate, for instance, in presence of singularities. They are defined as: • Strong energy conditions (SEC): Gravity is always attractive and therefore ρ + 3 p ≥ • Weak energy conditions (WEC): Energy density should always be positive, i.e., ρ ≥ , ρ + p ≥ • Null energy condition (NEC): Minimum requirement for the fulfilment of SEC and WEC, i.e., ρ + p ≥ • Dominant energy conditions (DCE): Energy density is always positive and independent of the observer’s referenceframe, i.e., ρ ≥ , | p | ≤ ρ .Energy conditions for both the teleparallel gravity models are presented in Fig. 16-17. SECNECDEC - - -
10 000010 00020 00030 00040 00050 000 z FIG. 16: ECs as a function redshift z for β = 31 . , α = − . m = 0 . n = − SECNECDEC - - -
10 000010 00020 00030 00040 00050 000 z FIG. 17: ECs as a function redshift z for β = 31 . , α = − . , b = − d = 0 . VII. OBSERVATIONAL CONSTRAINTS
In order to find the best fit value of the model parameters of our obtained models, we need to constrain the pa-rameters with some available datasets. Here, we use three datasets, namely, Hubble datasets with 57 datapoints,Supernovae datasets consisting of 580 data points from Union2 . A. Hubble parameter H(z)
Recently, Sharov and Vasiliev [28] compiled a list of 57 points of measurements of the Hubble parameter at in theredshift range 0 . (cid:54) z (cid:54) .
42, measured by extraction of H ( z ) from line-of-sight BAO data including the analysis2of correlation functions of luminous red galaxies [29] and H ( z ) estimations from differential ages (cid:77) t of galaxies (DAmethod) [30]. (See the Appendix in [28] for full list of tabulated datasets). Chi square test is used to constrain themodel parameters parameters given by χ OHD ( p s ) = (cid:88) i =1 [ H th ( p s , z i ) − H obs ( z i )] σ H ( z i ) (27)where H th ( p s , z i ) denotes the Hubble parameter at redshift z i predicted by the models with p s denoting the parameterspace ( α here in our model), H obs ( z i ) is the i -th measured one and σ H ( z i ) is its uncertainty. We also take a prior as H = 67 . B. Type Ia Supernova
Further we consider, the 580 points of Union2 . χ SN ( µ , p s ) = (cid:88) i =1 [ µ th ( µ , p s , z i ) − µ obs ( z i )] σ µ ( z i ) , (28)where µ th and µ obs are correspondingly the theoretical and observed distance modulus for the model and the standarderror is σ µ ( z i ) . The distance modulus µ ( z ) is defined to be µ ( z ) = m − M = 5 LogD l ( z ) + µ , where m and M arethe apparent and absolute magnitudes of any standard candel (supernovae of type Ia here) respectively. Luminositydistance D l ( z ) and the nuisance parameter µ are given by D l ( z ) = (1 + z ) H (cid:82) z H ( z ∗ ) dz ∗ and µ = 5 Log (cid:16) H − Mpc (cid:17) + 25respectively. In order to calculate luminosity distance, we have restricted the series of H ( z ) up to tenth term onlyand then integrated the approximate series to obtain the luminosity distance. C. Baryon Acoustic Oscillations
Finally, we consider a sample of BAO distances measurements from surveys of SDSS(R) [32], 6dF Galaxy survey[33], BOSS CMASS [34] and three parallel measurements from WiggleZ [35]. In the context of BAO measurements,the distance redshift ratio d z is given as, d z = r s ( z ∗ ) D v ( z ) , (29)where r s ( z ∗ ) is the co-moving sound horizon at the time photons decouple and z ∗ indicates the photons decouplingredshift i.e. z ∗ = 1090 [36]. Moreover, r s ( z ∗ ) is assumed same as considered in the reference [37] together withthe dilation scale is given by D v ( z ) = (cid:0) d A ( z ) zH ( z ) (cid:1) , where d A ( z ) is the angular diameter distance. The χ BAO valuescorresponding to BAO measurements are discussed in details in [38] and the chi square formula is given by, χ BAO = A T C − A, (30)where the matrix A is given by A = d A ( z ∗ ) D v (0 . − . d A ( z ∗ ) D v (0 . − . d A ( z ∗ ) D v (0 . − . d A ( z ∗ ) D v (0 . − . d A ( z ∗ ) D v (0 . − . d A ( z ∗ ) D v (0 . − . , C − representing the inverse of covariance matrix C given as in the reference [38] adopting the correlationcoefficients presented in [39] as C − = . − . − . − . − . − . − . . − . − . − . − . − . − . . − . − . − . − . − . − . . − . . − . − . − . − . . − . − . − . − . . − . . . Below, we have shown a comparision of our obtained model with the ΛCDM model together with error bars due tothe 57 points of H ( z ) datasets and the 580 points of Union2 . z H ( z ) z μ ( z ) ( a ) ( b )FIG. 18: Figures (a) and (b) are respectively the error bar plots of 57 points of H ( z ) datasets and 580 points of Union2 . CDM model (black dashed lies).
Next, we have shown the likelihood contours for the model parameter α and Hubble constant H with errors at1- σ , 2- σ and 3- σ levels in the α - H plane. The best fit constrained values of α and H are found to be α = − . H = 63 . H ( z ) datasets only with χ = 31 . α = − . H = 63 . H ( z ) + SN eIa + BAO with χ = 650 . - - - - - α H - - - - - - α H ( a ) ( b )FIG. 19: Figures (a) shows the maximum likelihood contours in the α - H plane for H ( z ) datasets only while figure (b) showsthe maximum likelihood for H ( z ) + SNeIa + BAO datasets jointly. The three contour regions shaded with dark, light shadedand ultra lightshaded in both the plots are with errors at 1- σ , 2- σ and 3- σ levels. The black dots represent the best fit valuesof model parameter α and H in both the plots. VIII. RESULTS AND DISCUSSIONS
The manuscript communicates the phenomena of late time acceleration in the framework of hybrid and logarithmicteleparallel gravity. To obtain the exact solutions of the field equations, we employ a parametrization of decelerationparameter first proposed in [20]. In this section, we shall discuss the energy conditions and the cosmological viabilityof the underlying teleparallel gravity models.In Section VI, we show the temporal evolution of SEC, NEC and WEC for both the teleparallel gravity models. Notethat in order to suffice the late time acceleration, the SEC has to violate [40]. This is due to the fact that for anaccelerating universe compatible with observations [1], the EoS parameter ω (cid:39) − ρ (1 + 3 ω ) < q ) as a function of redshift. The plot of the deceleration parameter q ( z ) clearly shows that our model suc-cessfully generates late time cosmic acceleration along with a decelerated expansion in the past for − > α > −
2. Thedeceleration parameter undergoes a signature flipping at the redshift z tr (cid:39) . α = − . q = − . z tr are consistent withvalues reported by other authors [42]. From Figs. 8 & 12 the values of EoS ω at z = 0 for both our models are obtainedas − . − . ω behave in concordance with standard cosmological modelpredictions (with Plank data, ω LCDMeff ∼ − .
68 at z = 0 as in Ref. [41]).In Fig. 5,6,9 and 10, we plot the energy density for both the models as a function of redshift. We chose the modelparameters so as to satisfy the WEC. Fulfillment of WEC ensures the cosmological pressure has to be negative toaccount for negative EoS parameter and therefore the cosmic acceleration. It is interesting to note that no knownentity has the remarkable property of negative pressure and can only be achieved by exotic matter or by modificationsto general relativity.The EoS parameter is an important cosmological parameter which has sparked great deal of interest among cosmolo-gists. Owing to the mysterious nature of the cosmological entity responsible for this acceleration, various dark energymodels have been devised to suffice the observations. To investigate the nature of the dark energy model representedby the equation (13), we study in Section VI, the { r, s } and { r, q } plane and Om ( z ). We observe that the value of α dictates the evolution of the r - s and r - q trajectories. We find the model to deviate significantly from the ΛCDM atearly times. However, at late times the model is observed to coincide with ( r = 1 , s = 0) and therefore consistent withΛCDM cosmology. This result is further re-assured from the r - q plane in Fig. 14. However, discrepancy arises fromFig. 15, where for none of the values of α we obtain a constant Om which clearly does not reflect a dark energy whichis time independent. Furthermore, the nature of dark energy represented by the equation (13) changes from being anQuintessence to Phantom as α changes from α ≤ − α > − . α and Hubbleconstant H with errors at 1- σ , 2- σ and 3- σ levels in the α - H plane is shown separately for H ( z ) datasets only andjoint datasets H ( z ) + SN eIa + BAO . The best fit constrained values of α and H are found to be α = − . H = 63 . H ( z ) datasets with χ = 31 . α = − . H = 63 . H ( z )+ SN eIa + BAO datasets with χ = 650 . Acknowledgments
S.M. acknowledges Department of Science & Technology (DST), Govt. of India, New Delhi, for awarding JuniorResearch Fellowship (File No. DST/INSPIRE Fellowship/2018/IF180676). SB thanks Biswajit Pandey for helpfuldiscussions. PKS acknowledges CSIR, New Delhi, India for financial support to carry out the Research project[No.03(1454)/19/EMR-II Dt.02/08/2019]. We are very much grateful to the honorable referee and the editor for theilluminating suggestions that have significantly improved our work in terms of research quality and presentation. [1] A. G. Riess et al., Astron. J. , 1009 (1998); S. Perlmutter et al., Astrophys. J. , 565 (1999); P. deBernardis et al.,Nature , 955 (2000); S. Perlmutter et al., Astrophys. J. , 102 (2003); M. Colless et al., Mon. Not. R. Astron. Soc. , 1039 (2001); M. Tegmark et al., Phys. Rev. D , 103501 (2004); S. Cole et al., Mon. Not. R. Astron. Soc. , 505(2005); V.Springel et al., Nature (London) , 1137 (2006); P. A. R. Ade et al., Astron. 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