Transit cosmological models with observational constraints in f(Q, T) gravity
aa r X i v : . [ phy s i c s . g e n - ph ] O c t Transit cosmological models with observational constraints in f ( Q, T ) gravity Anirudh Pradhan , Archana Dixit , Department of Mathematics, Institute of Applied Sciences & Humanities, GLA University,Mathura -281 406, Uttar Pradesh, India Email:[email protected] Email:[email protected]
Abstract
This cosmological model is a study of modified f ( Q, T ) theory of gravity which was recently proposedby Xu et al. (Eur. Phys. J. C , 708 (2019)). In this theory of gravity, the action contains anarbitrary function f ( Q, T ) where Q is non-metricity and T is the trace of energy-momentum tensorfor matter fluid. In our research, we have taken the function f ( Q, T ) quadratic in Q and linear in T as f ( Q, T ) = αQ + βQ + γT where α , β and γ are model parameters, motivated by f ( R, T ) gravity. We haveobtained the various cosmological parameters in Friedmann-Lemaitre-Robertson-walker (FLRW) Universeviz. Hubble parameter H , deceleration parameter q etc. in terms of scale-factor as well as in terms ofredshift z by constraining energy-conservation law. For observational constraints on the model, we haveobtained the best-fit values of model parameters using the available data sets like Hubble data sets H ( z ),Joint Light Curve Analysis (JLA) data sets and union 2 . R -test formula. We have calculated the present values of various observational parameters viz. H , q , t and statefinder parameters ( s, r ), these values are very close to the standard cosmological models.Also, we have observed that the deceleration parameter q ( z ) shows signature-flipping (transition) pointwithin the range 0 . ≤ z t ≤ .
668 through which it changes its phase from decelerated to acceleratedexpanding universe with equation of state (EoS) − . ≤ ω − .
96 for 0 ≤ z ≤ Keywords:
FLRW universe, Modified f ( Q, T ) gravity, Transit universe, Observational constraints.
Even if, Einstein’s general relativity (GR) proposed in 1915 is still considered to be one of the most suc-cessful theories, but in the wake of new observational advances in cosmology, there appears some limitationson standard GR in explaining those phenomena. The observations in [1]-[7] have strong evidence that ourpresent universe is undergoing an accelerated expansion phase. These same observations predict the existenceof two unknown components in the Universe, called dark matter (DM) and dark energy (DE) along with theusual baryonic matter. The heavy negative pressure due to DE is supposed to be the possible cause of theaccelerated expansion.To adapt new findings, there are mainly two ways, one to provide alternative theories of gravity and theother to modify the Einstein’s GR. Theories like Brans and Dicke [8] , Nordtvedt [9], Wagoner [10], Dunn[11], Saez and Ballester [12], Barber [13] are some of the prominent alternating theories. To understandthe dark energy problem in a better way, time-to-time various cosmologists have proposed some well-knownmodified theories of gravitation viz. f ( G ) gravity [14], f ( R ) gravity [15], f ( R, T ) gravity [16], f ( T ) gravity[17], f ( G, T ) [18], f ( R, T, R µν T µν ) gravity [19], f ( Q ) theory [20] and recently proposed f ( Q, T ) gravity [21].In standard GR the action for formulation of Einstein field equations is given by S = R ( R κ + L m ) √− gd x ,here g = det ( g µν ) is defined the determinant of metric tensor g µν , R is the Ricci scalar-curvature, L m is the1atter Lagrangian and κ = 8 πGc − is Einstein’s constant. In the attempt to modify Einstein’s GR, one ofthe simplest way is to replace R by some more general entity. A modified theory of gravity is formulated byreplacing the Ricci scalar-curvature R by an arbitrary function f ( R ) in the Einstein–Hilbert action, called as f ( R ) gravity has proposed by Nojiri and Odintsov [15]. The different cosmological features of f ( R ) gravityhave been studied by several researchers in different perspectives [22]-[32]. An extension of f ( R ) gravityis proposed by Harko et al. [17] by including the trace of energy-momentum tensor T ij in f ( R ) functioncalled as f ( R, T ) gravity. The theory of f ( R, T ) gravity has been rigorously studied by several authors. Thereferences [33]-[37] are a small list of the study on f ( R, T ) gravity.Einstein’ GR is basically a geometric theory based on Reimannian geometry. In one of the classicalapproach to modify GR is to a apply some more general geometric structures that could describe the gravi-tational field. The first attempt in this approach is due to Weyl [38], but the main aim of the Weyl geometrywas to unify electromagnetism and gravitation. On the similar line, Gauss-Bonnet gravity, or f ( G ) gravityis an interesting modified theory, in which gravitation Lagrangian is obtained by adding a generic func-tion f ( G ) in the Hilbert-Einstein action, where G = R αβµν R αβµν − R αβ R αβ + R is the Gauss-Bonnetinvariant ( R αβµν , R αβ and R are Riemann tensor, Ricci tensor and Ricci scalar respectively) [14]. f ( G )gravity is further studied by many researchers. See [39]-[43]. Sharif and Ikram [18] presented an extensionof f ( G ) gravity theory named as f ( G, T ) gravity by including the trace of the energy-momentum tensor T along with G in the modified Hilbert-Einstein action. For more study on f ( G, T ) gravity see references [44, 45]In another approach, the so called teleparallel equivalent to general relativity (TEGR), the basic idea is toreplace the metric tensor g µν of the space-time, by a set of tetrad vectors e iµ . The curvature is replaced by thetorsion generated by the tetrad, which can be used to entirely describe gravitational effects [17, 46, 47, 48].The teleparallel or f ( T ) gravity are extensively used to explain the current cosmic accelerated expansionwithout using the concept of dark energy [49]-[60].In symmetric teleparallel gravity (STG), the basic geometric variable describing the properties of thegravitational interaction is represented by the non-metricity Q of the metric. The non-metricity tensor is thecovariant derivative of the metric tensor which geometrically describes the variation of the length of a vectorin the parallel transport. This approach is first introduced by Nester and Yo in (1999) [20]. The STG wasfurther developed as f ( Q ) gravity theory or non-metric gravity and various geometrical and physical aspectsof it have been studied in past few years. In the study of the cosmology of the f ( Q ) theory, it has been foundthat the accelerated expansion is the intrinsic property of the Universe without need of either exotic darkenergy or extra field [61]-[75].Recently, the f ( Q ) theory is further extended by Xu et al. [21] in the form of f ( Q, T ) theory by couplingthe non-metricity Q with the trace of the matter energy-momentum tensor T . The Lagrangian of the grav-itational field is assumed to be general function of both Q and T . The action in f ( Q, T ) theory is taken inthe form S = R (cid:0) π f ( Q, T ) + L m (cid:1) √− gd x . The field equations of the theory were obtained by varying thegravitational action with respect to both metric and connection. They investigated the cosmological impli-cations of the theory and obtained the cosmological evolution equations for a flat, homogeneous, isotropicgeometry, by assuming some simple functional form of f ( Q, T ). Arora et al. have recently studied f ( Q, T )gravity models with observational constraints [76].In the present study, we have also interpreted the cosmological model with Friedmann-Lemaitre-Robertson-walker (FLRW) Universe in f ( Q, T ) theory with f ( Q, T ) = αQ + βQ + γT and this supported by f ( R, T )-gravity form f ( R, T ) = R + αR + βT in which the presence of square term of R reveals the existence of darkenergy and dark matter. Using the field equations in a different approach, applying the energy conservationcondition ˙ ρ + 3 H ( ρ + p ) = 0, we have obtained the various cosmological parameters viz. Hubble parameter H ( z ), deceleration parameter q ( z ) etc. in terms of scale-factor a ( t ) as well as in terms of redshift z . Applying R -test formula and using the available observational data sets Hubble data sets [77]-[83], Joint Light CurveAnalysis (JLA) data sets [84] and union 2 . ρ and isotropic pressure p explicitly and then estimated the equation of state parameter ω = pρ .2he present paper is investigated in the following five sections: first section contains a brief introduction,in Sect. 2 some preliminary definitions and an overview of the f ( Q, T ) theory are given. Cosmologicalsolutions of the FLRW Universe in f ( Q, T ) theory is given in Sect. 3. Result analysis and discussions aregiven in Sect. 4. Finally, conclusions are summarized in Sect. 5.
The Einstein-Hilbert action for f ( Q, T ) gravity is defined in [21] as I = Z (cid:18) π f ( Q, T ) + L m (cid:19) d x √− g (1)where f ( Q, T ) is an arbitrary function of the non-metricity Q and the trace of the energy-momentum tensor T , and L m is the Lagrangian for matter source and g = det( g ij ) read as the determinant of the metric tensor g ij and Q ≡ − g ij ( L αβi L βjα − L αβα L βij ) (2)where L αβγ is known as deformation tensor defined by L αβγ = − g αλ ( ∇ γ g βλ + ∇ β g λγ − ∇ λ g βγ ) (3)The non-metricity Q and the energy-momentum tensor are defined respectively as Q α = Q iαi , T ij = − √− g δ ( √− gL m ) δg ij (4)and θ ij = g αβ δT αβ δg ij (5)By varying the action (1) with respect to the metric components, we obtain the following field equations: − √− g ∇ α ( f Q √− gP αij ) − f g ij + f T ( T ij + θ ij ) − f Q ( P iαβ Q αβj − Q αβi P αβj ) = 8 πT ij (6)where P αij is defined as the supper-potential of the model mentioned as in [21].Now, let us suppose the universe is isotropic and spatially flat and hence, it can be described by an FLRWmetric given by ds = − N ( t ) dt + a ( t ) ( dx + dy + dz ) (7)where a ( t ) is called scale factor which describes evolution of the universe. The increasing value of a ( t ) revealsthe expansion and decreasing value indicates the collapse of the universe and its rate is measured by Hubblefunction defined as H = ˙ aa . The non-metricity Q for FLRW metric is derived as Q = 6 H for lapse function N ( t ) = 1 [21]. Let us suppose that our universe is filled with perfect fluid and hence, the energy momentumtensor for perfect fluid is defined by T ij = ( ρ + p ) u i u j + pg ij and therefore, the θ ij is defined and obtained as θ ij = δ ij p − T ij = diag(2 ρ + p, − p, − p, − p ) (8)Let us introduce the following notations F ≡ f Q and 8 π ˜ G ≡ f T for simplicity of field equations. For theFLRW metric (7), the derived field equation (6) gives the following equations:8 πρ = f − F H − G G ( ˙ F H + F ˙ H ) (9)8 πp = − f F H + 2( ˙ F H + F ˙ H ) (10)3o obtain the evolution function for Hubble parameter, we have combined the Eqs. (9) and (10) as˙ H + ˙ FF H = 4 πF (1 + ˜ G )( ρ + p ) (11)In generalized f ( Q, T ) gravity the energy-conservation law is derived in [21] as˙ ρ + 3 H ( ρ + p ) = ˜ G π (1 + ˜ G )(1 + 2 ˜ G ) " ˙ S − (3 ˜ G + 2) ˙˜ G (1 + ˜ G ) ˜ G S + 6 HS (12)where S = 2( ˙ F H + F ˙ H ). Let us choose such a solution of the field equations (9) & (10) so that ˜ G = 0 and˙ ρ + 3 H ( ρ + p ) = 0 (13)Therefore, the Eq. (12) becomes ˙ S − (3 ˜ G + 2) ˙˜ G (1 + ˜ G ) ˜ G S + 6 HS = 0 (14)Here, we have taken f ( Q, T ) = αQ + βQ + γT which is obtained by replacing R by Q in f ( R, T )-gravityin which some authors have studied the form f ( R, T ) = R + αR + βT with R as Ricci scalar curvature and T is the trace of energy momentum tensor T ij and here, factor R is added to explain the late time accelerationin expanding universe and this study motivated us to investigate the above form of f ( Q, T ). Now, we obtain F = f Q = α + 2 βQ and 8 π ˜ G = f T = γ and also, ˙˜ G = 0 and using this in Eq. (14), we get˙ S + 6 HS = 0 (15)Integrating Eq. (15), we obtain the Hubble parameter H as H = 16 " β r α + 12 βka − αβ (16)where α , β , γ are model parameters and k is an integrating constant.The various observational data sets are available in terms of redshift z and it can be easily compared withtheoretical studies in terms of redshift. Hence, the various parameters need to derive in terms of redshift z and hence, we are used the relationship between the scale-factor and redshift give as a a = 1 + z (17)where a is the present value of scale factor and as a standard convention, we will take a = 1 throughoutthe study. Using Eq. (17) in (16), we get H ( z ) = 16 (cid:20) β p α + 12 βk (1 + z ) − αβ (cid:21) (18)Now, the deceleration parameter q ( t ) is defined as q ( t ) = − − ˙ HH and calculated as q ( t ) = − βka [ α + βka − α q α + βka ] (19)4his in terms of redshift obtained as q ( z ) = − βk (1 + z ) [ α + 12 βk (1 + z ) − α p α + 12 βk (1 + z ) ] (20)Now, from Eqs. (9) & (10) we have calculated the energy density ρ and isotropic pressure p respectively as ρ = − α π + γ ) H − β (4 π + γ ) H − γk π + γ )(8 π + γ ) a (21) p = α π + γ ) H + 27 β π + γ H − k (16 π + γ )4(4 π + γ )(8 π + γ ) a (22)From Eq. (16) one can see that for α = 0, we can find H ∝ a / , H ∝ a and H ∝ a and hence, ρ ∝ a which shows the existence of stiff matter in the universe. And in late time universe it converted into ordinarymatter and dark energy. In the above section, we have obtained the various parameters viz. Hubble function H ( z ), decelerationparameter q ( z ), energy density ρ and isotropic pressure p in terms of α , β , γ , k , z and H where α , β , γ , k , are called as model parameters. To explain the various observable properties of universe, we are to findthe best fit values of the model parameters α , β , γ , k with various observational data sets. Therefore, wehave found the best fit values of these model parameters for the best fit curve of Hubble function as well asapparent magnitude m ( z ) with observational data sets. The Hubble parameter shows the rate of geometrical evolution of the universe (expansion rate) and it can beestimated from observational data sets as in Table-1 (see [77]-[83]). The expression for the Hubble parameter H ( z ) is obtained as in Eq. (18). To obtain the best fit values of model parameters α , β and k , we havefound the best fit curve of Hubble function H ( z ) with the 29 observed values of Hubble constant as shownin Table-1, using the R -test formula as given below: R SN = 1 − P i =1 [( H i ) ob − ( H i ) th ] P i =1 [( H i ) ob − ( H i ) mean ] The case R = 1 shows the exact fit the value of model parameters α , β and k with observational data sets.Hence, we obtain the curve-fitting with greatest value of R . Thus, we have obtained the best fit curve ofthe Hubble function (18) shown as in figure 1. We have found the best fit values of model parameters α , β and k as α = − . β = 3 . × − and k = 1 . × (see Table-2) for maximum R = 0 . . H ( z ) ± .
13 and their R values only 9 .
02% far from the best one.5 H ( z ) σ H Reference z H ( z ) σ H Reference0 .
070 69 19 . .
600 87 . . .
100 69 12 [78] 0 .
680 92 8 [79]0 .
120 68 . . .
730 97 . .
170 83 8 [78] 0 .
781 105 12 [79]0 .
179 75 4 [79] 0 .
875 125 17 [79]0 .
199 75 5 [79] 0 .
880 90 40 [82]0 .
200 72 . . .
900 117 23 [78]0 .
270 77 14 [78] 1 .
037 154 20 [79]0 .
280 88 . . .
300 168 17 [78]0 .
350 76 . . .
363 160 33 . .
352 83 14 [79] 1 .
430 177 18 [78]0 .
400 95 17 [78] 1 .
530 140 14 [78]0 .
440 82 . . .
750 202 40 [78]0 .
480 97 62 [82] 2 .
300 224 8 [83]0 .
593 104 13 [79]Table 1: Hubble’s constant Table. z H ( z ) Observational H(z) data setsTheoretical H(z) for α = -3.041, β = 3.686 × -5 , k = 10 Figure 1: The best fit curve of Hubble function H ( z ) as in Eq. (18) with observed values of H ( z ) as shownin Table-1. The best fit values of model parameters α , β and k are obtained with 95% confidence level ofbounds.The Figure 1 represents the best fit curve of Hubble parameter H ( z ) for the obtained values of the con-stants α , β and k . For these values, it is observed that the curve is quiet consistent. The present value ofHubble parameter is calculated as H = 71 . ± .
13, which are very close to the recent observations [86].6arameters H ( z ) JLA SNe Ia α − . − . − . β . × − . × − . × − k . × . × . × H .
26 54 .
62 83 . R . . . . . . α , β and k for the best fit curve of Hubble function H ( z ) andapparent magnitude m ( z ) with different observational data sets ( H ( z ), JLA, SNe Ia) with 95% confidencelevel of bounds. The luminosity distance is measured the total flux of the source of light and is defined as D L = c (1+ z ) R z dzH ( z ) .Let us define the apparent magnitude in terms of D L to obtain the best fit values of model parameters α , β and k for the best fit curve of H ( z ) with SNe Ia data sets as m ( z ) = 16 .
08 + 5 log ( H D L . ). To find thebest fit curve of apparent magnitude m ( z ) with SNe Ia data sets we have considered 51 observed data setsof apparent magnitude m ( z ) from Joint Light curve Analysis (JLA) as in [84] and 580 observed data setsof apparent magnitude from union 2 . R -test formula: R SN = 1 − P Ni =1 [( m i ) ob − ( m i ) th ] P Ni =1 [( m i ) ob − ( m i ) mean ] where sums are run over 1 to N and N has values 51 for JLA data and 580 for union 2.1 compilation data sets.The ideal case R = 1 occurs when the observed data and theoretical function m ( z ) agree exactly. Onthe basis of maximum value of R , we get the best fit values of α , β and k for the apparent magnitude m ( z ) function which is given in Table 2 for both JLA and SNe Ia data sets. The best fit curve of m ( z ) withobserved values of m ( z ) are shown in figures 2 a & 2 b respectively.From Eq. (16) we can find the scale-factor a ( t ) in terms of Hubble constant as a ( t ) = (cid:20) βk ( α + 36 βH ) − α (cid:21) (23)One can calculate the value of scale-factor for any value of Hubble constant H using Eq. (23) and it cansee that as H → ∞ then a → z using curve-fitting or cosmographicstudy. Eq. (23) depicts that for β → a ( t ) → α = 0 the Hubbleparameter H ∝ a − which gives the power-law and exponential cosmology.7. z m ( z ) Observational m(z) JLA data setsTheoretical m(z) for α = - 2.814 β = 2.432 × -5 , k = 1.0063 × H = 54.62 b. z m ( z ) Observational m(z) SNe Ia data setsTheoretical m(z) for α = -2.940, β = 6.576 × -5 , k = 1.0575 × H = 83.53 Figure 2: a. The best fit curve of m ( z ) with observational JLA data sets [84] and b. best fit curve of m ( z )with union 2.1 compilation of supernovae data sets [85]. The best fit values of model parameters α , β and k are obtained with 95% confidence level of bounds. Deceleration Parameter
The deceleration parameter q ( z ) is the second of the observational parameter that mentions the nature ofevolution of geometrical of the universe. The expressions for the deceleration parameter (DP) q ( z ) in termsof scale-factor a ( t ) and redshift z are shown in Eqs. (19) & (20) respectively and Figure 3 shows the vari-ation of DP for the best fit values of model parameters α , β and k with three data sets. One can see that q ( z ) is an increasing function of redshift z and it shows signature-flipping (transition) point at z t within therange 0 . ≤ z t ≤ .
668 (as shown in Table-3), where q ( z t ) = 0 and q ( z ) < z < z t and q ( z ) > z > z t , also, q ( z ) → − z → − q ( z ) tends to a finite positive value as z → ∞ . The present valueDP is calculated as − ≤ q < q ( z ) over redshift z reveals that our universe is undergoing an accelerating phase at present and decelerating in early stageswhich are very close to the recent observations. Thus, our derived model depicts a transit phase model fromdecelerating to accelerating universe as predicted by the recent observational studies [1, 2, 3, 4].From Eq. (20) we can find the general expression for the transit value of redshift z = z t by taking q ( z ) = 0and solving for z as: z t = (cid:18) α βk (cid:19) − β → z t → ∞ i.e. the model of an ever accelerating universe. Also,we can see that for the finite value of model parameters α , β and k we obtain either an transit phase modelor decelerating universe by choosing suitable values of the model parameters. From the Table-3 we can seethat the transition point lies within the range 0 . ≤ z t ≤ .
668 for the best fit values of model parameterswith observational data sets. 8 -1 -0.5 0 0.5 1 1.5 2 2.5 3 q ( z ) -1-0.500.5 H(z)SNe IaJLA
Figure 3: The plot of deceleration parameter q ( z ) over redshift z for the best fit values α , β and k which ismentioned in Table-2. The third observational parameter is the age of the present universe t which is mentioned in Gyrs. The ageof the present Universe is calculated as Z tt dt = Z z dtdz dz (25)where t is the present age of the Universe and t ≤ t . On integration of left hand side of Eq. (25) and usingthe relation dzdt = − (1 + z ) H ( z ) in right hand side of (25), we obtain( t − t ) = Z z dz (1 + z ) H ( z ) (26)where H ( z ) is the Hubble parameter H ( z ) (see Eq. (18)) with three observational data sets. The plot ofcosmic time ( t − t ) H over redshift is shown in Figure 4. The parallel graph of time with z -axis shows ageof the present universe within the range 11 . ≤ t ≤ .
63 Gyrs which is mentioned in Table-3. One canconclude that it is very close to age of the present universe calculated in [86].Parameters H JLA SNe Ia q − . − . − . z t .
599 0 .
668 0 . H .
26 54 .
62 83 . t H . . . t .
38 16 .
63 11 . ( t - t ) H H(z)SNe IaJLA
Age of the present Universe
Figure 4: The plot of cosmic time ( t − t ) H over redshift z for the best fit values α , β and k as given inTable-2 Energy density ( ρ ) , Isotropic pressure ( p ) and EoS parameter ( ω ) The expression for energy density ρ is represented by the Eq. (21) and one can see that it has two singularitiesat γ ∈ {− π, − π } and ρ ≥ γ < − π and ρ ≤ γ > − π . Also, we can find that ρ is anincreasing function of redshift z with γ < − π which shows that as t → t ( z → ρ is decreasing functionof redshift which is consistent with cosmological studies in observational as well as in theoretical. Eq. (22)represents the expression for isotropic pressure p and we can see that the expression of p has two singularitiesat γ ∈ {− π, − π } and p ≤ γ except γ = − π, − π . We can find that the energy density ρ ≥ p ≤ γ < − π . Also, one can see that the derived modeltends to Einstein’s GR model for k = 0, β = 0 and γ = − π .a. z -1 -0.5 0 0.5 1 1.5 2 ρ × H(z)SNe IaJLA b. z -1 -0.5 0 0.5 1 1.5 2 2.5 3 p × -7-6-5-4-3-2-10 H(z)SNe IaJLA c. z -1 -0.5 0 0.5 1 1.5 2 2.5 3 ω -500-400-300-200-1000 H(z)SNe IaJLA
Figure 5: a. The plot of energy density ρ versus z , b. the plot of isotropic pressure p versus z , and c. the plotof Equation of state parameter (EoS) ω versus z for the best fit values of α , β and k mentioned in Table-2.Figure 5 a represents the evolution of energy density ρ for γ = − . π and it depicts that for z → ∞ , ρ → ∞ which shows the big-bang singularity and at late time it tends to zero, while figure 5 b reveals thebehaviour of isotropic pressure p over redshift z for γ = − . π and we can see that it is a decreasing functionof redshift and p → −∞ as z → ∞ . And figure 5 c depicts the variation of ω for γ = − . π over redshift10 and one can see that it shows ω is an increasing function of redshift z but ω ≤
0. From figure 5 c wecan obtain the present value of ω lies within the range − . ≤ ω ≤ − .
96 for 0 ≤ z ≤ ω is supported to an accelerated universe. In cosmology, we are known two geometrical parameters the Hubble parameter H = ˙ aa and the decelerationparameter q = − a ¨ a ˙ a where a ( t ) is the scale-factor and these parameters are describe the history of universe.There are another geometrical parameters called as statefinder diagnostic proposed in [91] which representthe geometric evolution of various stages of dark energy models [91, 92, 93]. The statefinder parameters r and s are defined in terms of scale-factor a ( t ) respectively as r = ... aaH s = r − q − ) (27)For the model we obtain statefinder parameters as r = 1 + 3 k [1 − α βk a − β k a ]2 a H q α + βka (28) s = 36 βk [1 − α βk a − β k a ][12 βk − a ( α + βka − α q α + βka )] (29)a. z -1 -0.5 0 0.5 1 1.5 2 2.5 3 r( z ) × -6-5-4-3-2-101 H(z)SNe IaJLA b. z -1 -0.5 0 0.5 1 1.5 2 2.5 3 s ( z ) × -0.500.511.522.5 H(z)SNe IaJLA
Figure 6: The plot of statefinder parameters r ( z ), s ( z ) for the best fit values of α , β and k mentioned inTable-2. Parameters H ( z ) JLA SNe Ia r − . − . − . s . . . s, r ) z →− (0 ,
1) (0 ,
1) (0 , r(z) -7000 -6000 -5000 -4000 -3000 -2000 -1000 0 1000 s ( z ) H(z)SNe IaJLA
X: 1Y: 0 b. q(z) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 r( z ) × -10-8-6-4-20 H(z)SNe IaJLA
Figure 7: The plot of statefinder parameters r ( z ), s ( z ) and q ( z ), r ( z ) for the best fit values of α , β and k mentioned in Table-2.The plot of r , s over z are represented in figure 6 a & 6 b and we can calculate the present value of r & s as − . ≤ r ≤ − . . ≤ s ≤ . z → − r → s →
0. Now, figure 7 a shows the plot of s ( z ) over r ( z ) and the variation of ( s, r ) represents the variouscosmological models [91, 92, 93] and we can see ( s, r ) → (0 ,
1) as z → − b represents the plot of r ( z ) over q ( z ) and it shows that as z → − q, r ) → ( − ,
1) and it confirms that our present and future universe is in accelerating phase of expansion.
In our research, we have taken the function f ( Q, T ) quadratic in Q and linear in T as f ( Q, T ) = αQ + βQ + γT where α , β and γ are model parameters. We have obtained the various cosmological parameters in Friedmann-Lemaitre-Robertson-walker (FLRW) Universe viz. Hubble parameter H , deceleration parameter q etc. interms of scale-factor as well as in terms of redshift z by constraining on energy conservation law. Forobservational constrains on the model, we have obtained the best fit values of model parameters using theavailable data sets like Hubble data sets H ( z ), Joint Light Curve Analysis (JLA) data sets and union 2 . R -test formula. We have calculated the present values of variousobservational parameters { t , q , H } . These are very close to the standard cosmological models. The mainfeatures of our model is as follows: • The derived Hubble function is constrained by observational data sets and the present value of Hubbleconstant is calculated in the range 54 . ≤ H ≤ .
53 which is compatible with [86]. • The deceleration parameter q shows the signature-flipping (transition) point within the range 0 . ≤ z t ≤ .
668 that depicts our universe has been undergoing in accelerating phase from t z Gyrs ago where8 . ≤ t z ≤ .
16 which is supported by [1, 2, 3, 4]. • Our derived model evolves from quintessence universe − < ω < − .
96 to phantom dominated universe − . < ω < − ω = − ≤ z ≤ • The age of the present universe is calculated in the range 11 . ≤ t ≤ .
63 compatible with [86]. • The statefinder diagnostic is also, analyzed and we obtained ( s, r ) → (0 ,
1) as z → − References [1] A. G. Riess, et al. , Observational Evidence from Supernovae for an Accelerating Universe and a Cosmo-logical Constant,
Astron. J. , 1009 (1998).[2] S. Perlmutter, et al. , Measurement of Ω and Λ from 42 high-redshift supernovae,
Astrophys. J. , 565(1999).[3] P. de Bernardis, et al. , A flat Universe from high-resolution maps of the cosmic microwave backgroundradiation,
Nature , 955 (2000).[4] A. G. Riess, et al. , A 3% solution: determination of the Hubble constant with the Hubble Space Telescopeand Wide Field Camera 3,
Astrophys. J. , 119 (2011).[5] P. A. R. Ade, et al. , Planck Collaboration, Planck 2015 results,
Astron. Astrophys. , A13 (2016).[6] Y. Akrami, et al. , Planck 2018 results. I. Overview and the cosmological legacy of Planck, arXiv:1807.06205 [astro-ph.CO] (2018).[7] N. Aghanim, et al. , Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 [astro-ph.CO](2018).[8] C. H. Brans and R. H. Dicke, Mach’s principle and a relativistic theory of gravitation,
Phys. Rev. ASer-2 , 925 (1961).[9] K. Nordtvedt, Post-Newtonian Metric for a General Class of Scalar-Tensor Gravitational Theories andObservational Consequences,
Astrophys. J. , 1059 (1970).[10] R. V. Wagoner, Scalar-tensor theory and gravitational waves,
Phys. Rev. D , 3209 (1970).[11] K. A. Dunn, A scalar-tensor theory of gravitation, J. Math. Phys. , 2229 (1974).[12] D. Saez and V. J. Ballester, A simple coupling with cosmological implications, Phys. Lett. A , 467(1986).[13] G. A. Barber, On two “self-creation” cosmologies,
Gen. Rel. Gravit. , 117 (1985).[14] S. Nojiri, S. D. Odintsov and M. Sasaki, Gauss-Bonnet dark energy, Phys. Rev. D , 123509 (2005).[15] S. Nojiri and S. D. Odintsov, Modified gravity with negative and positive powers of curvature: Unificationof inflation and cosmic acceleration, Phys. Rev. D et al. , f ( R, T ) gravity,
Phys. Rev. D , 024020 (2011).[17] Yi-Fu Cai, et al. , f ( T ) teleparallel gravity and cosmology, Rep. Prog. Phys. , 106901 (2016); arXiv:gr-qc/1511.07586.[18] M. Sharif and A. Ikram, Energy conditions in f ( G, T ) gravity,
Eur. Phys. J. C , 640 (2016).[19] I. Ayuso, J. B. Jimenez and A. de la Cruz-Dombriz, Consistency of universally nonminimally coupled f ( R, T, R µν T µν ) theories, Phys. Rev. D , 104003 (2013); arXiv: 1411.1636 [hep-th].[20] J. M. Nester and H. J. Yo, Symmetric teleparallel general relativity, Chin. J. Phys. , 113 (1999).[21] Y. Xu, G. Li, T. Harko, and S. D. Liang, f ( Q, T ) Gravity,
Eur. Phys. J. C
708 (2019).1322] S. Capozziello and S. Vignolo, On the well-formulation of the initial value problem of metric-affine f ( R )-gravity, Int. J. Geom. Meth. Mod. Phys. , 985 (2009).[23] S. Capozziello and S. Vignolo, The cauchy problem for metric-affine f ( R )-gravity in presence of aKlein–Gordon scalar field, Int. J. Geom. Meth. Mod. Phys. , 167 (2011).[24] S. Nojiri and S. D. Odintsov, Unified cosmic history in modified gravity: from f ( R ) theory to Lorentznon-invariant models, Phys. Rept. , 59 (2011).[25] S. Vignolo and L. Fabbri, Spin fluids in Bianchi-I f ( R )-cosmology with torsion, Int. J. Geom. Meth.Mod. Phys. f ( R )-gravity and inflation, Int. J. Geom. Meth. Mod. Phys. f ( R ) gravity, Int. J. Geom. Meth.Mod. Phys. f ( R )-gravity, Int. J. Geom.Meth. Mod. Phys. f ( R ) gravity in scalar-tensor theories, Int. J. Geom.Meth. Mod. Phys. Phys. Rev. D f ( R ) gravity, Int. J. Geom. Meth. Mod. Phys. et al. , Reconstructing f ( R ) gravity from a Chaplygin scalar field in de Sitter spacetimes, Int.J. Geom. Meth. Mod. Phys. F ( R, T ) gravity,
Eur. Phys. J. C , 2203 (2012).[34] D. R. K. Reddy, et al. , Kaluza-Klein universe with cosmic strings and bulk viscosity in f ( R, T ) gravity,
Astrophys. Space Sci. , 261 (2013).[35] R. L. Naidu, et al. , Bianchi type- V bulk viscous string cosmological model in f ( R, T ) gravity,
Astrophys.Space Sci. , 247 (2013).[36] M. F. Shamir, Bianchi type- I cosmology in f ( R, T ) gravity,
Jour. Exper. Theoret. Phys. , 705 (2016).[37] R. Zia, D. C. Maurya and A. Pradhan, Transit dark energy string cosmological models with perfect fluidin f ( R, T )-gravity,
Int. J. Geom. Meth. Mod. Phys. , 1850168 (2018).[38] H. Weyl, Gravitation und Elektrizit ¨ at, Sitzungsberg. Preuss. Akad. Wiss. (1918).[39] S. Nojiri, S. D. Odintsov and O. G. Gorbunova, Dark energy problem: from phantom theory to modifiedGauss-Bonnet gravity,
J. Phys. A , 6627 (2006).[40] K. Bamba, et al. , Finite-time future singularities in modified Gauss–Bonnet and f ( R, G ) gravity andsingularity avoidance,
Eur. Phys. J. C , 295 (2010).[41] K. Bamba, et al. , Energy conditions in modified f ( G ) gravity, Gen. Relativ. Gravit. , 112 (2017).[42] M. Sharif and H. I. Fatima, Noether symmetries in f ( G ) gravity, J. Exper. Theoret. Phys. , 121(2016).[43] V. K. Oikonomou, Singular bouncing cosmology from Gauss-Bonnet modified gravity,
Phys. Rev. D f ( G , T ) gravity, Eur. Phys. J. C , 1(2017).[45] M. Sharif and A. Ikram, Stability analysis of Einstein universe in f ( G, T ) gravity,
Int. J. Mod. Phys. D
26 (08) , 1750084 (2017).[46] C. Moller, The four-momentum of an insular system in general relativity,
Nuclear Physics , 330 (1964).[47] C. Pellegrini and J. Plebanski, Tetrad fields and gravitational fields, Mat. Fys. Skr. Dan. Vid. Selsk. ,4 (1963).[48] K. Hayashi and T. Shirafuji, New general relativity, Phys. Rev. D , 3524 (1979).[49] R. Ferraro and F. Fiorini, Modified teleparallel gravity: inflation without an inflaton, Phys. Rev. D ,084031 (2007).[50] R. Aldrovandi and J. G. Pereira, TeleparallelGravity, vol. 173 (Fundamental Theories of Physics)Springer , Heidelberg, (2013).[51] T. Harko, et al. , Nonminimal torsion-matter coupling extension of f ( T ) gravity, Phys. Rev. D , 124036(2014).[52] S. Capozziello, G. Lambiase and E. N. Saridakis, Constraining f ( T ) teleparallel gravity by big bangnucleosynthesis, Eur. Phys. J. C , 576 (2017).[53] R. D’Agostino and O. Luongo, Growth of matter perturbations in nonminimal teleparallel dark energy, Phys. Rev. D , 124013 (2018).[54] T. Koivisto and G. Tsimperis, The spectrum of teleparallel gravity, Universe , 80 (2019)[55] J. G. Pereira and Y. N. Obukhov, Gauge structure of teleparallel gravity, Universe , 139 (2019).[56] A. Paliathanasis, J. D. Barrow and P. G. L. Leach, Cosmological solutions of f ( T ) gravity, Phys. Rev.D , 023525 (2016).[57] R. C. Nunes, S. Pan and E. N. Saridakis, New observational constraints on f ( T ) gravity from cosmicchronometers, J. Cosmol. Astropart. Phys. , 011 (2016).[58] S. D. Odintsov, V. K. Oikonomou and E. N. saridakis, Superbounce and loop quantum ekpyrotic cos-mologies from modified gravity: f ( G ), f ( G ) and f ( T ) theories, arXiv:1501.06591 [gr-qc] (2015).[59] K. Bamba, S. D. Odintsov and D. Saez-Gomez, Conformal symmetry and accelerating cosmology inteleparallel gravity, Phys. Rev. D , 084042 (2013).[60] S. Chattopadhyay, A reconstruction scheme for f ( T ) gravity and its consequences in the perturbationlevel, Int. J. Geom. Meth. Mod. Phys. , 1850025 (2018).[61] M. Adak and O. Sert, A solution to symmetric teleparallel gravity, Turk. J. Phys. , 1 (2005).[62] M. Adak, The symmetric teleparallel gravity, Turk. J. Phys. , 379 (2006).[63] M. Adak, et al. , Symmetric teleparallel gravity: Some exact solutions and spinor couplings, Int. J. Mod.Phys. A , 1 (2013).[64] J. Beltran Jimenez and T. S. Koivisto, Spacetimes with vector distortion: Inflation from generalisedWeyl geometry, Phys. Lett. B , 400 (2016).[65] A. Golovnev, T. Koivisto and M. Sandstad, On the covariance of teleparallel gravity theories,
Class.Quant. Grav. , 145013 (2017).[66] I. Mol, The Non-Metricity Formulation of General Relativity, Adv. Appl. Clifford Algebras , 2607(2017). 1567] M. Adak, Gauge approach to the symmetric teleparallel gravity, Int. J. Geom. Methods Mod. Phys. ,1850198 (2018).[68] I. Soudi, et al. , Polarization of gravitational waves in symmetric teleparallel theories of gravity and theirmodifications, Phys. Rev. D , 044008 (2019); arXiv:1810.08220 [gr-qc].[69] A. Conroy and T. Koivisto, The spectrum of symmetric teleparallel gravity,
Eur. Phys. J. C , 923(2018).[70] T. Harko, et al. , Coupling matter in modified Q gravity, Phys. Rev. D , 084043 (2018).[71] M. Hohmann, et al. , Propagation of gravitational waves in symmetric teleparallel gravity theories, Phys.Rev. D , 024009 (2019).[72] K. F. Dialektopoulos, T. S. Koivisto and S. Capozziello, Noether symmetries in symmetric teleparallelcosmology, Eur. Phys. J. C , 606 (2019); arXiv:1905.09019 [gr-qc].[73] J. Lu, X. Zhao and G. Chee, Cosmology in symmetric teleparallel gravity and its dynamical system, Eur. Phys. J. C , 530 (2019).[74] F. S. N. Lobo, et al. , Novel couplings between nonmetricity and matter, arXiv:1901.00805 [gr-qc] (2019).[75] J. Beltran Jimenez, et al. , Cosmology in f ( Q ) geometry, Phys. Rev. D , 103507 (2020)arXiv:1906.10027 [gr-qc].[76] S. Arora, et al. , f ( Q, T ) gravity models with observational constraints,
Phys. of Dark Univ. Phys. Rev. D , 023532 (2013).[78] D. Adak, et al. , Reconstructing the equation of state and density parameter for dark energy fromcombined analysis of recent SNe Ia, OHD and BAO data, arXiv:1102.4726[astro-ph.CO] (2011).[79] L. Amendolaand S. Tsujikawa, Dark energy: theory and observations, Cambridge University Press (2010).[80] R. Aurich and F. Steiner, Dark energy in a hyperbolic universe,
Mon. Not. Roy. Astron. Soc. , 735(2002).[81] M. Arabsalmani, V. Sahni and T. D. Saini, Reconstructing the properties of dark energy using standardsirens,
Phys. Rev. D , 083001 (2013).[82] S. W. Allen, et al. , Improved constraints on dark energy from Chandra X-ray observations of the largestrelaxed galaxy clusters, Mon. Not. Roy. Astron. Soc. , 879 (2008); arXiv:0706.0033 [astro-ph].[83] L. Anderson, et al. , The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey:baryon acoustic oscillations in the Data Release 9 spectroscopic galaxy sample,
Mon. Not. Roy. Astron.Soc. , 3435 (2012) .[84] M. Betoule, et al. , Improved cosmological constraints from a joint analysis of the SDSS-II and SNLSsupernova samples, arXiv:1401.4064v2 [astro-ph.CO].[85] N. Suzuki, et al. , The Hubble Space Telescope cluster supernova survey. V. Improving the dark-energyconstraints above z >
Astrophys. J. Proc. Natl. Acad. Sci. USA , et al. , New constraints on Ω m , Omega and ω from an independent set of eleven high redshiftsupernovae observed with HST, Astrophys. J. , 102 (2003).1688] M. Tegmark et al. , The three-dimensional power spectrum of galaxies from the sloan digital sky survey,
Astrophys. J. , 702 (2004).[89] WMAP Collab. (G. Hinshaw et al. ), Five-yearWilkinson microwave anisotropy probe (WMAP) obser-vation: Likelihoods and parameters from theWMAP data,
Astrophys. J. Suppl. Ser. , 306 (2009).[90] E. Komatsu et al. , Five-Year Wilkinson microwave anisotropy probe (WMAP) observations: Cosmolog-ical interpretation,
Astrophys. J. Suppl. Ser. , 330 (2009).[91] V. Sahni, et al. , Statefinder-a new geometrical diagnostic of dark energy,
JETP Lett. , 201 (2003).[92] U. Alam, et al. , Exploring the expanding universe and dark energy using the Statefinder diagnostic, Mon. Not. R. Astron. Soc. , 1057 (2003).[93] M. Sami, et al. , Cosmological dynamics of a nonminimally coupled scalar field system and its late timecosmic relevance,
Phys. Rev. D86