Effects of Rastall parameter on perturbation of dark sectors of the Universe
EEffects of Rastall parameter on perturbation of dark sectors of the Universe A. H. Ziaie ∗ , S. Ghaffari † and H. Shabani ‡ Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),University of Maragheh, Maragheh, Iranand Physics Department, Faculty of Sciences, University of Sistan and Baluchestan, Zahedan, Iran
In recent years, Rastall gravity is undergoing a considerable surge in popularity. This theorypurports to be a modified gravity theory with a non-conserved energy-momentum tensor (EMT)and an unusual non-minimal coupling between matter and geometry. The present work looks forthe evolution of homogeneous spherical perturbations within the Universe in the context of Rastallgravity. Using the spherical top hat collapse model we seek for exact solutions in linear regime fordensity contrast of dark matter (DM) and dark energy (DE). We find that the Rastall parameteraffects crucially the dynamics of density contrasts for DM and DE and the fate of spherical collapseis different in comparison to the case of general relativity (GR). Numerical solutions for perturbationequations in non-linear regime reveal that DE perturbations could amplify the rate of growth of DMperturbations depending on the values of Rastall parameter.
I. INTRODUCTION
The Rastall theory of gravity has been firstly proposedin 1972 [1] and then in [2] which suggests that one mayrelax the conservation of the energy momentum tensor(EMT), i.e., it assumes ∇ ν T µν (cid:54) = 0 for energy sources [1–5]. The core motivation for this choice is that the con-servation laws have been only tested on the Minkowskispacetime or quasistatic gravitational fields [5]. There-fore, one can still modify this presumption accountingfor non-zero curvature of spacetime. Another justifica-tion for using the theories which violate the EMT con-servation is to provide a room to discuss the process ofparticle production. It is known that the conservationof EMT does not lead to the particle creation [6, 7]. Inthe Rastall picture, the conservation law is proportionalto the gradient of the Ricci scalar, i.e., ∇ ν T µν = λ ∇ µ R .Such a relation can be introduced because of quantumeffects [8]. It is shown that quantum effects may resultin a “gravitational anomaly” for which the usual conser-vation of EMT gets violated [7, 12]. Such a phenomenonmay have an important impact on black hole Hawkingradiation [13]. In the past decades many attempts havebeen performed to explore the physical contents of theRastall gravity. Cosmological consequences of the theoryhave been discussed in [14–21], G¨odel-type solutions havebeen investigated in [22], the Brans-Dicke scalar field hasbeen considered in the Rastall background [23, 24] andstatic spherically symmetric solutions have been intro-duced in [25–30]. The Rastall proposal has been in- ∗ [email protected] † sh.ghaff[email protected] ‡ [email protected] Since, in the Rastall gravity the Ricci scalar is related to the traceof energy momentum tensor, one may classify this theory as aparticular form of f ( R, T ) gravity (see, e.g., Ref. [9]). In f ( R, T )theories the implementation of T can be justified by quantumeffects[10, 11]. spected from Mach’s principle landscape however, it isshown that it is compatible with this principle [31], seealso [32–47] for other works within the framework ofRastall gravity.As some new theories are invented to expand theold ones, there may be some motivations to extend theRastall gravity. For example, in [17] it is shown that theRastall theory is consistent with current observations onthe Universe expansion whenever a DE fluid along with apressure-less fluid fill the background. Surprisingly, thiscompatibility is seen at background as well as linear per-turbation levels, and furthermore, the DE candidate mayalso cluster under the shadow of the Rastall non-minimalcoupling between the geometry and cosmic fluids [17].A recent generalization of the Rastall gravity has beenproposed in [5] where a variable Rastall parameter (indi-cated by λ (cid:48) ) is utilized instead of the usual one ( λ ) whichis used in the original theory. More exactly, the authorsof [5] have modified the non-conservation equation as ∇ ν T µν = ∇ µ ( λ (cid:48) R ). This modification reasonably allowsa smooth variation of coupling between energy momen-tum source and geometry, because of the cosmic evolu-tion. They concluded that a primary inflationary phasecan exist even in an empty flat FRW spacetime. Also,the authors of [48] have shown that in the context of thegeneralized Rastall theory a complete cosmological sce-nario, (including pre and post inflationary eras) can beunderstood, without resorting to a cosmological constant.The primordial collapsed regions serve as the initialcosmic seeds from which the large scale structures likegalaxies, clusters, supernovae, quasars etc., are devel-oped [49–51]. Investigation of the DE perturbations inthe linear and non-linear level is of great importance.In these cases, DE perturbations may form halo struc-tures influencing the (dark) matter collapsed region non-linearly [52]. Note that, different DE scenarios may leadto the same background expansion rate, nevertheless,they behave differently in the perturbation level. Studyof the mutual interactions between dark sectors in theperturbation level helps us to understand nature of the a r X i v : . [ phy s i c s . g e n - ph ] S e p DE. Influence of the DE on structure formation bothat the background level (no fluctuations) and the per-turbed level has been investigated in various scenarios,see e.g., [52–59]. In the Rastall gravity, we also needDE in order to describe the current acceleration of theUniverse [21]. Hence the study of the effects of DE fluc-tuations on matter clustering in Rastall gravity can beof interest. A rather simple way to deal with this issueis to utilize the Top-Hat Spherical Collapse (SC) model.This approach was initially employed in Einstein-de Sit-ter background in the standard Cold-DM scenario [60],and later in ΛCDM model [61]. Work along this line hasbeen extended to the study of quintessence fields [62],decaying vacuum models [63], f ( R ) gravity theories [64],DE models with constant equation of state (EoS) [52, 65],coupled DE models [66] and agegraphic DE cosmolo-gies [67].In this work we consider the issue of structure forma-tion in the framework of the Rastall gravity. The paperis organized as follows. In Sec. II we briefly review thefield equations of the Rastall gravity. In Sec. III, themain evolutionary equations of DE and DM perturba-tions are obtained. In Sec. IV, we discuss the linear be-havior of fluctuations in matter as well as DE dominatederas. Sec. V devoted to inspecting the non-linear effectsand finally in Sec. VI we summarize our results. II. FIELD EQUATIONS OF RASTALL GRAVITY
According to the original idea of Rastall, the diver-gence of EMT is proportional to the covariant derivativeof Ricci curvature scalar as ∇ µ T µν = λ ∇ ν R, (1)where λ is the Rastall parameter. The Rastall field equa-tions are then given by [3, 4] G µν + γg µν R = κT µν , (2)where γ = κλ is the Rastall dimensionless parameter and κ being the Rastall gravitational coupling constant. Theabove equation can be rewritten in an equivalent form as G µν = κT eff µν , T eff µν = T µν − γT γ − g µν , (3)where T eff µν is the effective energy momentum tensorwhose components are given by [1, 34] T ≡ − ρ eff = − (3 γ − ρ + γ ( p r + 2 p t )4 γ − , (4) T ≡ p eff r = (3 γ − p r + γ ( ρ − p t )4 γ − , (5) T = T ≡ p eff t = (2 γ − p t + γ ( ρ − p r )4 γ − . (6)We note that in the limit of λ → T eff µν = T µν leading to G µν = κT µν . Therefore, theGR solutions for T = 0, or equivalently R = 0, are alsovalid the Rastall gravity [3, 68]. III. SPHERICAL COLLAPSE
For a spatially flat, homogeneous and isotropic Uni-verse filled with dark matter and dark energy, equation(3) can be put into the form H = 8 πG γ − (cid:88) k [3 γ (1 + w k ) − ρ k = 8 πG γ −
1) [(3 γ − ρ m + ρ de ) + 3 γw de ρ de ] , (7)¨ aa = − πG γ − (cid:88) k [3(2 γ − w k + (6 γ − ρ k = − πG γ −
1) [3(2 γ − w de ρ de + (6 γ − ρ m + ρ de )] , (8)where, an over-dot denotes derivative with respect totime, κ = 2(4 γ − κ G / (6 γ − κ G = 4 πG , k = { m , de } labels DM and DE components, H = ˙ a/a is the Hubbleparameter, w de = p de /ρ de is the EoS parameter of DEand ρ m , ρ de and p de are the (background) energy densi-ties of DM and DE and the pressure of DE, respectively.The Bianchi identity for Eq. 3 leaves us with the follow-ing continuity equation in Rastall gravity, as˙ ρ j + 3 Hβ j ρ j = 0 , β j = β ( γ, w j ) = (cid:20) (1 + w j )(4 γ − γ (1 + w j ) − (cid:21) . (9)This equation describes the density evolution of a singleperfect fluid labeled by j with background density ρ j andpressure p j = w j ρ j . Consider now a spherically symmet-ric region of radius r filled with a homogeneous density ρ c j (a top-hat distribution). The SC model describes aspherical region with a top-hat profile and uniform den-sity so that at time time t , ρ c j ( t ) = ρ j ( t ) + δρ j . This re-gion initially undergoes a small perturbation of the back-ground fluid density, i.e., δρ j and is immersed within ahomogeneous Universe with energy density ρ j . If δρ j > p c j = w c j ρ c j ˙ ρ c j + 3 hβ c j ρ c j = 0 , β c j = β ( γ, w c j ) = (cid:34) (1 + w c j )(4 γ − γ (1 + w c j ) − (cid:35) , (10)where, h = ˙ r/r denotes the local expansion rate insidethe spherical perturbed region and w c j denotes the EoSin this region. The Friedmann equation equations forspherical region take the form h = 8 πG γ − (cid:88) k [3 γ (1 + w c k ) − ρ c k = 8 πG γ −
1) [(3 γ − ρ cm + ρ cde ) + 3 γw cde ρ cde ] , (11)¨ rr = − πG γ − (cid:88) k [3(2 γ − w c k + (6 γ − ρ c k = − πG γ −
1) [3(2 γ − w cde ρ cde + (6 γ − ρ cm + ρ cde )] , (12)where the second equation governs the dynamics of radius r of the collapsing region. We note, in general, that thedensities and pressures obey different EoSs inside andoutside the spherical region, i.e., w c j (cid:54) = w j . Indeed, thedifference between the local and background EoSs, δw j ≡ w c j − w j can be related to the effective sound speed of thefluid, C j = δp j /δρ j . This relation can be re-expressedthrough introducing the density contrast of a single fluidspecies labeled by jδ j = ρ c j ρ j − δρ j ρ j . (13)We therefore have w c j = p c j ρ c j = p j + δp j ρ j + δρ j = w j + (cid:0) C j − w j (cid:1) δ j δ j . (14)The above equation provides a relation between EoSwithin the perturbed region and that of the background,the effective sound speed and the size of perturbations. Inthe present model, we consider the case in which the EoSsinside the collapsing region and the background are iden-tical. We therefore take δw j = 0 leading to C j = w j and β c j = β j . Differentiating Eq. (13) with respect totime gives ˙ δ j = 3(1 + δ j )( H − h ) β j , (15)whereby differentiating again with respect to time leavesus with the following equation for amplitude of the per-turbations¨ δ j = (cid:20) ˙ w j d ln β j dw j − H (cid:21) ˙ δ j + 4 πGβ j γ − δ j ) (cid:88) k [3(2 γ − w k + 6 γ − ρ k δ k + 3 β j + 13 β j ˙ δ j δ j , (16)where use has been made of Eqs. (7), (8), (12), (12) and(15). We note that for γ = 0, we have β j = 1+ w j and Eq.(16) reduces to its counterpart given in [52]. For a mix-ture of DM (here we do not distinguish between DM andbaryons) and DE gravitationally interacting with each other, the top-hat spherical regions evolve according tothe following system of differential equations¨ δ m + 2 H ˙ δ m − (15 γ −
4) ˙ δ γ − δ m )= 3 H (4 γ − γ − γ −
1) (1 + δ m ) (cid:104) (6 γ − m δ m + (cid:16) γ − w de + 6 γ − (cid:17) Ω de δ de (cid:105) , (17)for the density contrast in DM component i.e., δ m , and¨ δ de + (cid:20) H − ˙ w de d ln β de dw de (cid:21) ˙ δ de − (cid:20) w de )(5 γ − − w de )(4 γ − (cid:21) ˙ δ δ de = 3 H (1 + w de )(4 γ − δ de )2(6 γ − (cid:16) γ (1 + w de ) − (cid:17) (cid:104) (6 γ − m δ m + (cid:16) γ − w de + 6 γ − (cid:17) Ω de δ de (cid:105) , (18)for density contrast in DE component i.e., δ de . In theseequations we have set w m = 0 and DM and DE densityparameters are defined respectively asΩ m = 8 πGρ m H , Ω de = 8 πGρ de H . (19) IV. SOLUTIONS IN LINEAR REGIME
In order to extract some physical results from Eqs.(17) and (18) we rewrite them for constant value of w de along with neglecting the terms containing O ( δ ). Wethen have¨ δ m + 2 H ˙ δ m = 3(4 γ − H γ − γ − (cid:104) (6 γ − m δ m + (cid:16) γ − w de + 6 γ − (cid:17) Ω de δ de (cid:105) , (20)¨ δ de + 2 H ˙ δ de = 3(1 + w de )(4 γ − H γ − (cid:16) γ (1 + w de ) − (cid:17) (cid:104) (6 γ − m δ m + (cid:16) γ − w de + 6 γ − (cid:17) Ω de δ de (cid:105) . (21)The linear approximation of cosmological perturbationsis valid for all scales during the radiation dominated eraand for most scales during the matter dominated era upuntil very recently. Therefore, the initial stages of struc-ture formation can be adequately investigated within thelinear approximation. A. Matter dominated era
In principle, one can utilize any suitable parameteriza-tion for DE as a function of time or redshift. However,to obtain analytical solutions, we consider Eqs. (20) and(21) within the matter dominated epoch ( z = 10 ), whenthe density parameters for DM and DE can be approx-imated as Ω m ≈ de ≈
0, respectively. We thenhave δ (cid:48)(cid:48) m + 3 δ (cid:48) m a − γ − γ − a δ m = 0 (22) δ (cid:48)(cid:48) de + 3 δ (cid:48) de a − w de )(4 γ − a (cid:16) γ (1 + w de ) − (cid:17) δ m = 0 , (23)where a prime denotes a derivative with respect to a anduse has been made of Eqs. (8). It is straightforwardto find the analytic solutions of the above equations. Wecan firstly solve Eq. (22) for matter density contrast withthe solution given as δ m ( a ) = C a α + C a α , (24)where α , = − (cid:20) ± (cid:114) γ − γ − (cid:21) , (25)and C and C are integration constants. We then realizethat for those values of Rastall parameter which belongto the set S = { γ | γ ∈ R : γ < / ∨ γ > / } , we have α < α >
0, always. Therefore, as the Uni-verse expands, the first term in Eq. (24) decays but thesecond one increases leading to a growing matter den-sity contrast. The matter density contrast decreases for1 / < γ ≤ /
99 as for this case both the exponents ofscale factor assume negative values. However, this casecannot be of interest as we are dealing with matter dom-inated era. We note that for γ = 0, the solution obtainedin [52] is recovered. Now, if we neglect the decaying modein Eq. (24) and substitute the result into Eq. (23), weobtain the following solution for the amplitude of DEperturbations as δ de ( a ) = C − C √ a + ξ ( γ, w de ) δ m ( a ) , (26)where ξ ( γ, w de ) = (3 γ − w de )3 γ (1 + w de ) − . (27)At first glance we observe that the evolution of densitycontrast of DE depends on the Rastall parameter as wellas the EoS of DE. It is natural to choose the adiabaticinitial condition for DE density contrast [58], i.e., setting C = 0. We note that the adiabatic condition is differentin the usual DE models for which w de > − w de < −
1. Let us first consider the case γ = 0 for which ξ (0 , w de ) = 1 + w de .Neglecting the decaying term, we see that for phantommodels, adiabatic initial conditions mean that, any ini-tial over-density in DM ( δ m >
0) is accompanied by anunder-density in DE ( δ de <
0) and vice versa. The case C (cid:54) = 0 implies a non-adiabatic initial condition, i.e., theperturbations bear an isocurvature component. In thiscase, if we assume initially positive densities for DM andDE perturbations, namely δ im > δ ide >
0, we have δ de ( a ) = δ ide + (1 + w de ) (cid:0) δ m ( a ) − δ im (cid:1) . (28)For γ ∈ S , we always have a growing density contrast forDM, hence δ m ( a ) ≥ δ im . Therefore, if initially the DEperturbations are positive, then the pressure gradients,due to non-adiabatic perturbations, will make the DEhalo to decay, till a critical value for the scale factor, forwhich δ de ( a cr ) = 0, is reached. This critical value is givenby a cr = δ ide | w de | + δ im , w de < − . (29)We note that for a vanishing Rastall parameter, α = − / α = 1, where the former corresponds to adecaying mode and the latter corresponds to a growingmode for DM perturbations. For a > a cr , the DE densitycontrast turns to negative values leaving thus the scenariowith a DE void [52]. Next, we proceed to study Eq. (26)for non-vanishing Rastall parameter. Assuming adiabaticinitial conditions ( C = 0) together with neglecting thedecaying term, we can deduce the following propositions:I ) γ > ∧ ξ ( γ, w de ) < ∧ − < w de < − ,or equivalently (cid:110) < γ ≤ ∧ − < w de < − (cid:111) ∨ (cid:110) γ > ∧ − < w de < − γ γ (cid:111) ,II ) γ < ∧ ξ ( γ, w de ) < ∧ w de < − (cid:110) γ < ∧ − γ γ < w de < − (cid:111) ∨ (cid:110) ≤ γ < ∧ w de < − (cid:111) ,III ) γ ∈ S ∧ ξ ( γ, w de ) > ∧ − < w de < − ,or equivalently (cid:110) γ > ∧ − γ γ < w de < − (cid:111) ∨ (cid:110) γ < ∧ − < w de < − (cid:111) ,IV ) γ ∈ S ∧ ξ ( γ, w de ) > ∧ w de < − (cid:110) γ > ∧ w de < − (cid:111) ∨ (cid:110) γ < ∧ w de < − γ γ (cid:111) .The items I and III deal with usual DE models and thoseof II and IV deal with phantom models. The first itemcorresponds to the case in which an initial over-densityin DM component is matched by an under-density in DEand as the collapse proceeds, voids of DE can formed.However, it is still possible to have an initial over-densityfor DE component with w de > −
1. As indicated in caseIII, if initially there is an over-density in matter, i.e., δ im >
0, we then have δ ide >
0. This case is the coun-terpart of case I but with different evolution for DE per-turbations. More interestingly, as shown in case IV, aninitial overdensity for DE can also occur for phantom DEmodels. For this case, any initial overdensity in DM leadsto an initial overdensity in DE component, hence, over-dense regions of DE would be more and more overdenseas time elapses. The case II provides similar situation asreported in [52]. For non-adiabatic initial conditions Eq.(26) can be re-expressed as δ de ( a ) = δ ide + ξ ( γ, w de ) (cid:0) δ m ( a ) − δ im (cid:1) . (30)Now consider again the cases I and II. If we take δ ide > δ im >
0, then the pressure gradients give rise to DEdecay, turning it into DE void through switching the signof DE perturbation at a critical value of the scale factorfor which δ de ( a (cid:63) cr ) = 0 a (cid:63) cr = (cid:20) δ im + δ ide | ξ | (cid:21) α . (31)It is worth mentioning that, this situation can occur forboth phantom and usual DE models, in comparison tothe situation presented in [52] which can occur only forphantom models. We therefore conclude that the pres-ence of Rastall parameter could crucially alter the evolu-tion of DM and DE perturbations in matter dominatedera. In Fig. (1) we have encapsulated the conditionsI-IV where the allowed regions for the pair ( w de , γ ) arepresented. B. Dark Energy Dominated era
In order to study the effects of DE perturbations onthe evolution of DM perturbations, we consider Eqs. (20)and (21) in DE dominated period. Let us begin with Eq.(8) which can be put into the following form¨ a = − aH γ − (cid:104) (6 γ − m + Ω de ) + 3(2 γ − w de Ω de (cid:105) . (32) Case ICase III Case III - - - - - - - γ w de Case IICase IV Case IV - - - - - - - γ w de FIG. 1: The allowed values for DE equation of state andRastall parameters, subject to the conditions given in casesI-IV. The red dashed arrow corresponds to γ = 0. The whiteregion is not allowed for the model parameters. Now, considering the following transformations forderivatives˙ δ = aHδ (cid:48) , ¨ δ = ¨ aδ (cid:48) + a H δ (cid:48)(cid:48) , (33)together with using Eq. (32) we can re-express Eqs. (20)and (21) in terms of scale factor derivatives as a δ (cid:48)(cid:48) m + ab δ (cid:48) m − b δ m = b δ de , (34) a δ (cid:48)(cid:48) de + ab δ (cid:48) de − b δ de = b δ m , (35)where b = 2 − (cid:20) Ω m + (cid:18) γ − γ − w de (cid:19) Ω de (cid:21) ,b = 3(4 γ − γ −
1) Ω m ,b = 3(4 γ − (cid:16) γ − w de + 6 γ − (cid:17) γ − γ −
1) Ω de ,b = 3(1 + w de )(4 γ − γ (1 + w de ) −
1) Ω m ,b = (1 + w de )(3 γ − b γ (1 + w de ) − . (36)The system of differential equations (34) and (35) admitsthe following solutions for DM and DE perturbations as δ m ( a ) = C a − ( b + B − + C a ( B − b +1) + C a − ( b + B − + C a ( B − b +1) , (37) δ de ( a ) = 12 b (cid:104) C Aa − ( b + B − + C Aa ( B − b +1) + C A a − ( b + B − + C A a ( B − b +1) (cid:105) , (38)where B = (cid:104) b − (cid:112) ( b − b ) + 4 b b − b + 2 b + 2 b + 1 (cid:105) ,B = (cid:104) b + 2 (cid:112) ( b − b ) + 4 b b − b + 2 b + 2 b + 1 (cid:105) ,A = b − (cid:112) ( b − b ) + 4 b b − b ,A = b + (cid:112) ( b − b ) + 4 b b − b . (39)Note that in the limit of γ → b → −
12 Ω m −
12 (1 + 3 w de )Ω de ,b →
32 Ω m , b →
32 (1 + 3 w de )Ω de ,b →
32 (1 + w de )Ω m , b →
32 (1 + w de )(1 + 3 w de )Ω de . (40)The unknown constants C − C can be determined usingthe adiabatic initial conditions [52, 59] dδ m dz (cid:12)(cid:12)(cid:12) z = z i = − α δ m ( z i )1 + z i , dδ de dz (cid:12)(cid:12)(cid:12) z = z i = − α ξδ m ( z i )1 + z i ,δ de ( z i ) = ξδ m ( z i ) , (41) whence we finally obtain solutions (37) and (38) in termsof redshift 1 + z = a − , as δ m ( z ) = (1 + z i ) − b ) δ m ( z i ) BB ( A − A ) (cid:34) B (2 ξb − A ) × (cid:32) q (1 + z ) ( b + B − (1 + z i ) ( B − b +3) − q (1 + z ) ( b − B − (1 + z i ) − ( B +3 b − (cid:33) + q B (2 ξb − A ) (1 + z ) ( b − B − (1 + z i ) − ( B +3 b − − q B (2 ξb − A ) (1 + z ) ( b + B − (1 + z i ) ( B − b +3) (cid:35) , (42) δ de ( z ) = (1 + z i ) − b ) δ m ( z i ) BB b ( A − A ) (cid:34) AB (cid:18) ξb − A (cid:19) × (cid:32) q (1 + z ) ( b + B − (1 + z i ) ( B − b +3) − q (1 + z ) ( b − B − (1 + z i ) − ( B +3 b − (cid:33) + 12 q A B (2 ξb − A ) (1 + z ) ( b − B − (1 + z i ) − ( B +3 b − − q A B (2 ξb − A ) (1 + z ) ( b + B − (1 + z i ) ( B − b +3) (cid:35) , (43)where q = α + 12 ( b + B − , q = α + 12 ( b − B − ,q = α + 12 ( b + B − , q = α + 12 ( b − B − . (44)We can also integrate Eq. (34) for δ de = 0, in orderto obtain the behavior of matter perturbations in theabsence of DE perturbations. By doing so we get˜ δ m ( z ) = δ im b − q − (cid:34) ( α + b − q − (cid:18) z z i (cid:19) q − ( α + q ) (cid:18) z z i (cid:19) b − q − (cid:35) , (45)where q = 12 (cid:20) b − (cid:113) b − b + 4 b + 1 − (cid:21) , (46)and use has been made of the initial conditions givenin Eq. (41). The obtained expressions (42), (43) and(45) provide a wide class of solutions for density per-turbations, depending on Rastall and equation of statefor DE parameters. Firstly, from Eq. (34) we realizethat DE perturbations can act as a source for matterperturbations in such a way that an overdensity in DEcomponent could reduce ( b <
0) or enhance ( b > γ, w de ) where the FEC D - - - - - - - - γ w de FIG. 2: The allowed values for DE equation of state andRastall parameters for which b > b < γ = 0and w de = − / allowed regions for positive (shaded region) and negative(gray region) values of b coefficient are plotted. Inter-estingly we observe that not always a DE overdensity de-creases matter clustering and indeed, for certain values of( γ, w de ) parameters we could have matter clustering for w de < − /
3. Such a scenario occurs for the shaded regionconfined between the blue arrow and the curves CD and EF . Moreover, an underdensity in DE perturbations doesnot necessarily lead to an overdensity in matter compo-nent, and instead, can provide both overdense ( b < b >
0) regions of matter distribution.We also note that for all the points lying on the red arrow( γ = 0), a DE overdensity decreases matter clustering for w de < − / FIG. 3: Evolution of density contrasts δ m (solid curve), δ de (dashed curve) and ˜ δ m (dot-dashed curve) for γ = − .
263 and w de = − .
11 (upper panel) and γ = − .
975 and w de = − . γ, w de ) from gray and shaded regions of Fig. (2),respectively. For density parameters we have set Ω de = 1 − Ω m = 3 / . V. NON-LINEAR REGIME WITH VARYING w de When considering the non-linear regime, things getslightly more complicated. Our aim is again to determinethe density contrast, however, now we take into accountthe nonlinear terms within equations (17) and (18). Bydoing so, these equations for a varying DE state param-eter then read a δ (cid:48)(cid:48) m + ab δ (cid:48) m − b (1 + δ m ) δ m − b (1 + δ m ) δ de − a b δ (cid:48) δ m = 0 , (47) a δ (cid:48)(cid:48) de + ab δ (cid:48) de − b (1 + δ de ) δ m − b (1 + δ de ) δ de + a δ (cid:48) de w (cid:48) de (1 + w de )(3 γ (1 + w de ) − − a b δ (cid:48) δ de = 0 , (48)where b = 15 γ − γ − , b = 3(1 + w de )(5 γ − − w de )(4 γ − . (49)The above equations can be re-expressed in terms of theredshift using the following relations δ (cid:48) = − (1 + z ) dδdz , w (cid:48) de = − (1 + z ) dw de dzδ (cid:48)(cid:48) = − (1 + z ) d δdz + 2(1 + z ) dδdz . (50)Now if we take the following parametrization for DE stateparameter [69] w de ( z ) = w + w z z , (51)we can solve Eqs. (47) and (48) using numerical meth-ods. The constants w and w can be chosen so thatthey are consistent with observational constraints [70].As we expect the numerical solution for non-linear regimeis different and depends crucially on the model parame-ters and initial data. We choose the initial density con-trast for matter component to be a finite value at z = 2.Therefore, the formation of matter structures commencesat this redshift and evolves, along with the evolution ofDE perturbations, until the present time ( z = 0). Wefind that in response to non-linear perturbations in DE,matter perturbations grow at a faster rate and reach abigger amplitude than expected in linear regime. In Fig.(4) we have plotted the evolution of dark matter pertur-bations in the presence and absence of DE perturbationswithin upper and lower panels, respectively. For γ = 0(Black solid curve), the DM density contrast grows mono-tonically and reaches a finite value at the present time,while for negative values of Rastall parameter, the matterperturbations grow faster so that dark matter structuresmay form before reaching the present epoch. For posi-tive values of Rastall parameter (dashed and dot-dashedcurves) we observe even more a rate of growth in matterperturbations in such a way that massive objects such assuper-clusters can be born within the Universe. As weobserve in the lower panel, matter perturbations start togrow from their initial values and diverge as we reachthe present time. However, the rate of growth in densitycontrast for γ = 0 (black solid curve) is lesser than thecase in which the Rastall parameter is nonzero. Hence,in comparison to GR, we could have massive structuresthat form faster in Rastall gravity. VI. CONCLUDING REMARKS
In the present work we studied the evolution of DMand DE perturbations in the context of Rastall theory. γ = γ =- γ =- γ =- γ =- γ = γ = z δ m γ = γ =- γ =- γ =- γ = γ = γ = z δ ˜ m FIG. 4: Evolution of matter perturbations with (upper panel)and without (lower panel) taking into account the DE pertur-bations. The initial value of matter density contrast has beenchosen as δ m ( z = 2) = 10. For parametrization of DE equa-tion of state we have set w = − .
75 and w = 0 .
4. Fordensity parameters we have set Ω de = 1 − Ω m = 3 / . In order to simplify analysis, we restricted ourselves tospherically symmetric perturbations. Thus, for a spher-ically symmetric top hat collapse, we investigated dy-namics of density contrast for dark components in bothlinear and non-linear regimes. In the linear regime, weobserved that the Rastall parameter could play an im-portant role in the growth of DM perturbations. More-over, DM perturbations could in turn provide a settingfor enhancing or decreasing growth of DE perturbations.In DE dominated era, we obtained exact solutions forthe set of differential equations that govern the dynam-ics of density contrasts of DE and DM. We found thatDE perturbations could increase the rate of growth ofDM perturbations so that they grow even faster than thecase in which DE perturbations are absent. Numericalsolutions to perturbation equations in nonlinear regimeshow a different scenario. Matter perturbations couldgrow more rapidly compared with the linear case so thatstructures which collapse in this manner could be denserthan those of linear regime. Hence, the collapse processin non-linear regime could lead to the formation of su-per clusters of DM. We further note that depending onthe values and signs of the pair ( γ, w de ) other types ofsolutions can be obtained, in which, DE and DM pertur-bations experience oscillations with different frequenciesand amplitudes. One then can interpret such a behavioras the ability of spacetime and matter fields to couple toeach other in non-minimal way, that the representativeof which is the Rastall parameter. We therefore observethat the DE and DM perturbations in Rastall theorycould lead to different fates in comparison to GR. Acknowledgments
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