Growth of matter perturbations in an interacting dark energy scenario emerging from metric-scalar-torsion couplings
GGrowth of matter perturbations in an interacting dark energy scenarioemerging from metric-scalar-torsion couplings
Mohit Kumar Sharma ∗ and Sourav Sur † Department of Physics & AstrophysicsUniversity of Delhi, New Delhi - 110 007, India
Abstract
We study the growth of linear matter density perturbations in a modified gravity approach of scalarfield couplings with metric and torsion. In the equivalent scalar-tensor formulation, the matter fields in theEinstein frame interact as usual with an effective dark energy component, whose dynamics is presumablygoverned by a scalar field that sources a torsion mode. As a consequence, the matter density ceases tobe self-conserved, thereby making an impact not only on the background cosmological evolution but alsoon the perturbative spectrum of the local inhomogeneities. In order to estimate the effect on the growthof the linear matter perturbations, with the least possible alteration of the standard parametric form ofthe growth factor, we resort to a suitable Taylor expansion of the corresponding exponent, known as thegrowth index, about the value of the cosmic scale factor at the present epoch. In particular, we obtain anappropriate fitting formula for the growth index in terms of the coupling function and the matter densityparameter. While the overall parametric formulation of the growth factor is found to fit well with thelatest redshift-space-distortion (RSD) and the observational Hubble (OH) data at low redshifts, the fittingformula enables us to constrain the growth index to well within the concordant cosmological limits, thusensuring the viability of the formalism.
Keywords:
Cosmological perturbations, dark energy theory, modified gravity, torsion, cosmology of theoriesbeyond the SM.
The effect of the evolving dark energy (DE) on the rate of the large-scale structure (LSS) formation hasbeen a prime area of investigation in modern cosmology, particularly from the point of view of asserting thecharacteristics of the respective DE component [1–5]. While the observations grossly favour such a componentto be a cosmological constant Λ [6–12], a stringent fine-tuning problem associated with the correspondingmodel, viz. ΛCDM (where CDM stands for cold dark matter), has prompted extensive explorations of adynamically evolving DE from various perspectives. Moreover, certain observational results do provide somescope of a plausible dynamical DE evolution, albeit upto a significant degree of mildness. In this context, itis worth noting that however mild the DE dynamics may be, at the standard Friedmann-Robertson-Walker(FRW) background cosmological level, there may be substantial effects of such dynamics on the spectrum ofthe linear matter density perturbations. Hence, the analysis of the observational data on the evolution ofsuch perturbations, or the LSS growth data, is crucial for constraining dynamical DE models of all sort. ∗ email: [email protected] † email: [email protected], [email protected] a r X i v : . [ g r- q c ] F e b part from the commonly known dynamical DE models involving scalar fields (such as quintessence,kessence, and so on [13–21]), a considerable interest has developed in recent years on the cosmological sce-narios emerging from scalar-tensor equivalent modified gravity (MG) theories [22–26] that stretch beyond thestandard principles of General Relativity (GR). Such scenarios are particularly useful for providing plausibleresolutions to the issue of cosmic coincidence which one usually encounters in scalar field DE models and inthe concordant ΛCDM model. One resolution of course comes from the consideration of plausible contactinteraction(s) between a scalar field induced DE component and the matter field(s) [1, 3, 27–47], which thescalar-tensor formulations naturally lead to, under conformal transformations [48–57]. A DE-matter (DEM)interaction makes the background matter density ρ ( m ) ( z ) drifting from its usual ( dust -like) evolution withredshift z , thereby affecting the drag force on the matter perturbations. The evolution of the matter densitycontrast δ ( m ) ( z ) := δρ ( m ) ( z ) /ρ ( m ) ( z ) and the growth factor f ( z ) of the matter perturbations are thereforenot similar to those in the non-interacting models, in which the field perturbations decay out in the sub-horizon regime, while oscillating about a vanishing mean value. Actually, the decaying nature persists in theinteracting scenarios as well, however with the oscillations about a value proportional to the amount of theinteraction, measured by the strength of the scalar field and matter coupling. As such, the field perturbationscontribute to the velocity divergences of the matter, affecting in turn the evolution of δ ( m ) ( z ) [3, 58]. Strik-ingly enough, a DEM interaction can make the growth factor f ( z ) acquiring a value > z , whichnecessitates the modifications of the commonly known f ( z ) parametrizations in the literature [59–64], suchas the well-known parametrization f ( z ) = (cid:2) Ω ( m ) ( z ) (cid:3) γ ( z ) , where Ω ( m ) ( z ) is the matter density parameter and γ ( z ) is the so-called growth index [60–72]. Our objective in this paper is to attempt such a modification anddemonstrate its utilization in constraining a DEM scenario emerging from a typical scalar-tensor equivalent‘geometric’ alternative of GR, viz. the metric-scalar-torsion (MST) cosmological theory, formulated recentlyby one of us (SS) [73–75], on the basis of certain considerations drawn from robust argumentations that havebeen prevailing for a long time [76–80].MST essentially forms a class of modified (or ‘alternative’) gravity theories that contemplates on theappropriate gravitational coupling(s) with scalar field(s) in the Riemann-Cartan ( U ) space-time geometry,endowed with curvature as well as torsion . The latter being an inherent aspect of a general metric-compatibleaffine connection, is considered as the entity that naturally extends the geometric principles of GR, notonly from a classical viewpoint, but also from the perspective a plausible low energy manifestation of afundamental (quantum gravitational) theory . Nevertheless, conventional U theories (of Einstein-Cartantype) are faced with a stringent uniqueness problem while taking the minimal couplings with scalar fields intoconsideration [76–80]. Such couplings are simply not conducive to any unambiguous assertion of equivalentLagrangians upon eliminating boundary terms in the usual manner. The obvious wayout is the considerationof explicit non-minimal (or, contact) couplings of the scalar field(s) with, most appropriately, the entire U Lagrangian given by the U curvature scalar (cid:101) R [73]. For any particular non-minimal coupling of a scalarfield φ with (cid:101) R , the resulting (MST) action turns out to be equivalent to the scalar-tensor action, as thetrace mode of torsion, T µ , gets sourced by the field φ , by virtue of the corresponding (auxiliary) equationof motion. On the other hand, torsion’s axial (or, pseudo-trace) mode A µ can lead to an effective potential,for e.g. a mass term m φ (with m = constant) in that scalar-tensor equivalent action, upon implementinga norm-fixing constraint ( A µ A µ = constant) as in the Einstein-aether theories [151–153], or incorporating a φ -coupled higher order term ( A µ A µ ) [73]. Such a mass term is shown to play a crucial role in giving rise to aviable cosmological scenario marked by a φ -induced DE component with a weak enough dynamical evolutionamounting to cosmological parametric estimations well within the corresponding observational error limits forΛCDM. This also corroborates to the local gravitational bounds on the effective Brans-Dicke (BD) parameter w , which turns out to be linear in the inverse of the MST coupling parameter β [73].Particularly intriguing is the MST cosmological scenario that emerges under a conformal transformation See the hefty literature on the vast course of development of the torsion gravity theories in various contexts, the physicalimplications and observable effects of torsion thus anticipated, as well as searched extensively over several decades [80–150]. a priori in the form of dust ). Nevertheless, the crude estimate of β (or of the parameter s = 2 β , that appears in the exact solution of the Friedmann equations), obtained under the demand of asmall deviation from the background ΛCDM evolution [73], requires a robust reconciliation at the pertur-bative level. On the other hand, the methodology adopted here can in principle apply to any scalar-tensorcosmological scenario, once we resort to the dynamics in the Einstein frame.Now, the methodology of our analysis purports to fulfill our objective mentioned above. Specifically, wetake the following course, and organize this paper accordingly: in section 2, we review the basic tenets of MSTcosmology in the standard FRW framework, and in particular, the exact solution of the cosmological equationsin the Einstein frame that describes a typical interacting DE evolution. Then in the initial part of section 3, weobtain the differential equations for δ ( m ) ( z ) and f ( z ), and get their evolution profiles by numerically solvingthose equations for certain fiducial settings of the parameters s = 2 β and Ω ( m ) ≡ Ω ( m ) (cid:12)(cid:12) z =0 . Thereafter,in subsection 3.1, we resort to a suitable growth factor parametrization, demanding that an appropriateexpansion of the growth index γ ( z ) about the present epoch ( z = 0) should adhere to the observationalconstraints on the growth history predictions at least upto z (cid:39) f σ (8) ( z ), where f ( z ) is as given by its chosenparametrization, and σ (8) ( z ) is the root-mean-square amplitude of matter perturbations within a sphere ofradius 8 Mph − . Finally, in section 4, we estimate the requisite parameters s , Ω ( m ) and σ (8) ≡ σ (8) (cid:12)(cid:12) z =0 ,and hence constrain the model by fitting f σ (8) ( z ) with a refined sub-sample of the redshift-space-distorsion(RSD) data, and its combination with the observational Hubble data [176]. In section 5, we conclude with asummary of the work, and an account on some open issues. Conventions and Notations : We use metric signature ( − , + , + , +) and natural units (with the speed of light c = 1), and denote the metric determinant by g , the Planck length parameter by κ = √ πG N (where G N is the Newton’s gravitational constant) and the values of parameters or functions at the present epoch by anaffixed subscript ‘0’. As mentioned above, an intriguing scenario of an effective DEM interaction emerges from a typical scalar-tensor equivalent MG formulation, viz. the one involving a non-minimal metric-scalar-torsion (MST) coupling,in the Einstein frame [73]. Let us first review briefly the main aspects of such a formalism, and the emergentcosmological scenario in the standard FRW framework.Torsion, by definition, is a third rank tensor T αµν which is anti-symmetric in two of its indies ( µ and ν ), be-cause of being the resultant of the anti-symmetrization of a general asymmetric affine connection ( (cid:101) Γ αµν (cid:54) = (cid:101) Γ ανµ ),that characterizes the four-dimensional Riemann-Cartan (or U ) space-time geometry. The latter howeverdemands the metric-compatibility, viz. the condition (cid:101) ∇ α g µν = 0, where (cid:101) ∇ α is the U covariant derivativedefined in terms of the corresponding connection (cid:101) Γ αµν . Such a condition leads to a lot of simplification in theexpression for the U curvature scalar equivalent, (cid:101) R , which is usually considered as the free U Lagrangianin analogy with the free gravitational Lagrangian in GR, viz. the Riemannian (or R ) curvature scalar R .Specifically, (cid:101) R gets reduced to a form given by R , plus four torsion-dependent terms proportional to thenorms of irreducible modes, viz. the trace vector T µ ≡ T αµα , the pseudo-trace vector A µ := (cid:15) αβγµ T αβγ andthe (pseudo-)tracefree tensor Q αµν := T αµν + ( δ αµ T ν − δ αν T µ ) − (cid:15) αµνσ A α , as well as the covariant divergenceof T µ [80]. In absence of sources (or the generators of the so-called canonical spin density ), all the torsionterms drop out, and hence the U theory effectively reduces to GR. The situation remains the same forminimal couplings with scalar fields as well. However, such minimal couplings are themselves problematic,when it comes to assigning the effective Lagrangian uniquely upon eliminating the boundary terms [76–80].3n easy cure is to resort to distinct non-minimal couplings of a given scalar field φ , in general, with each ofthe constituent terms in (cid:101) R [80]. However, this implies the involvement of more than one arbitrary couplingparameters, which may affect the predictability and elegance of the theory. Hence, it is much reasonable toconsider a non-minimal φ -coupling with the entire (cid:101) R , so that there is a unique (MST) coupling parameter(to be denoted by β , say) [73].Eliminating boundary terms, we obtain the auxiliary equation (or constraint) T µ = 3 φ − ∂ µ φ , whichimplies that the (presumably primordial, and a priori massless) scalar field φ acts as a source of the tracemode of torsion. Considering further, a mass term m φ induced by torsion’s axial mode A µ , via one of thepossible ways mentioned above (in the Introduction), we get the effective MST action [73]: S = (cid:90) d x √− g (cid:20) βφ R − − β g µν ∂ µ φ ∂ ν φ − m φ + L ( m ) ( g µν , { ψ } ) (cid:21) , (1)which is nothing but the scalar-tensor action in presence of minimally coupled matter fields ( { ψ } ) describedby the Lagrangian L ( m ) , in the Jordan frame.Under a conformal transformation g µν → (cid:98) g µν = ( φ/φ ) g µν and field redefinition ϕ := φ ln ( φ/φ ) , with φ = ( κ √ β ) − — the value of φ at the present epoch t = t , one obtains the Einstein frame MST action (cid:98) S = (cid:90) d x (cid:112) − (cid:98) g (cid:34) (cid:98) R κ − (cid:98) g µν ∂ µ ϕ ∂ ν ϕ − Λ e − ϕ/φ + (cid:98) L ( m ) ( (cid:98) g µν , ϕ, { ψ } ) (cid:35) , (2)where (cid:98) R is the corresponding (Ricci) curvature scalar, and κ = √ πG N denotes the gravitational couplingfactor . The parameter Λ = m φ , which amounts to the effective field potential at t = t , and (cid:98) L ( m ) ( (cid:98) g µν , ϕ, { ψ } ) = e − ϕ/φ L ( m ) ( g µν , { ψ } ) , (3)is the transformed matter Lagrangian, which depends on the field ϕ both explicitly as well as implicitly(since g µν = g µν ( (cid:98) g µν , ϕ )). It is in fact this ϕ -dependence which leads to the DEM interaction in the standardcosmological setup, as we shall see below. Note also that, by definition, ϕ (cid:12)(cid:12) t = t = 0.Dropping the hats ( (cid:98) ), we express the gravitational field equation and the individual matter and field(non-)conservation relations in the Einstein frame as follows: R µν − g µν R = κ (cid:104) T ( m ) µν + T ( ϕ ) µν (cid:105) , (4) ∇ α (cid:16) g αν T ( m ) µν (cid:17) = − ∇ α (cid:16) g αν T ( ϕ ) µν (cid:17) = − T ( m ) ∂ µ ϕφ , (5)where T ( m ) µν and T ( ϕ ) µν are the respective energy-momentum tensors for matter and scalar field and T ( m ) ≡ g µν T ( m ) µν denotes the trace of T ( m ) µν .Considering the matter to be a priori in the form of a pressure-less fluid (viz. ‘dust’), we have in thestandard spatially flat FRW framework, T µ ( m ) ν = diag (cid:2) − ρ ( m ) , , , (cid:3) , so that − T ( m ) = ρ ( m ) is just thematter density, which is purely a function of the cosmic time t . Because of the interaction (5), the matterdensity ceases to have its usual dust-like evolution (i.e. ρ ( m ) ( t ) (cid:54)∝ a − ( t ), where a ( t ) is the FRW scale factor).Nevertheless, the above Eqs. (4) and (5) are shown to be solvable in an exact analytic way, for the configuration ϕ ( t ) = s φ ln [ a ( t )] , ρ ( m ) ( t ) ∝ a − (3+ s ) ( t ) , (6) Ignoring of course, any external source for the tensorial mode Q αµν , which therefore vanishes identically. This is particularlyrelevant to what we intend to study here, viz. a homogeneous and isotropic cosmological evolution in presence of torsion, whichis plausible only when the latter’s modes are severely constrained. One such constraint is Q αµν = 0 [97]. This can be actually be retrieved from the relationship κ = ( φ/φ ) κ eff ( φ ) , where κ eff ( φ ) = ( φ √ β ) − is the effective (running)gravitational coupling one has in the Jordan frame. s = 2 β [73]. Consequently, the matter density parameter Ω ( m ) ( a )is expressed as Ω ( m ) ( a ) := ρ ( m ) ( a ) ρ ( a ) = (3 − s ) Ω ( m ) a − (3 − s ) ( m ) ( a − (3 − s ) −
1) + (3 − s ) , (7)where ρ ( a ) is the total (or critical) density of the universe and Ω ( m ) is the value of Ω ( m ) at the present epoch( t = t , whence a = 1). Using the Friedmann and Raychaudhuri equations we can then express the Hubbleparameter and total EoS parameter of the system, respectively, as H ( a ) := ˙ aa = H (cid:16) − s (cid:17) − / (cid:104) Ω ( m ) a − (3+ s ) + (cid:16) − s − Ω ( m ) (cid:17) a − s (cid:105) / , (8) w ( a ) := p ( a ) ρ ( a ) = − ( m ) ( a ) + 2 s , (9)where H = H ( a = 1) is the Hubble constant, and p ( a ) denotes the total pressure. Note that in thelimit s →
0, the above equations reduce to the corresponding ones for ΛCDM. Therefore one can directlyestimate the extent to which the MST cosmological scenario can deviate from ΛCDM, by demanding thatsuch a deviation should not breach the corresponding 68% parametric margins for ΛCDM. This would in turnprovide an estimation of the parameter s , which has actually been carried out in [73], using the Planck 2015and the WMAP 9 year results. The upper bound on s , thus obtained, is of the order of 10 − . Nevertheless, arather robust reconciliation is required from an independent analysis, for instance, using the RSD and H ( z )observations, which we endeavor to do in this paper. In this section, we discuss the evolution of linear matter density perturbations in the deep sub-horizon regimefor the aforementioned Einstein frame background MST cosmological scenario. The perturbations can bestudied in the well-defined conformal Newtonian gauge. The metric in this gauge is given as [3] ds = e N [ − (1 − H − dN + (1 + 2Φ) δ ij dx i dx j ] , (10)where N := ln a ( t ) is the number of e-foldings, H is the conformal Hubble parameter and Φ is the Bardeenpotential. Note that we have taken the same potential Φ in both temporal and spatial part of the metricunder the assumption of a vanishing anisotropic stress.The evolution of the matter density contrast δ ( m ) depends on the divergence or convergence of the peculiarvelocity (cid:126)vvv ( m ) via the perturbed continuity equation dδ ( m ) dN = − θ ( m ) , where θ ( m ) := ∇ · (cid:126)vvv ( m ) . (11)On the other hand, the Euler equation for matter perturbations is given by dθ ( m ) dN = − (cid:34) θ ( m ) (cid:18) − w − κ √ s dϕdN (cid:19) + (cid:98) λ − (cid:18) Φ + κ (cid:114) s δϕ (cid:19)(cid:35) , (12)where (cid:98) λ ≡ H /k (with k being the comoving wavenumber), andΦ (cid:39) (cid:98) λ Ω ( m ) δ ( m ) , and < δϕ > (cid:39) (cid:98) λ (cid:114) s ( m ) δ ( m ) , (13)considering only the mean value of δϕ , as it shows a damped oscillatory behavior in the sub-horizon regime.5 and δϕ both being proportional to (cid:98) λ , become negligible in the deep sub-horizon limit ( (cid:98) λ (cid:28) θ ( m ) ( N ), because of the (cid:98) λ − pre-factorin the second term of Eq. (12). As a consequence, the DE perturbation δϕ which itself is negligible in thesub-horizon regime (despite being scale-dependent) may, by virtue of its coupling with matter, lead to asignificant effect on the growth of matter density perturbations.Eqs. (9), (11) and (12) yield the second-order differential equation dδ ( m ) dN + (cid:34) − s ) − ( m ) (cid:35) dδ ( m ) dN = 3(1 + s )2 Ω ( m ) δ ( m ) , (14)which can be reduced to the following first-order differential equation: dfdN + f + (cid:34) − s ) − ( m ) (cid:35) f = 3(1 + s )2 Ω ( m ) , (15)by defining the so-called growth factor f ( N ) := d [ln δ ( m ) ] /dN [154–158]. Due to the pre-factor (1 + s ) in ther.h.s. of Eq. (15), the function f ( N ) can cross the unity barrier at high redshifts (whence Ω ( m ) → f ( z ) for a fixed Ω ( m ) = 0 . s , including s = 0 (the ΛCDM case). Fig. (1b), on the other hand, depicts the evolution of δ ( m ) ( z ), whichtends to increase with s for a fixed Ω ( m ) = 0 . s = s = s = zf ( z ) Ω ( m ) = (a) Growth factor evolution for fiducial Ω ( m ) and s . s = s = s = z δ ( m ) ( z ) Ω ( m ) = (b) Density contrast evolution for fiducial Ω ( m ) and s . Figure 1:
Functional variations of the growth factor and the matter density contrast, f ( z ) and δ ( m ) ( z ) respectively, in theredshift range z ∈ [0 , ( m ) = 0 . s = 0, 0 .
01, and 0 . As mentioned earlier, following the well-known prescription of [59, 60] we may consider parametrizing thegrowth factor f ( z ) as [Ω ( m ) ( z )] γ ( z ) . However, such a parametrization does not explain the crossing of f ( z )from < > f ( z ) always6estricted within the range [0 ,
1] at all redshifts which in our case is not true. So to alleviate this limitation,we propose the ansatz: f ( z ) = (1 + s ) (cid:104) Ω ( m ) ( z ) (cid:105) γ ( z ) , (16)which evidently implies f ( z ) approaching 1 + s at large redshifts (whence Ω ( m ) → γ ( z ).In particular, choosing to express the growth index as a function of the scale factor a , we in this paper resortto the following truncated form of its Taylor expansion about a = 1 (which corresponds to the present epoch): γ ( a ) = γ + γ (1 − a ) , with γ , γ := constants , (17)as in [62, 70]. Note that this parametrization is valid atleast upto a redshift z (cid:39) z = 1. In fact, it is rather convenient for us to re-write Eq. (17) as γ ( N ) = γ + γ (1 − e N ) with γ = γ ( N ) (cid:12)(cid:12) N =0 , γ = dγ ( N ) dN (cid:12)(cid:12)(cid:12) N =0 , (18)where γ = γ ( N ) (cid:12)(cid:12) N =0 = 1ln Ω ( m ) ln (cid:18) f s (cid:19) [ f = f | N =0 ] , (19) γ = dγ ( N ) dN (cid:12)(cid:12)(cid:12) N =0 = 1ln Ω ( m ) (cid:104) γ ( s − ( m ) ) + (1 + s )(Ω ( m ) ) γ + 2(1 − s ) − (cid:16) Ω ( m ) + (Ω ( m ) ) − γ (cid:17)(cid:105) , (20)by Eqs. (15) and (16).For the ΛCDM case ( s = 0), assuming Ω ( m ) = 0 .
3, one gets γ (cid:39) .
555 and γ (cid:39) − . γ can discriminate between various DE models and modified gravity theories. For instance, theminimal level Dvali-Gabadadze-Porrati (DGP) model predicts (0 . < γ < . Let us now focus on determining the parametric set p ( θ ) = { s, Ω ( m ) , σ (8) , γ , γ } . While the form of theparameter γ is already obtained in terms of s , Ω ( m ) and γ , we require to assert the form of γ in the firstplace. However, as we see from Eq. (19), γ depends on s and Ω ( m ) as well. Hence we resort to solvingnumerically the differential equation (15), by taking s ∈ [0 , .
1] and Ω ( m ) ∈ [0 . , .
4] (which are of coursefairly wide range of values), and for a step-size of 0 .
01. Using Eq. (19) thereafter, we obtain the following fit: γ (cid:39) . ( m ) ] . − . s Ω ( m ) . (21)In order to verify the validity of this fitting, let us take the Ω ( m ) = 0 .
3, say, and the limit s →
0. Eq. (21)then gives γ (cid:39) .
555 which is precisely what we had estimated theoretically, for the ΛCDM case, in the lastsubsection, by using Eqs. (15) and (19). The goodness of the fit is illustrated in Figs. (2a) and (2b), in whichwe have plotted the fractional error in the fitting, viz. E f ( z )( = [ f F ( z ) − f ( z )] /f ( z ) with z ∈ [0 , . ( m ) = 0 . s , and for a fixed s = 0 .
01 and a range of fiducial values ofΩ ( m ) , respectively. In both the cases, the error turns out to be (cid:39) .
2% at z (cid:39)
1, indicating a fair amount ofthe accuracy of the fit. 7 = s = s = s = s = - - - - - - z E f ( z ) Ω ( m ) = (a) Growth factor fitting error for fixed Ω ( m ) and variable s . Ω ( m ) = Ω ( m ) = Ω ( m ) = Ω ( m ) = Ω ( m ) = - - - - - - - z E f ( z ) s = (b) Growth factor fitting error for fixed s and variable Ω ( m ) . Figure 2:
Functional variations of the growth factor fitting error, E f ( z ), in the redshift range z ∈ [0 , . After formulating γ and γ in terms of s and Ω ( m ) , we are left with only three parameters s , Ω ( m ) and σ (8) in hand. So in order to estimate them from the observations use the f σ (8) ( z ) observations from variousgalaxy data surveys [159–169], we will now proceed to perform the statistical analysis, in particular MCMCsimulation to estimate our model parameters. Theoretically, ( f σ (8) ) th ( z ) can be written as [170–175]( f σ (8) ) th ( z ) = f ( z ) σ (8) δ ( m ) ( z ) δ ( m ) , where σ (8) = σ (8) | z = z . (22)which can be explicitly written as( f σ (8) ) th ( N ) = σ (8) (1 + s )(Ω ( m ) ) γ ( N ) exp (cid:20) (1 + s ) (cid:90) N (Ω ( m ) ) γ ( N ) dN (cid:21) , (23)where we have used Eq. (16). Since our parameter s is presumably positive definite and small, it is convenientfor us to write s = | (cid:101) s | , where (cid:101) s can take both positive and negative values.In order to perform the standard χ minimization, we use the growth data observations: A obs ≡ ( f σ (8) ) obs along with the theoretical predicted values: A th ≡ ( f σ (8) ) th in the standard definition of the χ function χ := V m C − mn V n , (24)where V := A obs − A th and C − mn is the inverse of the covariance matrix between three WiggleZ data points [175].As we have already shown in fig. (2) that the parametric form (18) tends to diverge in case of interactingDE from its numerical solution (16) at high redshifts, we therefore restrict ourselves for the observationsupto z = 1 for the datasts: GOLD-2017 [172] , and H ( z ) data set [176]. Also, we set the range of priorsas follows: ( i ) − ≤ (cid:101) s ≤
1, ( ii ) 0 . ≤ Ω ( m ) ≤ .
6, ( iii ) 0 . ≤ σ (8) ≤ . iv ) 0 . ≤ h ≤ .
9, where h := H / [100 Km s − M pc − ]. The obtained contour plots between parameters upto 3 σ level are shown inFigs. 3a and 3b. 8 . . . . . ( m ) . . . . . s . . . . ( ) . . . . . ( m )0 . . . . (8)0 (a) . . . . ( m ) . . . . . h . . . . . s . . . . . ( ) . . . . ( m )0 . . . . . h . . . . . (8)0 (b) H ( z ) data. Figure 3:
The 1 σ -3 σ contour levels for Gold dataset (left), and its combination with the Hubble dataset (right). The solid blueline denotes the best-fit and dashed lines correspond to the 1 σ level. (best fit & 68% limits) χ /dof datasets Ω ( m ) σ (8) h (cid:101) s GOLD . +0 . − . . +0 . − . - 0 . +0 . − . . GOLD+ H ( z ) . +0 . − . . +0 . − . . +0 . − . . +0 . − . . Best fit values with 1 σ confidence limits of parameters Ω ( m ) , σ (8) , h and (cid:101) s together with their corresponding χ /dof for GOLD and GOLD + H ( z ) dataset. The estimations are shown in table (1) in which one can see that the best-fit of (cid:101) s for both sets of data(GOLD and GOLD+ H ( z )) is insignificant (as expected, since observations mostly prefer the ΛCDM model),but even then within 1 σ limits its domain can reach upto significantly large value i.e. O (10 − ) which showsa reasonable large deviation from the ΛCDM model. This indicates from the low-redshift data we can stillobserve a convincing amount of DEM interaction even at the 1 σ level. We have formulated the growth of linear matter density perturbations in a parametric form for a DE modelwhich stems out from a modified gravity approach consists of metric and torsion as two basic entities of thespace-time geometry. In the formalism, we have briefly demonstrated that a non-minimal coupling of metricand torsion with scalar field can give rise to a scalar-tensor action of DE in the Jordan frame which uponconformal transformation to the Einstein frame naturally makes scalar field non-minimally coupled with thematter sector. Due to this coupling, matter and scalar field exchange their energies between each other whichceases their individual energy densities to be self-conserved. The latter, thus, has direct influence on theunderline matter density contrast and its evolution, which we have explored in this work.We have demonstrated that in the perturbed FRW space-time, the scalar field and matter couplingenhances the growth of matter density perturbations in the sub-horizon regime, allowing it to cross the upperbarrier of unity at large redshifts. Since this effect is unique in the interacting DEM scenarios it requiresa slight modifications in the standard parametric ansatz of growth factor. With suitable modification wepropose a slightly different growth factor ansatz to make the parametric formulation compatible with thetheoretical predictions. Also, in view of the time evolving growth index, which is even encountered for theΛCDM model, we have chosen an appropriate functional form i.e. first order Taylor expansion about present-day value of the scale factor a ( t ). This simple but well defining form of the growth index indeed illustrates theparametric formulation of growth factor close to its actual evolution atleast upto z (cid:39)
1. Since the present-dayvalue of growth index itself depends on the background model parameters, therefore in order to choose itsexplicit function form we have numerically obtained its fitting formula in terms of coupling as well as energydensity parameter which we have shown to be a well approximation for a wide range of parameters.As to the parametric estimations, we have constrained parameters (cid:101) s , Ω ( m ) and σ (8) by using the RSD aswell as its combination with the Hubble data. We have found that for the GOLD datset the (cid:101) s and hence s parameter can show mildly large deviation from the ΛCDM model upto 1 σ , which is comparatively smallerfor the combined dataset, as expected. The consistency in our estimations with the theoretical predictionsconfirms the validity of our fitting function. However, to explain growth history for redshifts >
1, the aboveparametrization still requires further modifications to deal with various DE models as well as modified gravitytheories, which we will shall endeavor to report in near future.10
Acknowledgments
The work of MKS was supported by the Council of Scientific and Industrial Research (CSIR), Governmentof India.
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