Constraints to dark energy using PADE parameterisations
Mehdi Rezaei, Mohammad Malekjani, Spyros Basilakos, Ahmad Mehrabi, David F. Mota
aa r X i v : . [ a s t r o - ph . C O ] J un D RAFT VERSION O CTOBER
12, 2018
Preprint typeset using L A TEX style AASTeX6 v. 1.0
CONSTRAINTS TO DARK ENERGY USING PADE PARAMETERISATIONS
M. R
EZAEI , M. M ALEKJANI , S. B ASILAKOS , A. M EHRABI , AND
D. F. M
OTA Department of Physics, Bu-Ali Sina University, Hamedan 65178, Iran Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efessiou 4, 11-527 Athens, Greeceand Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, Norway
AbstractWe put constraints on dark energy properties using the PADE parameterisation, and compare it to the sameconstraints using Chevalier-Polarski-Linder (CPL) and Λ CDM, at both the background and the perturbationlevels. The dark energy equation of state parameter of the models is derived following the mathematicaltreatment of PADE expansion. Unlike CPL parameterisation, the PADE approximation provides differentforms of the equation of state parameter which avoid the divergence in the far future. Initially, we performa likelihood analysis in order to put constraints on the model parameters using solely background expansiondata and we find that all parameterisations are consistent with each other. Then, combining the expansion andthe growth rate data we test the viability of PADE parameterisations and compare them with CPL and Λ CDMmodels respectively. Specifically, we find that the growth rate of the current PADE parameterisations is lowerthan Λ CDM model at low redshifts, while the differences among the models are negligible at high redshifts.In this context, we provide for the first time growth index of linear matter perturbations in PADE cosmologies.Considering that dark energy is homogeneous we recover the well known asymptotic value of the growth index,namely γ ∞ = w ∞ − w ∞ − , while in the case of clustered dark energy we obtain γ ∞ ≃ w ∞ (3 w ∞ − w ∞ − w ∞ − . Finally,we generalize the growth index analysis in the case where γ is allowed to vary with redshift and we find thatthe form of γ ( z ) in PADE parameterisation extends that of the CPL and Λ CDM cosmologies respectively. INTRODUCTIONVarious independent cosmic observations includingthose of type Ia supernova (SnIa) (Riess et al. 1998;Perlmutter et al. 1999; Kowalski et al. 2008), cos-mic microwave background (CMB) (Komatsu et al.2009; Jarosik et al. 2011; Komatsu et al. 2011;Planck Collaboration XIV 2016), large scale structure(LSS), baryonic acoustic oscillation (BAO) (Tegmark et al.2004; Cole et al. 2005; Eisenstein et al. 2005; Percival et al.2010; Blake et al. 2011b; Reid et al. 2012), high redshiftgalaxies (Alcaniz 2004), high redshift galaxy clusters(Wang & Steinhardt 1998a; Allen et al. 2004) and weakgravitational lensing (Benjamin et al. 2007; Amendola et al.2008; Fu et al. 2008) reveal that the present universe expe-riences an accelerated expansion. Within the framework ofGeneral Relativity (GR), the physical origin of the cosmicacceleration can be described by invoking the existenceof an exotic fluid with sufficiently negative pressure, theso-called Dark Energy (DE). One possibility is that DEconsists of the vacuum energy or cosmological constant Λ with constant EoS parameter w Λ = − (Peebles & Ratra2003). Alternatively, the fine-tuning and cosmic coincidence problems (Weinberg 1989; Sahni & Starobinsky 2000;Carroll 2001; Padmanabhan 2003; Copeland et al. 2006) ledthe scientific community to suggest a time evolving energydensity with negative pressure. In those models, the EoSparameter is a function of redshift, w ( z ) (Caldwell et al.1998; Erickson et al. 2002; Armendariz-Picon et al. 2001;Caldwell 2002; Padmanabhan 2002; Elizalde et al. 2004).A precise measurement of EoS parameter and its variationwith cosmic time can provide important clues about the dy-namical behavior of DE and its nature (Copeland et al. 2006;Frieman et al. 2008; Weinberg et al. 2013; Amendola et al.2013).One possible way to study the EoS parameter of dynam-ical DE models is via a parameterisation. In literature, onecan find many different EoS parameterisations. One of thesimplest and earliest parameterisations is the Taylor expan-sion of w de ( z ) with respect to redshift z up to first orderas: w de ( z ) = w + w z (Maor et al. 2001; Riess et al.2004). It can also be generalized by considering the sec-ond order approximation in Taylor series as: w de ( z ) = w + w z + w z (Bassett et al. 2008). However, thesetwo parameterisations diverge at high redshifts. Hence the R EZAEI ET AL .well-known Chevallier-Polarski-Linder (CPL) parameterisa-tion, w de ( z ) = w + w (1 − a ) = w + w z/ (1 + z ) , wasproposed (Chevallier & Polarski 2001; Linder 2003). TheCPL parameterisation can be considered as a Taylor serieswith respect to (1 − a ) and was extended to more gen-eral case by assuming the second order approximation as: w de ( a ) = w + w (1 − a ) + w (1 − a ) (Seljak et al.2005). In addition to CPL formula, some purely phenomeno-logical parameterisations have been proposed more recently.For example w de ( z ) = w + w z/ (1 + z ) α , where α isfixed to 2 (Jassal et al. 2005). In this class, the power law w de ( a ) = w + w (1 − a β ) /β (Barboza et al. 2009) and log-arithmic w de ( a ) = w + w ln a (Efstathiou 1999) parame-terisations have been investigated. Another logarithm param-eterisation is w de ( z ) = w / [1 + b ln (1 + z )] α , where α istaken to be or (Wetterich 2004). Notice that although theCPL is a well-behaved parameterisation at early ( a ≪ ) andpresent ( a ∼ ) epochs, it diverges when the scale factor goesto infinity. This is also a common difficulty for the above phe-nomenological parameterisations. Recently to solve the di-vergence, several phenomenological parameterisations havebeen introduced (see Dent et al. 2009; Frampton & Ludwick2011; Feng et al. 2012, for more details). Notice that most ofthese EoS parameterisations are ad hoc and purely written byhand where there is no mathematical principle or fundamen-tal physics behind them. In this work we focus on the PADEparameterisation ( see section 2), which from the mathemati-cal point of view seems to be more stable: it does not divergeand can be employed at both small and high redshifts. Usingthe different types of PADE parameterisations to express theEoS parameter of DE w de in terms of redshift z , we study thegrowth of perturbations in the universe.DE not only accelerate the expansion rate of universebut also change the evolution of growth rate of matterperturbations and consequently the formation epochs oflarge scale structures of universe (Armendariz-Picon et al.1999; Garriga & Mukhanov 1999; Armendariz-Picon et al.2000; Tegmark et al. 2004; Pace et al. 2010; Akhoury et al.2011). Moreover, the growth of cosmic structures arealso affected by perturbations of DE when we dealwith dynamical DE models with time varying EoS pa-rameter w de = − (Erickson et al. 2002; Bean & Doré2004; Hu & Scranton 2004; Basilakos & Voglis 2007;Ballesteros & Riotto 2008; Basilakos et al. 2009a;Koivisto & Mota 2007; Mota et al. 2007; Gannouji et al.2010; Basilakos et al. 2010; Sapone & Majerotto 2012;Batista & Pace 2013; Dossett & Ishak 2013; Basse et al.2014; Pace et al. 2014c; Batista 2014; Basilakos 2015;Pace et al. 2014b; Nesseris & Sapone 2014; Mehrabi et al.2015c,b; Malekjani et al. 2015; Mehrabi et al. 2015a;Malekjani et al. 2017).In addition to the background geometrical data the datacoming from the formation of large scale structures provide avaluable information about the nature of DE. In particular, we can setup a more general formalism in which the backgroundexpansion data including SnIa, BAO, CMB shift parameter,Hubble expansion data joined with the growth rate data oflarge scale structures in order to put constraints on the param-eters of cosmology and DE models (see Cooray et al. 2004;Corasaniti et al. 2005; Basilakos et al. 2010; Blake et al.2011b; Nesseris et al. 2011; Basilakos & Pouri 2012;Yang et al. 2014; Koivisto & Mota 2007; Mota et al. 2007;Gannouji et al. 2010; Mota et al. 2008; Llinares et al. 2014;Llinares & Mota 2013; Contreras et al. 2013; Chuang et al.2013; Li et al. 2014; Basilakos 2015; Mehrabi et al. 2015a,b;Basilakos 2016; Mota et al. 2010; Malekjani et al. 2017; Fay2016; Bonilla Rivera & Farieta 2016).In this work, following the lines of the above studies andusing the latest observational data including the geometricaldata set (SnIa, BAO, CMB, big bang nucleosynthesis (BBN), H ( z ) ) combined with growth rate data f ( z ) σ , we performan overall likelihood statistical analysis to place constraintsand find best fit values of the cosmological parameters wherethe EoS parameter of DE is approximated by PADE param-eterisations. Previously, the PADE parameterisations havebeen studied from different observational tests in Cosmology.For example in Gruber & Luongo (2014), the cosmographyanalysis has been investigated using the PADE approxima-tion. In Wei et al. (2014), the authors proposed several pa-rameterisations for EoS of DE on the basis of PADE approx-imation. Confronting these EoS parameterisations with thelatest geometrical data, they found that the PADE parameter-isations can work well (for similar studies, see Aviles et al.2014; Zaninetti 2016; Zhou et al. 2016). Here, for the firsttime, we study the growth of perturbations in PADE cos-mologies. After introducing the main ingredients of PADEparameterisations in Sect.2, we study the background evolu-tion of the universe in Sect.(3). We implement the likelihoodanalysis using the geometrical data to put constraints on thecosmological and model parameters in PADE parameterisa-tions. In Sect.(4), the growth of perturbations in PADE cos-mologies is investigated. Then we perform an overall like-lihood analysis including the geometrical + growth rate datato place constraints and obtain the best fit values of the cor-responding cosmological parameters. Finally we provide themain concussions in Sect.(5). PADE PARAMETERISATIONSFor an arbitrary function f ( x ) , the PADE approximate oforder ( m, n ) is given by the following rational function (Pade1892; Baker & Graves-Morris 1996; Adachi & Kasai 2012) f ( x ) = a + a x + a x + ... + a n x n b + b x + b x + ... + b n x m , (1)where exponents ( m, n ) are positive and the coefficients ( a i , b i ) are constants. Obviously, for b i = 0 (with i ≥ )the current approximation reduces to standard Taylor expan-sion. In this study we focus on three PADE parameterisationsADE PARAMETERISATIONS
PADE (I)
Based on Eq. (1), we first expand the EoS parameter w de with respect to (1 − a ) up to order (1 , as follows (see alsoWei et al. 2014): w de ( a ) = w + w (1 − a )1 + w (1 − a ) . (2)From now on we will call the above formula as PADE (I)parameterisation. In terms of redshift z , Eq. (2) is written as w de ( z ) = w + ( w + w ) z w ) z . (3)As expected for w = 0 Eq. (2) boils down to CPL pa-rameterisation. Unlike CPL parameterisation, here the EoSparameter with w = 0 avoids the divergence at a → ∞ (orequivalently at z = − ). Using Eq. (2) we find the followingspecial cases regarding the EoS parameter (see also Wei et al.2014) w de = w + w w , for a → z → ∞ , early time) ,w , for a = 1 ( z = 0 , present) , w w , for a → ∞ ( z → − , far future) , (4)where we need to set w = 0 and − . Therefore, we arguethat PADE (I) formula is a well-behaved function in therange of ≤ a ≤ ∞ (or equivalently at − ≤ z ≤ ∞ ).2.2. simplified PADE (I) Clearly, PADE (I) approximation has three free parameters w , w and w . Setting w = 0 we provide a simplifiedversion of PADE (I) parameterisation, namely w de ( a ) = w w (1 − a ) . (5)Notice, that in order to avoid singularities in the cosmic ex-pansion w needs to lie in the interval − < w < .2.3. PADE (II)
Unlike the previous cases, here the current parameterisa-tion is written as a function of N = ln a . In this context, theEoS parameter up to order ( , ) takes the form w de ( a ) = w + w ln a w ln a , (6) Table 1 . The statistical results for the various DE parameterisationsused in the analysis. These results are based on the expansion data.The concordance Λ CDM model is shown for comparison.Model PADE I simp. PADE I PADE II CPL Λ CDM k χ where w , w and w are constants (see also Wei et al. 2014).In PADE (II) parameterisation, we can easily show that w de = w w , for a → z → ∞ , early time) ,w , for a = 1 ( z = 0 , present) , w w , for a → ∞ ( z → − , far future) , (7)Notice, that in order to avoid singularities at the above epochswe need to impose w = 0 . BACKGROUND HISTORY IN PADEPARAMETERISATIONSIn this section based on the aforementioned parameterisa-tions we study the background evolution in PADE cosmolo-gies. Generally speaking, for isotropic and homogeneousspatially flat FRW cosmologies, driven by radiation, non-relativistic matter and an exotic fluid with an equation of state p de = w de ρ de , the first Friedmann equation reads H = 8 πG ρ r + ρ m + ρ de ) , (8)where H ≡ ˙ a/a is the Hubble parameter, ρ r , ρ m and ρ de arethe energy densities of radiation, dark matter and DE, respec-tively. In the absence of interactions among the three fluidsthe corresponding energy densities satisfy the following dif-ferential equations ˙ ρ r + 4 Hρ r = 0 , (9) ˙ ρ m + 3 Hρ m = 0 , (10) ˙ ρ de + 3 H (1 + w de ) ρ de = 0 , (11)where the over-dot denotes a derivative with respect to cos-mic time t . Based on Eqs. (9) and (10), it is easy to derivethe evolution of radiation and pressure-less matter, namely ρ r = ρ r0 a − and ρ m = ρ m0 a − . Inserting Eqs . (2), (5) and(6) into equation (11), we can obtain the DE density of thecurrent PADE parameterisations (see also Wei et al. 2014) ρ (PADEI)de = ρ (0)de a − w w w w ) [1 + w (1 − a )] − w − w w w w ) , (12) ρ (simp . PADEI)de = ρ (0)de a − w w w ) [1 + w (1 − a )] − − w w w w ) , (13) ρ (padeII)de = ρ a − w w w ) (1 + w ln a ) w − w w w ) . (14) R EZAEI ET AL . −1.6 −0.8 0.0 0.8 w h −1.00−0.75−0.50 w Ω m −1.6−0.80.00.8 w h −1.00 −0.75 −0.50 w −0.80−0.72−0.64−0.56 w h −1.0−0.8−0.6 w −1.2−0.60.00.6 w Ω m −0.80−0.72−0.64−0.56 w h −1.0 −0.8 −0.6 w −1.2−0.6 0.0 0.6 w −0.48−0.42−0.36−0.30 w h −0.96−0.88−0.80−0.72 w Ω m −0.48−0.42−0.36−0.30 w h −0.96−0.88−0.80−0.72 w −0.48−0.40−0.32−0.24 w h −1.05−0.90−0.75−0.60 w −0.50.00.51.0 w Ω m −0.48−0.40−0.32−0.24 w h −1.05−0.90−0.75−0.60 w −0.5 0.0 0.5 1.0 w Figure 1 . The σ , σ and σ likelihood contours for various cosmological parameters using the latest expansion data. The upper left (upperright) panel shows the results for CPL (PADE I) parameterisation. The lower left (lower right) panel shows the results for simplified PADE I(PADE II) parameterisation. Also, combining Eqs.(12, 13, 14) and Eq.(8) we derivethe dimensionless Hubble parameter E = H/H (see alsoWei et al. 2014). Specifically, we find E = Ω r0 a − + Ω m0 a − + (1 − [Ω r0 + Ω m0 ]) × a − w w w w ) × (1 + w − aw ) − w − w w w w ) , (15) E = Ω r0 a − + Ω m0 a − + (1 − [Ω r0 + Ω m0 ]) × a − w w w ) × (1 + w − aw ) − − w w w w ) , (16) E = Ω r0 a − + Ω m0 a − + (1 − [Ω r0 + Ω m0 ]) × a − w w w ) × (1 + w ln a ) w − w w w ) , (17) where Ω m0 (density parameter), Ω r0 (radiation parameter)and Ω de0 = 1 − Ω m0 − Ω r0 (dark energy parameter). More-over, following the above lines in the case of CPL parameter-isation we have ρ CPLde = ρ (0)de a − w + w ) exp {− w (1 − a ) } (18)and E = Ω r0 a − + Ω m0 a − + (1 − Ω m0 − Ω r0 ) × a − w + w ) exp[ − w (1 − a )] , (19)Bellow, we study the performance of PADE cosmo-logical parameterisation against the latest observationaldata. Specifically, we implement a statistical analy-ADE PARAMETERISATIONS -2-1.8-1.6-1.4-1.2-1-0.8-0.6 0 0.5 1 1.5 2 2.5 3 3.5 4 w de z Λ CDMCPLPADE(I)PADE(II)Simp.PADE(I) -3-2-101234 0 0.5 1 1.5 2 2.5 3 3.5 4 (cid:1) E ( % ) z Λ CDMCPLPADE(I)PADE(II)Simp.PADE(I) (cid:1) de z (cid:2) CDMCPLPADE(I)PADE(II)Simp.PADE(I)
Figure 2 . The redshift evolution of various cosmological quantities,namely dark energy EoS parameter w de ( z ) ( top panel), relativedeviation ∆ E (%) = [( E − E Λ ) /E Λ ] × (middle panel) and Ω de ( z ) ( bottom panel). The different DE parameterisations arecharacterized by the colors and line-types presented in the innerpanels of the figure. sis using the background expansion data including thoseof SnIa (Suzuki et al. 2012), BAO (Beutler et al. 2011;Padmanabhan et al. 2012; Anderson et al. 2013; Blake et al.2011a), CMB (Hinshaw et al. 2013), BBN (Serra et al.2009; Burles et al. 2001), Hubble data (Moresco et al. 2012;Gaztanaga et al. 2009; Blake et al. 2012; Anderson et al.2014). For more details concerning the expansion data, the χ ( p ) function, the Markov chain Monte Carlo (MCMC)analysis, the Akaike information criterion (AIC) and -0.4-0.2 0 0.2 0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 q z Λ CDM CPL PADE(I) PADE(II) Simp.PADE(I)
Figure 3 . The evolution of the deceleration parameter q for differ-ent PADE parameterisations considered in this work. The CPL and Λ CDM models are shown for comparison. the Bayesian information criterion (BIC) we refer thereader to Mehrabi et al. (2015b) (see also Basilakos et al.2009b; Hinshaw et al. 2013; Mehrabi et al. 2015a, 2017;Malekjani et al. 2017). In this framework, the joint likeli-hood function is the product of the individual likelihoods: L tot ( p ) = L sn × L bao × L cmb × L h × L bbn , (20)which implies that the total chi-square χ is given by: χ ( p ) = χ + χ + χ + χ + χ , (21)where the statistical vector p includes the free parametersof the model. In our work the above vector becomes (a) p = { Ω DM0 , Ω b0 , h, w , w , w } for PADE (I) and (II) pa-rameterisations, (b) p = { Ω DM0 , Ω b0 , h, w , w } for sim-plified PADE (I) and (c) p = { Ω DM0 , Ω b0 , h, w , w } inthe case of CPL parameterisation. Notice that we utilize Ω m0 = Ω DM0 +Ω b0 and h = H / , while the energy den-sity of radiation is fixed to Ω r0 = 2 . × − h − (1 . (Hinshaw et al. 2013).Additionally, we utilize the well know information criteria,namely AIC (Akaike 1974) and BIC (Schwarz 1978) in orderto test the statistical performance of the cosmological modelsthemselves. In particular, AIC and BIC are given by AIC = − L max + 2 k , BIC = − L max + k ln N , (22)where k is the number of free parameters and N is the to-tal number of observational data points. The results of ourstatistical analysis are presented in Tables (1) and (2) respec-tively. Although the current DE parameterisations providelow AIC values with respect to those of Λ CDM, we find ∆AIC = AIC − AIC Λ < hence, the DE parameterisa-tions explored in this study are consistent with the expansiondata. In order to visualize the solution space of the model pa-rameters in Fig.(1) we present the σ , σ and σ confidencelevels for various parameter pairs. Using the best fit modelparameters [see Table 2] in Fig. (2) we plot the redshift evo- R EZAEI ET AL .lution of w de (upper panel), ∆ E (%) = [( E − E Λ ) /E Λ ] × (middle panel) and Ω de (lower panel). The different parame-terisations are characterized by the colors and line-types pre-sented in the caption of Fig. (2). We find that the EoS pa-rameter of PADE II evolves only in the quintessence regime( − < w de < − / ). For the other DE parameterisations weobserve that w de varies in the phantom region ( w de < − )at high redshifts, while it enters in the quintessence regime( − < w de < − / ) at relatively low redshifts. Notice,that the present value of w de can be found in Table (2).From the middle panel of Fig. (2), we observe that the rel-ative difference ∆ E is close to − . at low redshifts( z ∼ . ), while in the case of PADE (II) we always have E PADEII ( z ) > E Λ ( z ) . Lastly, in the bottom panel of Fig.(2)we show the evolution of Ω de , where its current value can befound in Table (2). As expected, Ω de tends to zero at highredshifts since matter dominates the cosmic fluid. In the caseof PADE parameterisations we observe that Ω de is larger thanthat of the usual Λ cosmology.Finally, we would like to estimate the transition redshift z tr of the PADE parameterisations by utilizing the decelerationparameter q ( z ) = − − ˙ H/H . Following, standard lines itis easy to show ˙ HH = − (cid:16) w de ( z )Ω de ( z ) (cid:17) (23)which implies that q ( z ) = 12 + 32 w de ( z )Ω de ( z ) (24)Using the best fit values of Table (2), we plot in Fig. (3)the evolution of q for the current DE parameterisations. Inall cases, including that of Λ CDM, q tends to 1/2 at earlyenough times. This is expected since the universe is mat-ter dominated ( Ω de ≃ ) at high redshifts. Now solving the q ( z tr ) = 0 we can derive the transition redshift, namely theepoch at which the expansion of the universe starts to accel-erate. In particular, we find z tr = 0 . (PADE I), z tr = 0 . (simplified PADE ), z tr = 0 . (PADE II), z tr = 0 . (CPL)and z tr = 0 . ( Λ CDM). The latter results are in good agree-ment with the measured z tr based on the cosmic chronome-ter H ( z ) data Farooq et al. (2017) (see also Capozziello et al.2014, 2015). GROWTH RATE IN DE PARAMETERISATIONSIn this section, we study the linear growth of matterperturbations in PADE cosmologies and we compare themwith those of CPL and Λ CDM respectively. In this kindof studies the natural question to ask is the following: how DE affects the linear growth of matter fluctuations?
In order to treat to answer this question we need to in-troduce the following two distinct situations, which havebeen considered within different approaches in the liter-ature (Armendariz-Picon et al. 1999; Garriga & Mukhanov1999; Armendariz-Picon et al. 2000; Erickson et al. 2002; Bean & Doré 2004; Hu & Scranton 2004; Abramo et al.2007, 2008; Ballesteros & Riotto 2008; Abramo et al. 2009;Basilakos et al. 2009a; de Putter et al. 2010; Pace et al.2010; Akhoury et al. 2011; Sapone & Majerotto 2012;Pace et al. 2012; Batista & Pace 2013; Dossett & Ishak2013; Batista 2014; Basse et al. 2014; Pace et al. 2014a,c,b;Malekjani et al. 2015; Naderi et al. 2015; Mehrabi et al.2015c,b,a; Nazari-Pooya et al. 2016; Malekjani et al. 2017):(i) the scenario in which the DE component is homogeneous( δ de ≡ ) and only the corresponding non-relativistic matteris allowed to cluster ( δ m = 0 ) and (ii) the case in which thewhole system clusters (both matter and DE). Owing to thefact that we are in the matter phase of the universe we canneglect the radiation term from the Hubble expansion.4.1. Basic equations
The basic equations that govern the evolution of non-relativistic matter and DE perturbations are given by(Abramo et al. 2009) ˙ δ m + θ m a = 0 , (25) ˙ δ de + (1 + w de ) θ de a + 3 H ( c − w de ) δ de = 0 , (26) ˙ θ m + Hθ m − k φa = 0 , (27) ˙ θ de + Hθ de − k c θ de (1 + w de ) a − k φa = 0 , (28)where k is the wave number and c eff is the effective soundspeed of perturbations (Abramo et al. 2009; Batista & Pace2013; Batista 2014). Combining the Poisson equation − k a φ = 32 H [Ω m δ m + (1 + 3 c )Ω de δ de ] , (29)with Eqs. (27 & 28), eliminating θ m and θ de and chang-ing the derivative from time to scale factor a , we obtainthe following stystem of differential equations (see alsoMehrabi et al. 2015a; Malekjani et al. 2017) δ ′′ m + 32 a (1 − w de Ω de ) δ ′ m = 32 a [Ω m δ m + Ω de (1 + 3 c ) δ de ] , (30) δ ′′ de + Aδ ′ de + Bδ de = 32 a (1 + w de )[Ω m δ m + Ω de (1 + 3 c ) δ de ] . (31)Bellow we set c eff ≡ which means that the whole sys-tem (matter and DE) fully clusters. Moreover, we remindreader that for homogeneous DE models we have δ de ≡ ,hence Eq.(30) reduces to the well known differential equa-tion of Peebles (1993) [see also Pace et al. (2010) and refer-ences therein]. Concerning the functional forms of A and B we have A = 1 a [ − w de − aw ′ de w de + 32 (1 − w de Ω de )] ,B = 1 a [ − aw ′ de + aw ′ de w de w de − w de (1 − w de Ω de )] . (32)ADE PARAMETERISATIONS Table 2 . A summary of the best-fit parameters for the various DE parameterisations using the background data.Model PADE I simplified PADE I PADE II CPL Λ CDM Ω (0)m . ± .
010 0 . ± .
010 0 . ± . . ± . . ± . h . ± .
012 0 . ± .
012 0 . ± .
013 0 . ± .
012 0 . ± . w − . ± . − . ± . − . ± . − . ± . − w − . +0 . − . − . +0 . − . − . +0 . − . − w − . +0 . − . − . ± . − . +0 . − . − − w de ( z = 0) − . − . − . − . − . de ( z = 0) 0 .
714 0 .
730 0 . . . −1.6 −0.8 0.0 0.8 w h −1.2−1.0−0.8−0.6−0.4 w Ω m −1.6−0.80.00.8 w h −1.2−1.0−0.8−0.6−0.4 w BackgroundClustered DEHomogenous DE −0.80−0.72−0.64−0.56 w h −1.2−1.0−0.8−0.6 w −1.2−0.60.00.61.2 w Ω m −0.80−0.72−0.64−0.56 w h −1.2 −1.0 −0.8 −0.6 w −1.2−0.6 0.0 0.6 1.2 w BackgroundClustered DEHomogenous DE −0.48−0.42−0.36−0.30 w h −0.96−0.88−0.80−0.72 w Ω m −0.48−0.42−0.36−0.30 w h −0.96−0.88−0.80−0.72 w BackgroundClustered DEHomogenous DE −0.48−0.42−0.36−0.30−0.24 w h −1.05−0.90−0.75−0.60 w −0.50.00.51.0 w Ω m −0.48−0.42−0.36−0.30−0.24 w h −1.05−0.90−0.75−0.60 w −0.5 0.0 0.5 1.0 w BackgroundClustered DEHomogenous DE
Figure 4 . The σ and σ likelihood contours for various planes using the solely expansion data (blue), combined expansion and growth ratedata for clustered (green) and homogeneous (red) DE parameterisations. The upper left (upper right) shows the results for CPL (PADE I)parameterisation. The lower left (lower right) shows the results for simplified PADE I (PADE II) parameterisation. R EZAEI ET AL . Table 3 . The statistical results for homogeneous (clustered) DE parameterisations used in the analysis. These results are based on the expan-sion+growth rate data. The concordance Λ CDM model is shown for comparison.Model PADE I simplified PADE I PADE II CPL Λ CDM k χ Table 4 . A summary of the best-fit parameters for homogeneous (clustered) DE parameterisations using the background+growth rate data.
Model PADE I simplified PADE I PADE II CPL Λ CDM Ω (0)m . ± .
010 (0 . ± . . ± .
010 (0 . ± . . ± . . ± . . ± . . ± . . ± . h . ± .
012 (0 . ± . . ± .
012 (0 . ± . . ± .
012 (0 . ± . . ± .
011 (0 . ± . . ± . w − . ± .
088 ( − . +0 . − . ) − . ± .
038 ( − . ± . − . ± .
075 ( − . ± . − . +0 . − . ( − . ± . − w . +0 . − . (0 . +0 . − . ) − . +0 . − . (0 . +0 . − . ) − . +0 . − . ( − . +0 . − . ) − w − . +0 . − . ( − . +0 . − . ) − . ± .
034 ( − . ± . − . +0 . − . ( − . +0 . − . ) − − σ . ± .
015 (0 . ± . . ± .
015 (0 . ± . . ± .
015 (0 . ± . . ± .
015 (0 . ± . . ± . w de ( z = 0) − . − . − . − . − . − . − . − . − . de ( z = 0) 0 . . . . . . . . . ADE
PARAMETERISATIONS f z Λ CDMHomogen eous CPLHomogen eous PADE(I)Homogeneous PADE(II)Homogeneous Simp.PADE(I)Clustered CPLClustered PADE(I)Clustered PADE(II)Clustered Simp.PADE(I) -5-4-3-2-1012 0 0.5 1 1.5 2 2.5 3 3.5 4 (cid:1) f ( % ) z Λ CDMClustered CPLClustered PADE(I)Clustered PADE(II)Clustered Simp.PADE(I)Homogeneous CPLHomogeneous PADE(I)Homogeneous PADE(II)Homogeneous Simp.PADE(I)
Figure 5 . The redshift evolution of the growth rate function f ( z ) (top-panel) and the corresponding fractional difference ∆ f (%) =100 × [ f ( z ) − f Λ ( z )] /f Λ ( z ) (bottom panel). The different DE pa-rameterisations are characterized by the colors and line-types pre-sented in the inner panels of the figure. z Data (cid:4)
CDMClustered CPLClustered PADE(I)Clustered PADE(II)Clustered Simp.PADE(I)Homogeneous CPLHomogeneous PADE(I)Homogeneous PADE(II)Homogeneous Simp.PADE(I)
Figure 6 . Comparison of the observed and theoretical evolution ofthe growth rate f ( z ) σ ( z ) as a function of redshift z . Open squarescorrespond to the data. Line-types and colors are explained in theinner plot of the figure. In order to perform the numerical integration of the abovesystem (30 & 31) it is crucial to introduce the appropriateinitial conditions. Here we utilize (see also Batista & Pace 2013; Mehrabi et al. 2015a; Malekjani et al. 2017) δ ′ mi = δ mi a i ,δ dei = 1 + w dei − w dei δ mi ,δ ′ dei = 4 w ′ dei (1 − w dei ) δ mi + 1 + w dei − w dei δ ′ mi , (33)where we fix a i = 10 − and δ mi = 1 . × − . Infact using the aforementioned conditions we verify that mat-ter perturbations always stay in the linear regime.From thetechnical viewpoint, using w de , Ω de we can solve the sys-tem of equations (30 & 31) which means that the fluctu-ations ( δ de , δ m ) can be readily calculated, and from them f ( z ) = d ln δ m /d ln a , σ ( z ) = δ m ( z ) δ m ( z =0) σ ( z = 0) (rmsmass variance at R = 8 h − M pc ) and f ( z ) σ ( z ) immedi-ately ensue.Now we perform a joint statistical analysis involving theexpansion data (see Sect. 3) and the growth data. In princi-ple, this can help us to understand better the theoretical ex-pectations of the present DE parameterisations, as well as totest their behaviour at the background and at the perturbationlevel. The growth data and the details regarding the likeli-hood analysis ( χ , MCMC algorithm etc) can be found inSect. 3 of our previous work (Mehrabi et al. 2015a). Briefly,in order to obtain the overall likelihood function we need toinclude the likelihood function of the growth data in Eq.(20)as follows L tot ( p ) = L sn × L bao × L cmb × L h × L bbn × L gr , (34)hence χ ( p ) = χ + χ + χ + χ + χ + χ , (35)where the statistical vector p contains an additional free pa-rameter, namely σ ≡ σ ( z = 0) .In Tables (3) and (4) we show the resulting best fit val-ues for various DE parameterisations under study, in whichwe also provide the observational constraints of the clusteredDE parameterisations. Furthermore, in Fig. (4) we presentthe σ and σ contours for various parameter pairs. The bluecontour represents the confidence levels based on geometri-cal data and green ( red) contours show the confidence levelsbased on geometrical + growth rate data for clustered (homo-geneous) DE parameterisations. Comparing the latter resultswith those of see Sect. 3 we conclude that the observationalconstraints which are placed by the expansion+growth dataare practically the same with those found by the expansiondata. Therefore, we can use the current growth data in or-der to put constrains only on σ , since they do not signifi-cantly affect the rest of the cosmological parameters. Thismeans that the results of see Sect. 3 concerning evolutionof the main cosmological functions ( w de , E ( z ) and Ω de ) re-main unaltered. To this end, in Fig. (5) we plot the evolutionof growth rate f ( z ) as a function of redshift (upper panel)0 R EZAEI ET AL .and the fractional difference with respect to that of Λ CMDmodel (lower panel), ∆ f (%) = 100 × [ f ( z ) − f Λ ( z )] /f Λ ( z ) .Specifically, in the range of ≤ z ≤ we find: • for homogeneous (or clustered) PADE I parameter-isation the relative difference is ∼ [ − , ( or ∼ [ − . , ) • in the case of simplified PADE I we have ∼ [ − , . and ∼ [ − , . for homogeneousand clustered DE respectively • for homogeneous (or clustered) PADE II DE the rela-tive deviation lies in the interval ∼ [ − . , − . (or ∼ [ − , − . ). Finally, in the case of CPLparameterisation we obtain ∼ [ − . , . (homo-geneous) and ∼ [ − , . (clustered).In this context, we verify that at high redshifts the growthrate tends to unity since the universe is matter dominated,namely δ m ∝ a . Moreover, we observe that the evolution of ∆ f has one maximum/minimum and one zero point. As ex-pected, this feature of ∆ f is related to the evolution of ∆ E (see middle panel of Fig. 2). Indeed, we verify that large val-ues of the normalized Hubble parameter E ( z ) correspond tosmall values of the growth rate. Also, looking at Fig. 2 ( mid-dle panel) and Fig.5 (bottom panel) we easily see that when ∆ E has a maximum the growth rate ∆ f has a minimum andvice versa. We also observe that if ∆ E < then ∆ f > and vice versa. Finally, in Fig. (6), we compare the observed f σ ( z ) with the predicted growth rate function of the currentDE parameterisations [for curves see caption of Fig. (6)].We find that all parameterisations represent well the growthdata. As expected from AIC and BIC analysis (see Table 3)the current DE parameterisations and standard Λ CDM cos-mology are all consistent with current observational data.4.2.
The growth index
We would like to finish this section with a discussion con-cerning the growth index of matter fluctuations γ , which af-fects the growth rate of clustering via the following relation(first introduced by Peebles 1993) f ( z ) = d ln δ m d ln a ( z ) ≃ Ω γ m ( z ) . (36)The theoretical formula of the growth index has beenstudied for various cosmological models, including scalarfield DE (Silveira & Waga 1994; Wang & Steinhardt1998b; Linder & Jenkins 2003; Lue et al. 2004;Linder & Cahn 2007; Nesseris & Perivolaropoulos 2008),DGP (Linder & Cahn 2007; Gong 2008; Wei 2008;Fu et al. 2009), Finsler-Randers (Basilakos & Stavrinos2013), running vacuum Λ( H ) (Basilakos & Sola 2015), f ( R ) (Gannouji et al. 2009; Tsujikawa et al. 2009), f ( T ) (Basilakos 2016), Holographic DE (Mehrabi et al. 2015a) and Agegraphic DE (Malekjani et al. 2017) If we com-bine equations (25-28), (29) and using simultaneously dδ m dt = aH dδ m da then we obtain (see also Abramo et al. 2007,2009; Mehrabi et al. 2015a) a δ ′′ m + a HH ! δ ′ m = 32 Ω m µ , (37)where ˙ HH = d ln Hd ln a = − − w de ( a )Ω de ( a ) , (38)and Ω de ( a ) = 1 − Ω m ( a ) . The quantity µ ( a ) characterizesthe nature of DE in PADE parametrisations, namely µ ( a ) = Homogeneous PADE Ω de ( a )Ω m ( a ) ∆ de ( a )(1 + 3 c ) Clustered PADE(39)where we have set ∆ de ≡ δ de /δ m . Obviously, if we use c = 0 then Eq.(37) reduces to Eq.(30), while in the case ofthe usual Λ CDM model we need to a priori set δ de ≡ .Furthermore, substituting Eq.(36) and Eq.(38) in Eq.(37)we arrive at − (1+ z ) dγdz ln(Ω m )+Ω γ m +3 w de Ω de (cid:18) γ − (cid:19) + 12 = 32 Ω − γ m µ . (40)Regarding the growth index evolution we use thefollowing phenomenological parameterisation (seealso Polarski & Gannouji 2008; Wu et al. 2009;Bueno Belloso et al. 2011; Di Porto et al. 2012;Ishak & Dossett 2009; Basilakos 2012; Basilakos & Pouri2012) γ ( a ) = γ + γ [1 − a ( z )] . (41)Now, utilizing Eq.(40) at the present time z = 0 and with theaid of Eq.(41) we obtain (see also Polarski & Gannouji 2008) γ = Ω γ m0 + 3 w de0 ( γ − )Ω de0 + − Ω − γ m0 µ lnΩ m0 , (42)where µ = µ ( z = 0) and w de0 = w de ( z = 0) . Clearly,in order to predict the growth index evolution in DE modelswe need to estimate the value of γ . For the current parame-terisation it is easy to show that at high redshifts z ≫ theasymptotic value of γ ( z ) is written as γ ∞ ≃ γ + γ , whilethe theoretical formula of γ ∞ is given by Steigerwald et al.(2014) γ ∞ = 3( M + M ) − H + N )2 + 2 X + 3 M (43)where the following quantities have been defined: M = µ | ω =0 , M = dµdω (cid:12)(cid:12)(cid:12)(cid:12) ω =0 (44)and N = 0 , H = − X w de ( a ) | ω =0 , (45)ADE PARAMETERISATIONS ω = lnΩ m ( a ) . Obviously, for z ≫ we get Ω m ( a ) → [or Ω de ( a ) → ] which implies ω → . For more detailsregarding the theoretical treatment of (43) we refer the readerto Steigerwald et al. (2014). It is interesting to mention thatthe asymptotic value of the equation of state parameter forthe current PADE cosmologies is written as w ∞ ≡ w de ( a →
0) = w + w w , for PADE I w w , for Sim . PADE I w w , for PADE II . (46)At this point we are ready to present our growth index re-sults: • Homogeneous PADE parameterisations: here weset µ ( a ) = 1 ( ∆ de ≡ ). From Eqs.(44) and (45)we find { M , M , H , X } = { , , w ∞ , − w ∞ } and thus Eq.(43) becomes γ ∞ = 3( w ∞ − w ∞ − . (47)Lastly, inserting γ ≃ γ ∞ − γ into Eq.(42) and utiliz-ing Eqs. (46-47) together with the cosmological con-straints of Table (4) we obtain ( γ , γ , γ ∞ ) = (0 . , − . , . , for PADE I(0 . , − . , . , for Sim . PADE I(0 . , − . , . , for PADE II . (48)For comparison we provide the results for the Λ CDMmodel and CPL parameterisation respectively. Specifi-cally, we find ( γ , γ , γ ∞ ) Λ ≃ (0 . , − . , . and ( γ , γ , γ ∞ ) CPL ≃ (0 . , − . , . . • Clustered PADE parameterisations: here the func-tional form of µ ( a ) is given by the second branch ofEq.(39) which means that we need to define ∆ de . FromEq.(33) we simply have ∆ de = de − de and thus µ ( a ) takes the following form µ ( a ) = 1 + (1 + 3 c ) Ω de Ω m (1 + w de )(1 − w de ) . (49)In this case, from Eqs.(44) and (45) we obtain (formore details see the Appendix) { M , M , H , X } = { , − (1 + w ∞ )(1 + 3 c )1 − w ∞ , w ∞ , − w ∞ } and from Eq.(43) we find γ ∞ ≃ − w ∞ )(1 − w ∞ ) − (1 + w ∞ )(1 + c )](6 w ∞ − w ∞ − . Notice, that in the case of fully clustered PADE param-eterisations ( c = 0 ) the above expression becomes γ ∞ ≃ w ∞ (3 w ∞ − w ∞ − w ∞ − . (50)(46-47) Now, utilizing Eqs.(46-50) and the cosmolog-ical parameters of Table (4) we find ( γ , γ , γ ∞ ) = (0 . , . , . , for PADE I(0 . , . , . , for Sim . PADE I(0 . , . , . , for PADE II . (51)To this end, if the CPL parameterisation is allowed tocluster then the asymptotic value of the growth indexis given by Eq.(50), where w ∞ = w + w . In this casewe find ( γ , γ , γ ∞ ) CPL ≃ (0 . , . , . .In Table (5), we provide a compact presentation ofour numerical results including the relative fractionaldifference ∆ γ (%) = [( γ − γ Λ ) /γ Λ ] × betweenall DE parameterisations and the concordance Λ cos-mology, in 3 distinct redshift bins. Overall, we findthat the fractional deviation lies in the interval ∼ [ − . , . . We believe that relative differencesof | ∆ γ | ≤ will be difficult to detect even withthe next generation of surveys, based mainly on Euclid(see Taddei & Amendola 2015). Using the latter fore-cast and the results presented in section 4, we can nowdivide the current DE parameterisations into those thatcan be distinguished observationally and those that arepractically indistinguishable from Λ CDM model. Theformer DE parameterisations are the following three:homogeneous PADE I, clustered Simplified PADE Iand clustered CPL. However, the reader has to remem-ber that these results are based on utilizing cosmologi-cal parameters that have been fitted by the present dayobservational data (see Table 4). Therefore, if futureobservational data would provide slightly different val-ues for the parameters of DE parameterisations thenthe growth rate predictions of the studied DE parame-terisations could be somewhat different than those de-rived here. CONCLUSIONSWe studied the cosmological properties of various DE pa-rameterisations in which the EoS parameter is given with theaid of the PADE approximation. Specifically, using differ-ent types of PADE parameterisation we investigated the be-haviour of various DE scenarios at the background and at theperturbation levels.2 R
EZAEI ET AL . Table 5 . Numerical results. The st and the nd columns indicate the status of DE and the corresponding parametrisation. rd , th and th columns show the γ , γ and ∆ de0 values. The remaining columns present the fractional relative difference between the DE parameterisationsand the Λ CDM cosmology based on the cosmological parameters appeared in Table 4.DE Status DE Parametrisation γ γ ∆ de0 ∆ γ (%) z < . . ≤ z < ≤ z < . Homogeneous PADE I 0.555 -0.031 -1.2 -1.8 -2.2Sim. PADE I 0.558 -0.021 -0.1 -0.4 -0.6PADE II 0.559 -0.017 0.15 -0.01 -0.15CPL 0.561 -0.02 0.3 -0.02 -0.2Clustered PADE I 0.547 0.005 0.035 -0.8 -0.5 -0.1Sim. PADE I 0.542 0.012 0.047 -1.4 -0.7 -0.2PADE II 0.549 0.003 0.028 -0.6 -0.4 -0.02CPL 0.539 0.013 0.055 -2 -1.5 -0.8
The main results of the present study are summarized asfollows: (i)
Initially, using the latest expansion data we performeda likelihood analysis in the context of Markov Chain MonteCarlo (MCMC) method. It is interesting to mention that thestatistical performance of the MCMC method has been dis-cussed in Capozziello et al. (2011) and references therein.Specifically, these authors showed that if we have a multi-dimensional space of the cosmological parameters then theMCMC algorithm provides better constraints than other pop-ular fitting procedures. The results of our analysis for theexplored PADE cosmologies, including those of CPL and Λ CDM can be found in Tables (1 & 2). Based on this anal-ysis we placed constraints on the model parameters and wefound that all DE parameterisations are consistent with theexpansion data. In this framework, using the best fit val-ues we found that only the PADE (II) parameterisation re-mains in the quintessence regime ( < w de < / ). Therest of the PADE parameterisations evolves in the phantomregion ( w de < − ) at high redshifts, while they enter in thequintessence regime at relatively low redshifts. Concerningthe cosmic expansion we found that prior to the present timethe Hubble parameter of the DE parameterisations (PADEand CPL) is ∼ − . larger than the Λ CDM cosmologicalmodel.We also showed that the transition redshift from de-celerating to accelerating expansion in the context of PADEparameterisations is consistent with that (Farooq et al. 2017)using the cosmic chronometer H ( z ) data. Notice, that similarresults found in the framework of modified theory of gravi-ties (Capozziello et al. 2014, 2015). (ii) Then, we studied for the first time the growth of per-turbations in homogeneous and clustered PADE cosmolo-gies. First we used a joint statistical analysis involving theexpansion data and the growth data in order to place con-straints on σ . Second, based on the best fit cosmological parameters we computed the evolution of the growth rate ofclustering f ( z ) . For the current DE parameterisations wefound that the growth rate function is lower than Λ CDMmodel at low redshifts, while the differences among the pa-rameterisations are negligible at high redshifts. Third, fol-lowing the notations of Steigerwald et al. (2014) we derivedthe functional form of the growth index ( γ ) of linear mat-ter perturbations. Assuming that DE is homogeneous wefound the well known asymptotic value of the growth in-dex, namely γ ∞ = w ∞ − w ∞ − [ w ∞ = w ( z → ∞ ) ], whilein the case of clustered DE parameterisations we obtained γ ∞ ≃ w ∞ (3 w ∞ − w ∞ − w ∞ − .Finally, utilizing the fractional deviation between all DEparameterisations and the concordance Λ cosmology wefound that ∆ γ ∼ [ − . , . . We concluded that rela-tive differences of | ∆ γ | ≤ will be difficult to detect evenwith the next generation of surveys, based on Euclid (seeTaddei & Amendola 2015). Combining the latter forecastand the results presented in section 4, we divided the currentDE parameterisations into those that can be distinguishedobservationally and those that are practically indistinguish-able from Λ CDM. The former DE parameterisations are thefollowing three: homogeneous PADE I, clustered SimplifiedPADE I and clustered CPL. ACKNOWLEDGEMENTSMM and AM acknowledge support from the Iran ScienceElites Federation. DFM acknowledges support from the Re-search Council of Norway, and the NOTUR facilities. SBacknowledges support by the Research Center for Astron-omy of the Academy of Athens in the context of the pro-gram ”Testing general relativity on cosmological scales” (ref.number 200/872).ADE
PARAMETERISATIONS A. M COEFFICIENT FOR CLUSTERED DARK ENERGY MODELSHere we provide some details concerning the coefficient M which appears in Eq.(43). This coefficient is given in terms of thevariable ω = lnΩ m (see section 4.1) which means that when a → ( z ≫ ) we get Ω m → (or ω → ). From Eq.(44) we have M = dµdω (cid:12)(cid:12)(cid:12)(cid:12) ω =0 = Ω m dµd Ω m (cid:12)(cid:12)(cid:12)(cid:12) Ω m =1 . Of course, in the case of homogeneous dark energy, namely µ ( a ) = 1 we simply find M = 0 . However, if dark energy isallowed to cluster then the situation becomes quite different.Indeed, using Eq.(49) we obtain after some calculations dµd Ω m = (1 + 3 c ) dd Ω m (cid:18) Ω de Ω m (cid:19) w de − w de + (1 + 3 c ) × Ω de Ω m dd Ω m (cid:18) w de − w de (cid:19) where Ω de = 1 − Ω m , dd Ω m (cid:18) Ω de Ω m (cid:19) = − ,dd Ω m (cid:18) w de − w de (cid:19) = dda (cid:18) w de − w de (cid:19) dad Ω m with d Ω m da = 3 a Ω m (1 − Ω m ) w de = 3 a Ω m Ω de w de . Obviously, based on the above equations we arrive at Ω m dµd Ω m = − (1 + 3 c ) 1 + w de Ω m (1 − w de ) + (1 + 3 c ) a m w de × dda (cid:18) w de − w de (cid:19) Taking the limit Ω m → ( a → ) of the latter expression we calculate M M = Ω m dµd Ω m (cid:12)(cid:12)(cid:12)(cid:12) Ω m =1 = − (1 + 3 c ) 1 + w de − w de . REFERENCES
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