Testing Einstein's gravity and dark energy with growth of matter perturbations: Indications for new Physics?
TTesting Einstein’s gravity and dark energy with growth of matter perturbations:Indications for new Physics?
Spyros Basilakos ∗ and Savvas Nesseris † Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efesiou 4, 11527, Athens, Greece Instituto de F´ısica Te´orica UAM-CSIC, Universidad Auton´oma de Madrid, Cantoblanco, 28049 Madrid, Spain (Dated: December 20, 2016)The growth index of matter fluctuations is computed for ten distinct accelerating cosmologicalmodels and confronted to the latest growth rate data via a two-step process. First, we implement ajoint statistical analysis in order to place constraints on the free parameters of all models using solelybackground data. Second, using the observed growth rate of clustering from various galaxy surveyswe test the performance of the current cosmological models at the perturbation level while eithermarginalizing over σ or having it as a free parameter. As a result, we find that at a statistical level,i.e. after considering the best-fit χ or the value of the Akaike information criterion, most modelsare in very good agreement with the growth rate data and are practically indistinguishable fromΛCDM. However, when we also consider the internal consistency of the models by comparing thetheoretically predicted values of ( γ , γ ), i.e. the value of the growth index γ ( z ) and its derivativetoday, with the best-fit ones, we find that the predictions of three out of ten dark energy modelsare in mild tension with the best-fit ones when σ is marginalized over. When σ is free we findthat most models are not only in mild tension, but also predict low values for σ . This could beattributed to either a systematic problem with the growth-rate data or the emergence of new physicsat low redshifts, with the latter possibly being related to the well-known issue of the lack of power atsmall scales. Finally, by utilizing mock data based on an LSST-like survey we show that with futuresurveys and by using the growth index parameterization, it will be possible to resolve the issue ofthe low σ but also the tension between the fitted and theoretically predicted values of ( γ , γ ). PACS numbers: 95.36.+x, 98.80.-k, 04.50.Kd, 98.80.Es
I. INTRODUCTION
The majority of studies in observational cosmologyconverge to the following general conclusion (see Refs.[1,2] and references therein), that the Universe is spatiallyflat and it contains ∼
30% of matter (luminous and dark),while the rest is the enigmatic dark energy (DE). Despitethe great progress made at theoretical level, up to nowthe nature of the DE has yet to be discovered and sev-eral unanswered questions remain, see Refs.[3], [4] for anoverview and a discussion of some of the problems. Asa matter of fact, the discovery of the underlying physicsof dark energy, thought to be driving the accelerated ex-pansion of the Universe, is considered one of the mostfundamental problems on the interface uniting astron-omy, cosmology and particle physics.In the literature there is a large family of cosmologicalscenarios that provide a mathematical explanation re-garding the accelerated expansion of the Universe. Gen-erally speaking, the cosmological models are mainly clas-sified in two large categories. The first group of DE mod-els is nested inside Einstein’s general relativity (GR) andit introduces new fields in nature (for review see [5] andreferences therein). Alternatively, modified gravity mod-els provide a theoretical platform which assumes that ∗ Electronic address: [email protected] † Electronic address: [email protected] the present accelerating epoch is due to the possibilityof gravity becoming weak at extragalactic scales. There-fore, DE has nothing to do with new fields and it appearsas a geometric effect [5].In this framework, the corresponding effectiveequation-of-state (EoS) parameter is allowed to take val-ues in the phantom regime, namely w < − WiggleZ etc (see our Table II andreferences therein). The growth of matter perturbationscan be also be used for self-consistency tests of generalrelativity, see Refs. [12],[13], even in a model independentfashion.However, from the theoretical viewpoint, it has beenproposed that the so-called growth index γ , first intro-duced by [14], can be used towards testing the natureof dark energy. Indeed, in the literature one can finda large body of studies in which the theoretical formof the growth index is provided analytically for variouscosmological models, including scalar field DE [15–20], a r X i v : . [ a s t r o - ph . C O ] D ec DGP [19, 21–23], f ( R ) [24, 25], Finsler-Randers [26],time varying vacuum models Λ( H ), [27]), Clustered DE[28], Holographic dark energy [29] and f ( T ) [30].In this article, we attempt to check the performanceof a large family of flat DE models (10 models) at theperturbation level. First, a joint likelihood analysis, in-volving the latest geometrical data (SNe type Ia, CMBshift parameter and BAO) is performed in order to de-termine the cosmological parameters of the DE models.Second, we attempt to discriminate the different DE cos-mologies by estimating the growth index γ and the corre-sponding redshift evolution. Then, by utilizing the avail-able growth rate data we show that the evolution of thegrowth index is a potential discriminator for a large frac-tion of the explored DE models.The structure of the manuscript is as follows: In Sec.II we present the main ingredients of the linear growth ofmatter fluctuations in the dark energy regime. In Secs.III and IV with the aid of a joint statistical analysis(based on SNe Ia, CMB shift parameter and BAO data)we constrain the DE model parameters. In Sec. V, wetest the DE cosmologies by comparing the correspondingtheoretical predictions of the growth index evolution withobservations. Finally, we summarize our conclusions inSec. VI. II. LINEAR GROWTH AND DARK ENERGY
In this section we provide the basic tools that are nec-essary in order to study the linear matter fluctuations.Since we are well inside in the matter dominated era wecan neglect the radiation term from the Hubble expan-sion. Now, for different types of dark energy the dif-ferential equation that governs the evolution of matterfluctuations at subhorizon scales is [18, 19, 24, 31–34]¨ δ m + 2˜ νH ˙ δ m − πGµρ m δ m = 0 . (1)As is well known, δ m ∝ D ( t ), where D ( t ) is the lineargrowth factor usually normalized to unity at the presenttime. It is clear that the nature of dark energy is reflectedin the quantities ˜ ν and µ ≡ G eff /G N . In the case ofscalar field dark energy models which adhere to Einstein’sgravity we have ˜ ν = µ = 1, while if we allow interactionsin the dark sector, in general we get ˜ ν (cid:54) = 1 and µ (cid:54) = 1. Foreither inhomogeneous dark energy models (inside GR) ormodified gravity models one can show that ˜ ν = 1 and µ (cid:54) = 1.Another important quantity in this kind of study is the However, in more complicated models, eg ones with couplingsbetween matter and geometry, one may have µ ≡ G eff /G N + β ( a, k ), where β ( a, k ) is a function that depends on derivativesof the Lagrangian of the model [35]. growth rate of clustering (first introduced by [14]) f ( a ) = d ln δ m d ln a (cid:39) Ω γm ( a ) . (2)Based on the above equation we can easily obtain thegrowth factor D ( a ) = exp (cid:20)(cid:90) a Ω m ( x ) γ ( x ) x dx (cid:21) , (3)with Ω m ( a ) = Ω m a − E ( a ) (4)and from which we define d Ω m da = − m ( a ) a (cid:18) d ln Ed ln a (cid:19) . (5)Notice, that E ( a ) = H ( a ) /H is the dimensionless Hub-ble parameter and γ is the so called growth index. There-fore, inserting Eq.(2) in Eq.(1) and with the aid of Eq.(5)we arrive at a dfda + (cid:18) ν + d ln Ed ln a (cid:19) f + f = 3 µ Ω m a ln(Ω m ) dγda +Ω γm − γ +2˜ ν − (cid:18) γ − (cid:19) d ln Ed ln a = 32 µ Ω − γm . (7) Another expression of the above equation is given bySteigerwald et al. [34] dωd ln a ( γ + ω dγdω ) + e ωγ + 2˜ ν + d ln Ed ln a = 32 µ e ω (1 − γ ) , (8)where ω = lnΩ m ( a ) which means that at z (cid:29) a → m ( a ) → ω → γ ∞ = 3( M + M ) − H + N )2 + 2 X + 3 M , (9)where the relevant quantities are M = µ | ω =0 , M = dµdω (cid:12)(cid:12)(cid:12)(cid:12) ω =0 (10)and N = d ˜ νdω (cid:12)(cid:12)(cid:12)(cid:12) ω =0 , H = − X d ( d ln E/d ln a ) dω (cid:12)(cid:12)(cid:12)(cid:12) ω =0 . (11)Concerning the functional form of the growth index weuse a Taylor expansion around a ( z ) = 1 (see [36–40]) γ ( a ) = γ + γ (1 − a ) . (12)Therefore, the asymptotic value reduces to γ ∞ (cid:39) γ + γ ,where we have set γ = γ (1). Now, writing Eq.(7) at thepresent epoch ( a = 1) − γ (cid:48) (1)ln(Ω m ) + Ω γ (1) m − γ (1) + 2˜ ν − γ −
12 ) d ln Ed ln a (cid:12)(cid:12)(cid:12)(cid:12) a =1 = 32 µ Ω − γ (1) m , (13)and using Eq.(12) we find γ = Ω γ m − γ + 2˜ ν − γ − ) d ln Ed ln a (cid:12)(cid:12) a =1 − µ Ω − γ m ln Ω m . (14)Notice, that a prime denotes a derivative with respect tothe scale factor, µ = µ (1) and ˜ ν = ˜ ν (1).To conclude this section it is important to realize thatthe growth of matter perturbations is affected by themain cosmological functions, namely E ( a ), Ω m ( a ), µ ( a )and ˜ ν ( a ). Therefore, for the benefit of the reader let usbriefly present the main steps that we follow in the restof the paper. • Suppose that we have a dark energy model thatcontains n -free cosmological parameters, given bythe cosmological vector θ i = ( θ , θ , ..., θ n ). Firstwe place constraints on θ i by performing an overalllikelihood analysis, involving the latest geometricaldata (standard candles and standard rulers). • For this cosmological model we know its basic cos-mological quantities which implies that we cancompute γ ∞ from Eq.(9). Then solving the sys-tem of γ ∞ = γ + γ and Eq.(14) we can write γ , in terms of the cosmological parameters θ i (Ω m ,etc). • Once, steps (i) and (ii) are accomplished, we finallytest the performance of the cosmological model atthe perturbation level utilizing the available growthdata.
III. LIKELIHOOD ANALYSIS
In this section we perform a joint statistical analysisusing the latest background data. Briefly, the total likeli-hood function is the product of the individual likelihoods: L tot ( θ i ) = L sn × L bao × L cmb , (15)thus the overall chi-square χ is written as χ ( θ i ) = χ + χ + χ . (16)In particular we use the JLA SNIa data of Ref. [41],the BAO from 6dFGS[42], SDDS[43], BOSS CMASS[44],WiggleZ[45], MGS[46] and BOSS DR12[47]. Finally, wealso use the CMB shift parameters based on the Planck2015 release [2], as derived in Ref. [48]. As we have already mentioned in the previous sec-tion, the cosmological vector θ i includes the free pa-rameters of the particular cosmological model whichare related with the cosmic expansion. In the presentanalysis, some of the relevant parameters are θ i =( α, β, Ω m , Ω d , Ω r , Ω b , H , ... ), where Ω m and Ω b arethe total matter (cold dark matter and baryons) andbaryon density parameters today, while α, β are the pa-rameters related to the stretch and color of the SNIadata. Assuming a spatially flat universe we have, Ω d =1 − Ω m − Ω r . Also, in the case of the CMB shift param-eter, the contribution of the radiation term Ω r and thebaryon density Ω b needs to be considered (see below).Here the radiation density at the present epoch is fixedto Ω r = Ω m a eq , where the scale factor at equality is a eq = . Ω m h ( T cmb / . K ) − .We also marginalize over the parameters M and δM of the JLA set as described in the appendix of Ref. [49].These parameters implicitly contain H and thus χ isindependent of H = h km/s/M pc , where accordingto Planck h (cid:39) .
67 [2]. However, we keep the parame-ters α, β free in our analysis. Therefore, using the afore-mentioned arguments the cosmological vector becomes θ i = ( α, β, Ω m , Ω b h , h, θ i +1 d ), where θ i +1 d contains thefree parameters which are related with the nature of thedark energy.The next step is to apply the Akaike Information Crite-rion (AIC) information criterion [50] in order to test thestatistical performance of the models themselves. Since, N/k (cid:29) − L max + 2 k , where L max is the maximum likelihood, N is the numberof data points used in the fit and k is the number of freeparameters. A smaller value of AIC indicates a bettermodel-data fit. In the case of Gaussian errors, χ min = − L max , one can show that the difference in AIC betweentwo models is written as ∆AIC = ∆ χ + 2∆ k . IV. CONSTRAINTS ON DARK ENERGYMODELS
Here we provide the basic properties of the most pop-ular DE models whose free parameters are constrainedfollowing the methodology of the previous section. Wemention that in all cases we assume a spatially flatFriedmann-Lemaˆıtre-Robertson-Walker (FLRW) geome-try.Notice, that for the study of matter perturbations inSecs. II and V the effect of radiation is not necessary.However, for the fitting of the current DE models to theBaryonic Acoustic Oscillations (BAO) and the CMB shiftparameter in Sec. III we need to include the radiationcomponent in the Hubble parameter. In order to dealwith this issue we replace the matter component Ω m a − in the normalized Hubble parameter E ( a ) with Ω m a − +Ω r a − . Accordingly, the present value of Ω d = 1 − Ω m is replaced by Ω d = 1 − Ω m − Ω r .In Table I, the reader may see a more compact presen-tation of the best fit values of cosmological parameters θ i , including also the various nuisance parameters α, β of the JLA SnIa data, the separate contribution of thebaryons Ω b h but also the best fit χ min and the corre-sponding value of the AIC. In what follows we will focuson Ω m and the various DE model parameters, but forcompleteness in Table I we give the best-fit values of theother parameters ( α, β, Ω b h , h ) as well. A. Constant equation of state ( w CDM model)
In this simple model the equation of state (hereafterEoS) parameter w = p d /ρ d is constant [51], where p d isthe pressure and ρ d is the density of the dark energy fluidrespectively. Although the quintessence scenario ( − ≤ w < − / w < − w CDM model adheres to GR and it does not allowinteractions in the dark sector, namely µ ( a ) = ˜ ν ( a ) = 1.Also, the dimensionless Hubble parameter is given by E ( a ) = Ω m a − + Ω d a − w ) , (17)where Ω d = 1 − Ω m . Therefore, from Eq.(17) we arriveat d ln Ed ln a = − − w [1 − Ω m ( a )] (18)and { M , M , H , X } = { , , w , − w } . If we substitute the above coefficients into Eq.(9) thenwe find (see also [15–20, 40]) γ ∞ = 3( w − w − . Note however, that the above expression neglects the ef-fects of DE perturbations, as discussed in Ref. [53]. Ofcourse for w = − γ (Λ) ∞ = 6 / θ i = (Ω m , w ). In this case the to-tal likelihood function peaks at Ω m = 0 . ± . w = − . ± .
012 with χ (Ω m , w ) (cid:39) . γ , γ ). Based on the procedure described at the endof section III and utilizing the best fit values of the cos-mological parameters (see Table I) we obtain ( γ , γ ) (cid:39) (0 . , − . m wefind Ω m = 0 . ± .
003 with χ (Ω m ) (cid:39) . γ , γ ) (cid:39) (0 . , − . m isin excellent agreement with the one obtained from thePlanck 2015 TT, TE, EE and lowP CMB data Ω P lanckm =0 . ± . . B. Parametric dark energy (CPL model)
This kind of phenomenological model was first intro-duced by Chevalier-Polarski-Linder [54, 55]. In particu-lar, the dark energy EoS parameter is parameterized asa first order Taylor expansion around the present epoch: w ( a ) = w + w (1 − a ) , where w and w are constants, while for an interestingextension of this model see Ref. [56]. The normalizedHubble parameter now becomes: E ( a ) = Ω m a − + Ω d a − w + w ) e w ( a − . Since the CPL model is inside GR and due to the absenceof dark matter/energy interactions we get µ ( a ) = ˜ ν ( a ) =1. Also the logarithmic derivative d ln E/d ln a is given byEq.(18) but here we have w = w ( a ). Using the abovefunctions we can derive the growth coefficients (see also[34]) { M , M , H , X } = { , , w + w )2 , − w + w ) } , which provide γ ∞ = 3( w + w − w + w ) − . In this case the cosmological vector contains three freeparameters θ i = (Ω m , w , w ) and the overall likeli-hood function peaks at Ω m = 0 . ± . w = − . ± . w = 0 . ± . χ (Ω m , w , w ) is 708.283 (AIC=722.283)and ( γ , γ ) (cid:39) (0 . , − . Note that the frequently quoted value of Ω
Planckm = 0 . ± . C. HDE model
Applying the holographic [57] principle within theframework of GR ˜ ν ( a ) = 1 one can show that w ( a ) = − − (cid:112) Ω d ( a )3 s and d lnΩ d d ln a = − w ( a )3 [1 − Ω d ( a )] , where Ω d ( a ) = 1 − Ω m ( a ) and s is a constant. It is easyto check that at high redshifts z (cid:29) a → d → w ∞ tends to − /
3. Also the dimensionless Hubble parameter, E ( a ) = H ( a ) /H is given by E ( z ) = Ω m a − − Ω d ( a ) . Again, the functional form of d ln E/d ln a is given byEq.(17). Obviously, the above three equations producea system whose solution gives the evolution of the maincosmological parameters, namely E ( a ), w ( a ) and Ω d ( a ),where θ i = (Ω m , s ).The quantity µ ( a ) that describes the intrinsic featuresof the HDE is written as [29] µ ( a ) = (cid:40) Ω d ( a )Ω m ( a ) ∆ d ( a )(1 + 3 c ) clustered HDE(19)where ∆ d = w ( a )1 − w ( a ) [29, 58] and c is the effective soundspeed of the dark energy.Here we consider the following two cases: • Homogeneous HDE (hereafter HHDE) in which µ ( a ) = 1. Therefore, in this case we find (see [29]) { M , M , H , X } = { , , w ∞ , − w ∞ } . where w ∞ (cid:39) − / γ ∞ = 47 . Our joint statistical analysis yields that the likeli-hood function peaks at Ω m = 0 . ± .
003 and s = 0 . ± .
006 with χ (Ω m , s ) (cid:39) . γ , γ ) (cid:39) (0 . , . • Clustered HDE (hereafter CHDE): here µ ( a ) isgiven by the second branch of Eq.(19). In thisframework, we obtain { M , M , H , X } = { , − (1 + 3 c )3 , ∞ , − ∞ } and for w ∞ = − / γ ∞ = 3(1 − c )7 . We restrict our analysis to c = 0, which impliesthat the sound horizon is small with respect to theHubble radius and thus DE perturbations grow ina similar fashion to matter perturbations [59]. Uti-lizing the aforementioned cosmological parameterswe compute ( γ , γ ) (cid:39) (0 . , − . D. Time varying vacuum ( Λ t CDM model)
Let us now consider the possibility of a decaying Λ-cosmology, that is, Λ = Λ( a ). The decaying vacuumequation of state does not depend on whether Λ is strictlyconstant or variable. Therefore, the EoS takes the nom-inal form, p Λ ( t ) = − ρ Λ ( t ) = − Λ( t ) / πG . In the cur-rent article we study a specific dynamical vacuum modelwhich is based on the renormalization group in quantumfield theory. It has been proposed that, the evolution ofthe vacuum is given byΛ( H ) = Λ + 3 ν ( H − H ) , where Λ ≡ Λ( H ) = 3Ω Λ0 H and ν is provided in theRenormalization Group (RG) context as a “ β -functionwhich determines the running of the cosmological “con-stant” (CC) within QFT in curved spacetime [60]. TheFriedmann equations are the same with those of the con-cordance ΛCDM model, while the current vacuum sce-nario matter is obliged to exchange energy with vacuumin order to fulfil the Bianchi identity which implies˙ ρ m + 3 Hρ m = − ˙ ρ Λ . Combining the Friedmann equations and the latter gen-eralized conservation law one can write the evolution ofthe normalized Hubble parameter E ( a ) = ˜Ω Λ0 + ˜Ω m a − − ν ) , (20)with d ln Ed ln a = −
32 (1 − ν ) ˜Ω m ( a ) , (21)where we have set ˜Ω m ( a ) = ˜Ω m a − − ν ) E ( a ) , ˜Ω m ≡ Ω m − ν and ˜Ω Λ0 ≡ − Ω m − ν − ν . Obviously, the cosmologi-cal vector includes the following free parameters θ i = For the fitting the radiation term is included in the Λ t CDMmodel as follows: we replace ˜Ω m a − − ν ) in Eq.(20) with˜Ω m a − − ν ) + ˜Ω r a − − ν ) , where ˜Ω m ≡ Ω m − ν , ˜Ω r ≡ Ω r − ν ,and ˜Ω Λ0 ≡ − Ω m − Ω r − ν − ν [61]. ( α, β, ˜Ω m , Ω b h , h, ν ). For more details concerning theglobal dynamics of the present time varying vacuummodel we refer the reader to the following Refs.[61–66].On the other hand, the growth index of matter pertur-bations has been investigated by Basilakos and Sola [27].Specifically, the quantities ˜ ν and µ are given by˜ ν = 1 + 32 ν (22)and µ ( a ) = 1 − ν − ν ˜Ω m ( a ) + 3 ν (1 − ν ) . (23)To this end, the growth coefficients are found to be [27] { M , M , H , X } = { − ν − ν , − − ν )2 , − ν ) } which provide γ ∞ = 6 + 3 ν − ν . Finally, using the cosmological data and the joint likeli-hood analysis we find (Ω m , ν ) = (0 . ± . , − . · − ± . · − ) for a best-fit χ (Ω m , ν ) (cid:39) . γ , γ ) = (0 . , − . E. Dvali, Gabadadze and Porrati (DGP) gravity
The first modified gravity model that we present is thatof Dvali, Gabadadze and Porrati [67]. In this scenario,one can obtain an accelerating expansion of the Universebased on the fact that gravity itself becomes weak at cos-mological scales (close to Hubble radius) because our fourdimensional spacetime survives into an extra dimensionalmanifold (see [67] and references therein). It has beenshown that the normalized Hubble parameter is writtenas E ( a ) = (cid:112) Ω m a − + Ω rc + (cid:112) Ω rc , (24)where Ω rc = (1 − Ω m ) / .From Eq.(24), we easily find d ln Ed ln a = − m ( a )1 + Ω m ( a ) . (25)For the DGP model the function µ ( a ) takes the form µ ( a ) = 2 + 4Ω m ( a )3 + 3Ω m ( a ) When we also include radiation, this changes to Ω rc = (1 − Ω m − Ω r ) / and ˜ ν ( a ) = 1. Inserting the above equations in Eqs.(10),(11) we have { M , M , H , X } = { , , − , } and from Eq.(9) the asymptotic value of the growth indexbecomes (see also [19, 21–23]) γ ∞ = 1116 . As in the concordance ΛCDM model, the cosmolog-ical vector contains the following free parameters θ i =( α, β, Ω m , Ω b h , h ). The overall statistical analysis pro-vides a best fit value of Ω m = 0 . ± . χ (Ω m ) (cid:39) .
654 (AIC=808.654), withrespect to that of ΛCDM cosmology. To this end, usingthe above and the calculations of section III we obtain( γ , γ ) (cid:39) (0 . , . F. Finsler-Randers dark energy model (FRDE)
In the last decade there have been quite interesting ap-plications of Finsler geometry in its Finsler-Randers ver-sion, in the topics of cosmology, astrophysics and generalrelativity [68] (and references therein). Recently, it hasbeen found [26] that the Finsler-Randers field equationsprovide a Hubble parameter which is identical with thatof DGP gravity. This means that Eqs.(24) and (25) arealso valid here. As expected, the joint analysis providesexactly the same statistical results.However, the two models (FRDE and DGP) deviateat the perturbation level since in the case of the FRDEmodel we have µ ( a ) = ˜ ν ( a ) = 1 [68]. Therefore, it is easyto show that { M , M , H , X } = { , , − , } γ ∞ = 916 . Again, solving the system γ ∞ = γ + γ and Eq.(14) forΩ m = 0 . ± .
008 we derive ( γ , γ ) (cid:39) (0 . , − . G. Power law f ( T ) gravity model Among the large family of modified gravity models, the f ( T ) gravity extends the old definition of the so calledteleparallel equivalent of general relativity [69–71], where T is the torsion scalar. In the current article we use thepower-law model of Bengochea and Ferraro [72], with f ( T ) = α ( − T ) b , where α = (6 H ) − b Ω F b − . In this framework, the Hubble parameter normalized tounity at the present time takes the form E ( a, b ) = Ω m a − + Ω d E b ( a, b ) , (26)where Ω d = 1 − Ω m . Obviously, for b = 0 the powerlaw f ( T ) model reduces to ΛCDM cosmology , namely T + f ( T ) = T −
2Λ (where Λ = 3Ω d H , Ω d = Ω Λ0 ).It has been shown that in order to have an acceleratedexpansion of the Universe which is consistent with thecosmological data one needs b (cid:28) E ( a, b ) around b = 0and thus we obtain an approximate normalized Hubbleparameter, namely E ( a, b ) (cid:39) E ( a ) + Ω d ln (cid:2) E ( a ) (cid:3) b + ... . (27)Recently, Basilakos [30] investigated the growth indexfor the power law f ( T ) gravity model. In brief, differen-tiating Eq.(26) and using Eq.(4) we arrive at d ln Ed ln a = −
32 Ω m ( a )[1 − bE b − Ω d ] (28)and for b (cid:28) d ln Ed ln a (cid:39) −
32 Ω m ( a ) (cid:20) d bE ( a ) + ... (cid:21) . (29)Notice that here we have ˜ ν = 0 and the quantity µ takesthe following form (see [30] and references therein) µ ( a ) = 11 + b Ω d (1 − b ) E − b ) (30)or µ ( a ) (cid:39) − Ω d E ( a ) b + · · · . (31)Now, based on the above equations the growth index co-efficients of (10) and (11) become (for more details see[30]) { M , M , H , X } = { , b, − − b )2 , − b ) } and thus the asymptotic value of the growth index is γ ∞ = 611 − b . As expected, for b = 0 we recover the ΛCDM value6 /
11. In this context, we find that the likelihood func-tion peaks at Ω m = 0 . ± . b = 0 . ± . χ (Ω m , b ) (cid:39) .
363 (AIC=720.363). Therefore,if we use the latter best fit solution then we estimate( γ , γ ) (cid:39) (0 . , − . Notice, that for b = 1 / H. f ( R ) gravity ( f CDM model)
Another modified gravity that we include in our anal-ysis is the popular f ( R ) model of Hu and Sawicki. How-ever, here we make use of the implementation with the b parameter as in Ref. [75]. This has two advantages: first,the deviation from ΛCDM is easily seen and second, bydoing a series expansion around b = 0 we can find ex-tremely accurate (better than 0 .
1% for b (cid:46) − % for b (cid:46) .
1) analytical approximations. Inthis formalism, the Lagrangian for the Hu and Sawickimodel can be equivalently written as [75]: f ( R ) = R − (cid:0) b Λ R (cid:1) n (32)where n is a parameter of the model, henceforth chosenas n = 1 without loss of generality.As mentioned we can perform a series expansion ofthe solution of the equations on motion around b = 0,i.e. ΛCDM, as H ( a ) = H ( a ) + M (cid:88) i =1 b i δH i ( a ) , (33)where H ( a ) H = Ω m a − + Ω r a − + (1 − Ω m − Ω r ) (34)and M is the number of terms we keep before truncatingthe series, but usually only the two first non-zero termsare more than enough for excellent agreement with thenumerical solution. Finally, δH i ( a ) is a set of algebraicfunctions that can be determined from the equations ofmotion, see Ref. [75] for the exact and quite long expres-sions.Studying the growth index in this class of models ismore complicated, as the modified Newton’s constant de-pends on both the time via the scale factor a and the scale k , ie G eff = G eff ( a, k ) [76]. More specifically we have G eff ( a, k ) G N = 1 F k a F ,R /F k a F ,R /F , (35)where F = f (cid:48) ( R ), F ,R = f (cid:48)(cid:48) ( R ), G N is the bare Newton’sconstant and we have normalized Eq. (35) so that for b = 0, i.e. for the ΛCDM we get G eff ( a,k ) G N = 1 as expected.We also follow Ref. [75] and set k = 0 . h Mpc − (cid:39) H .In the notation of the other sections we have: µ ( a, k ) = G eff ( a, k ) G N , ˜ ν ( a ) = 1 . (36)In Ref. [77] it was shown that these kinds of mod-els predict rather low and rather high values for theparameters γ and γ respectively or more specifically TABLE I: A summary of the best-fit background parameters for the various cosmological models used in the analysis. Thefifth and sixth columns show the specific DE model parameters.Model α β Ω m Ω b h h DE Params χ min AICΛCDM 0 . ± .
004 3 . ± .
011 0 . ± .
003 0 . ± . . ± . w = − w a = 0 708 .
592 718 . w CDM 0 . ± .
006 3 . ± .
010 0 . ± .
004 0 . ± . . ± . w = − . ± .
012 708 .
438 720 . w a = 0CPL 0 . ± .
004 3 . ± .
012 0 . ± .
005 0 . ± . . ± . w = − . ± .
009 708 .
283 722 . w a = 0 . ± . . ± .
004 3 . ± .
010 0 . ± .
003 0 . ± . . ± . s = 0 . ± .
006 713 .
218 725 . t CDM 0 . ± .
005 3 . ± .
004 0 . ± .
002 0 . ± . . ± . ν = ( − . ± . · − .
550 720 . . ± .
004 3 . ± .
010 0 . ± .
008 0 . ± . . ± .
004 798 .
654 808 . f ( T ) 0 . ± .
006 3 . ± .
015 0 . ± .
004 0 . ± . . ± . b = 0 . ± .
008 708 .
363 720 . f ( R ) 0 . ± .
009 3 . ± .
020 0 . ± .
007 0 . ± . . ± . b = 0 . ± .
009 708 .
526 720 . fσ ( z ) growth data.z fσ ( z ) Ref.0 .
02 0 . ± .
040 [80]0 .
067 0 . ± .
055 [81]0 .
10 0 . ± .
130 [82]0 .
17 0 . ± .
060 [83]0 .
35 0 . ± .
050 [84, 85]0 .
77 0 . ± .
180 [84, 86]0 .
25 0 . ± .
058 [87]0 .
37 0 . ± .
038 [87]0 .
22 0 . ± .
070 [88]0 .
41 0 . ± .
040 [88]0 .
60 0 . ± .
040 [88]0 .
60 0 . ± .
067 [89]0 .
78 0 . ± .
040 [88]0 .
57 0 . ± .
066 [90]0 .
30 0 . ± .
055 [89]0 .
40 0 . ± .
041 [89]0 .
50 0 . ± .
043 [89]0 .
80 0 . ± .
080 [91] ( γ , γ ) (cid:39) (0 . , − . k -dependence ofthe effective Newton’s constant in order to get the ex-act values for these parameters we need to solve Eq. (6)numerically to estimate γ (cid:39) ln( f (1))ln(Ω m ) , where f (1) is thegrowth rate at a = 1, and then use Eq. (14) to get γ .In this case the cosmological vector contains twofree parameters θ i = (Ω m , b ) and the overall likeli-hood function peaks at Ω m = 0 . ± .
007 and b =0 . ± . χ (Ω m , b ) is 708.526(AIC=720.526) and ( γ , γ ) (cid:39) (0 . , − . V. TESTING DARK ENERGY MODELS WITHGROWTH DATAA. Analysis with the real data
In this section we present the details of the statisticalmethod and on the observational sample that we adoptin order to test the performance of the dark energy mod-els at the perturbation level. Specifically, we utilize the recent growth rate data, namely A ≡ f ( z ) σ ( z ) where σ ( z ) is the redshift-dependent rms fluctuations of thelinear density field at at R = 8 h − Mpc. Notice that thesample contains N gr = 18 entries (see Table II and thecorresponding references). Following the standard anal-ysis we use the χ -minimization procedure, which in ourcase is defined as follows: χ ( φ µ ) = N gr (cid:88) i =1 (cid:20) A D ( z i ) − A M ( z i , φ µ ) σ i (cid:21) (37)where φ µ = ( σ ≡ φ , γ , γ ) is the statistical vectorat the background level (not to be confused with θ i ), A D ( z i ) and σ i are the growth data and the correspondinguncertainties at the observed redshift z i . Also, D and M indicate data and model respectively. The theoreticalgrowth-rate is given by: A M ( z, φ µ ) = f σ ( z, φ µ ) = σ D ( z )Ω m ( z ) γ ( z ) . (38)where for the latter equality we have set σ ( z ) = σ D ( z ),Ω m ( z ) = Ω m (1 + z ) /E ( z ), D ( z ) is the growth factornormalized to unity at the present time and γ ( z ) is thegrowth index.Now, we provide the basic steps towards marginaliz-ing χ gr over σ (see also [78]). Substituting the secondequality Eq.(38) into Eq.(37), we simply obtain χ = Γ − Bσ + Cσ , (39)where Γ = N gr (cid:88) i =1 A D ( z i ) σ i ,B = N gr (cid:88) i =1 A D ( z i ) D ( z i )Ω m ( z i ) γ σ i ,C = N gr (cid:88) i =1 D ( z i )Ω m ( z i ) γ σ i . In the case of Λ t CDM model we remind the reader that we needto replace Ω m ( z ) with ˜Ω m ( z ) = ˜Ω m (1 + z ) /E ( z ) The corresponding likelihood L gr = e − χ / is then givenby L gr ( D| φ µ , M ) = e − (cid:104) Γ − B C + C ( BC − σ ) (cid:105) , (40)where we have completed the square. Applying Bayes’stheorem and marginalizing over σ we find p ( φ µ |D , M ) = 1 p ( D|M ) e − (cid:104) Γ − B C (cid:105) · (cid:90) dσ p ( σ , φ µ |M ) e − C ( BC − σ ) . (41)Considering flat priors, namely p ( σ , φ µ |M ) = 1 and σ is within a range σ ∈ (0 , ∞ ) we arrive at p ( φ µ |D , M ) = 1 p ( D|M ) e − (cid:104) Γ − B C (cid:105) (cid:90) ∞ dσ e − C ( BC − σ ) . (42)Introducing the variable y = σ − BC we find p ( φ µ |D , M ) = 1 p ( D|M ) e − (cid:104) Γ − B C (cid:105) (cid:114) π C (cid:20) (cid:18) B √ C (cid:19)(cid:21) , (43)where erf( x ) = (cid:82) x dye − y , to which corresponds themarginalized ˜ χ function˜ χ = Γ − B C + ln C − (cid:20) (cid:18) B √ C (cid:19)(cid:21) . (44)Notice that we have ignored the constant − ln π . Thefirst two terms in ˜ χ , i.e. Γ − B C correspond to the casewhere σ is fixed in such a way that the original χ [seeEq.(37)] is minimized.In what follows we consider two approaches: first, weuse the marginalized ˜ χ function which is independent of σ and thus it contains only two free parameters ( γ , γ )and second, for the sake of comparison we also of mini-mize χ , given by Eq. (37), with respect to σ .Lastly, in order to compare the DE models we utilizethe Akaike information criterion for small sample sizewhich is defined for the case of Gaussian errors, as:AIC = ˜ χ , min + 2 k gr + 2 k gr ( k gr − N gr − k gr − , where k gr is the number of free parameters.Below and in Table III, we provide our statistical re-sults for the case when σ is marginalized over. • For the w CDM model: ˜ χ , min = 13 . γ = 0 . ± .
173 and γ = − . ± . γ = 0 . ± . γ = − . ± .
415 with ˜ χ , min = 13 . TABLE III: A summary of the best-fit parameters ( γ , γ ) forthe various cosmological models used in the analysis, with σ marginalized over and the χ given by Eq. (44).Model γ γ χ min AICΛCDM 0 . ± . − . ± .
415 13 .
456 17 . w CDM 0 . ± . − . ± .
426 13 .
454 17 . . ± . − . ± .
454 13 .
445 17 . . ± . − . ± .
592 13 .
412 17 . t CDM 0 . ± . − . ± .
426 13 .
455 17 . . ± . − . ± .
931 13 .
409 17 . f ( T ) 0 . ± . − . ± .
436 13 .
451 17 . f ( R ) 0 . ± . − . ± .
447 13 .
450 17 . γ , γ , σ )for the various cosmological models used in the analysis andthe χ given by Eq. (37).Model γ γ σ χ min AICΛCDM 0 . ± . − . ± .
532 0 . ± .
095 6 .
999 13 . w CDM 0 . ± . − . ± .
532 0 . ± .
097 6 .
998 13 . . ± . − . ± .
533 0 . ± .
099 6 .
995 13 . . ± . − . ± .
543 0 . ± .
104 6 .
992 13 . t CDM 0 . ± . − . ± .
537 0 . ± .
095 6 .
999 13 . . ± . − . ± .
555 0 . ± .
154 6 .
985 13 . f ( T ) 0 . ± . − . ± .
532 0 . ± .
098 6 .
998 13 . f ( R ) 0 . ± . − . ± .
535 0 . ± .
098 6 .
997 13 . • For the CPL model: ˜ χ , min = 13 . γ = 0 . ± .
176 and γ = − . ± . • For the HDE model: ˜ χ , min = 13 . γ = 0 . ± .
177 and γ = − . ± . • For the Λ t CDM model: ˜ χ , min = 13 . γ = 0 . ± .
066 and γ = 0 . ± . • For the DGP-FRDE gravity: ˜ χ , min = 13 . γ = 0 . ± .
341 and γ = − . ± . • For the f ( T ) model: ˜ χ , min = 13 . γ = 0 . ± .
175 and γ = − . ± . • For the f CDM model: ˜ χ , min = 13 . γ = 0 . ± .
176 and γ = − . ± . σ is a free parameter. • For the w CDM model: χ , min = 6 . γ = 0 . ± . γ = − . ± .
532 and σ = 0 . ± . γ = 0 . ± . γ = − . ± .
532 and σ =0 . ± .
095 with χ , min = 6 .
999 (AIC=13.856),which is agreement with that of [74, 79]; • For the CPL model: χ , min = 6 .
995 (AIC=13.852), γ = 0 . ± . γ = − . ± .
533 and σ =0 . ± . • For the HDE model: χ , min = 6 . γ = 0 . ± . γ = − . ± .
543 and σ = 0 . ± . • For the Λ t CDM model: χ , min = 6 . γ = 0 . ± . γ = − . ± .
537 and σ = 0 . ± . • For the DGP-FRDE gravity: χ , min = 6 . γ = 0 . ± . γ = − . ± .
555 and σ = 0 . ± . • For the f ( T ) model: χ , min = 6 . γ = 0 . ± . γ = − . ± .
532 and σ = 0 . ± . • For the f CDM model: χ , min = 6 . γ = 0 . ± . γ = − . ± .
535 and σ = 0 . ± . γ , γ )plane, with σ marginalized over and free respectively, inwhich the corresponding contours are plotted for the 1 σ ,2 σ and 3 σ confidence levels. On top of that we also plotthe theoretical ( γ , γ ) values (see section IV) of all DEmodels as indicated by the colored black and green dots.Overall, we find that in the case of a marginalized σ , the w CDM, CPL, ΛCDM, Λ t CDM, HDE, and f ( T )( γ , γ ) models are well within the 1 σ borders (∆ χ σ (cid:39) .
30; see light blue (inner) sectors in Fig. 1). The restof the DE models (DGP, FRDE and f CDM with n = 1)seem to be in mild tension with the theoretical predic-tions for ( γ , γ ). One the other hand, as seen in Fig. 2in the case of a free σ , we find that most models, exceptHDE, are in mild or in the case of the DGP in strongtension with their theoretically predicted ( γ , γ ) values.Testing further the performance of our results with re-spect to σ we find that there is a correlation between σ and γ , i.e. as γ becomes more negative σ becomessmaller, in agreement with the results of Refs.[74, 75].Finally, we also checked that a global fit of all thefree parameters, e.g. in the case of ΛCDM θ i =( α, β, Ω m , Ω b h , h, γ , γ , σ ), gives exactly the same fitas our two step process. This confirms our assumptionthat the background parameters ( α, β, Ω m , Ω b h , h )and perturbation order parameters ( γ , γ , σ ) areuniquely fixed by their corresponding data. B. Analysis with mock data
In this section we briefly discuss forecasts of ourmethodology with mock f σ data based on a ΛCDMcosmology with (Ω m , σ ) = (0 . , . f σ ( z ) for the ΛCDM cosmology, uniformly distributedin the range z ∈ [0 ,
2] divided into 10 equally spaced binsof step d z = 0 .
2. The f σ ( z i ) at each point was estimatedby adding its theoretical value to a Gaussian error andassigning an error of 1% of its value, which is in agree-ment with the expected LSST accuracy as described inRefs. [92, 93].Following the same analysis as before, we considerthe same two strategies for dealing with σ , i.e. firstby marginalizing over it and second by fitting it alongwith the other two parameters. In the first case wefind γ = 0 . ± .
019 and γ = − . ± . γ = 0 . ± . γ = − . ± .
111 and σ = 0 . ± . f σ data in the ( γ , γ ) plane, with σ marginal-ized over (left panel) and free (right panel) respectively,in which the corresponding contours are plotted for the1 σ , 2 σ and 3 σ confidence levels.As can be seen, the constraints on ( γ , γ ) are muchmore in line with the theoretical predictions (black dotsin Fig. 3), thus making it possible to discriminate be-tween DE models and proving that our methodology willbe extremely useful with the upcoming data in the nearfuture. VI. CONCLUSIONS
We studied the growth index beyond the concordanceΛCDM model by utilizing several forms for the dark en-ergy. In the first part of our article, we implementedan overall likelihood analysis using the most recent highquality cosmological data (SNIa, CMB shift parameterand BAOs), thereby putting tight constraints on themain cosmological free parameters. At the level of the re-sulting Hubble function, we showed that the majority ofdark energy models (apart from HDE, DGP and Finsler-Randers cosmologies), are statistically indistinguishable(within 1 σ ) from a flat ΛCDM model, as long as they areconfronted with the above background geometrical data.Of course, the DGP can readily be ruled out as it has a δχ ∼
90 from ΛCDM.At the perturbation level, not only do the aforemen-tioned DE models reproduce the ΛCDM Hubble expan-sion, but we also found that by using their χ and AICvalues, all models fit equally well the growth rate dataand are statistically indistinguishable from the ΛCDMmodel on the basis of their growth index evolution.However, it should be noted that in the case of a free σ - - Γ Γ L CDM 0.2 0.4 0.6 0.8 1.0 - - Γ Γ wCDM 0.2 0.4 0.6 0.8 1.0 - - Γ Γ CPL 0.2 0.4 0.6 0.8 1.0 - - Γ Γ L t CDM0.2 0.4 0.6 0.8 1.0 - - Γ Γ HDE 0.5 1.0 1.5 2.0 - - Γ Γ DGP (cid:144)
FRDE 0.2 0.4 0.6 0.8 1.0 - - Γ Γ f H T L - - Γ Γ f H R L FIG. 1: The plots of the 1 σ , 2 σ and 3 σ confidence levels in the ( γ , γ ) plane with σ marginalized over, for the ΛCDM, w CDM,CPL and Λ t CDM models (top row) and the HDE, DGP/FRDE, f ( T ) and f ( R ) models (bottom row). The red dots denotethe best-fit in each case, given in Table III, while the black dots denote the theoretical predictions as given in the text. Inaddition, the green dots correspond to the clustered HDE model with c = 0 and the DGP models. - - Γ Γ L CDM 0.2 0.4 0.6 0.8 1.0 - - Γ Γ wCDM 0.2 0.4 0.6 0.8 1.0 - - Γ Γ CPL 0.2 0.4 0.6 0.8 1.0 - - Γ Γ L t CDM0.2 0.4 0.6 0.8 1.0 - - Γ Γ HDE 0.2 0.4 0.6 0.8 1.0 1.2 - - Γ Γ DGP (cid:144)
FRDE 0.2 0.4 0.6 0.8 1.0 - - Γ Γ f H T L - - Γ Γ f H R L FIG. 2: The plots of the 1 σ , 2 σ and 3 σ confidence levels in the ( γ , γ ) plane with σ fixed to its best-fit value, for the ΛCDM, w CDM, CPL and Λ t CDM models (top row) and the HDE, DGP/FRDE, f ( T ) and f ( R ) models (bottom row). The red dotsdenote the best-fit in each case, given in Table IV, while the black dots denote the theoretical predictions as given in the text.In addition, the green dots correspond to the clustered HDE model with c = 0 and the DGP models. parameter the best fit values of ( γ , γ ) for most models,except HDE, are in mild to strong tension with their corresponding theoretically predicted values, see Fig. 2.On the contrary, this is not so apparent in the case of a2 - - Γ Γ - - Γ Γ FIG. 3: The plots of the 1 σ , 2 σ and 3 σ confidence levels in the ( γ , γ ) plane with σ either marginalized over (left) or fixedto its best-fit value (right) for the ΛCDM model and the mock LSST-like data. The red dots denote the best-fit in each case,given in the text, while the black dots denote the theoretical predictions as given in the text. marginalized σ , as the contours are now larger due tothe marginalization. In this case, after inspection of thecontours of Fig. 1 our results can be summarized in thefollowing statements (for more details see section V): • Three models, ie., DGP, FRDE and f CDM canbe distinguished since they are in tension with thegrowth data and they show strong and significantvariations with respect to the concordance Λ model. • Four DE models, namely, w CDM, CPL, Λ t CDM,HDE and f ( T ) are in agreement with the growthdata and they cannot be distinguished from theΛCDM model at any significant level.The reason we considered both approaches, i.e.marginalizing over and fitting σ , is that as this is aderived parameter, the uncertainty in its measurementis still rather high, thus affecting the interpretation ofour analysis. Furthermore, we found that the growthrate data consistently and for all models, prefer a ratherlow value for σ of approximately σ (cid:39) . ± . σ P lanck = 0 . ± . . σ discrepancy from the growth rate datapoint of view or 9 . σ from the Planck data point of view.It should be noted that the observed suppression of powerof the late Universe observables, e.g. low σ and thechronic tension between the CMB and low-redshift ob-servables has already been discussed in the literature,see e.g. Ref. [94]. To conclude, the main benefit of our analysis is thateven though at a first glance all of the models seem in-distinguishable at the statistical level given their χ and AIC values, see Tables III and IV, when comparedwith their theoretically predicted ( γ , γ ) ones, we can seethere is a significant inconsistency. Since the models areknown to be internally consistent this means either thatthere is a problem with the growth-rate data or that thereis new physics emerging at low redshifts. Also, by usingmock growth rate data we demonstrated that this willbe made more clear with future dynamical data (basedmainly on LSST ), which are expected to improve signifi-cantly the relevant constraints (especially on γ and σ )and thus possibly resolve the issue with the lack of powerat low redshifts discussed in Ref. [94], but also sheddinglight on the nature of dark energy on cosmological scales. Acknowledgements
The authors would like to thank W. Cardona for point-ing out a typo in the text.S.B. acknowledges support by the Research Center forAstronomy of the Academy of Athens in the context ofthe program “
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