A new approach to cosmological perturbations in f(R) models
aa r X i v : . [ a s t r o - ph . C O ] A ug Preprint typeset in JHEP style - HYPER VERSION
A new approach to cosmological perturbations in f ( R ) models Daniele Bertacca a,b,c , Nicola Bartolo a,b , Sabino Matarrese a,b a Dipartimento di Fisica Galileo Galilei, Universit`a di Padova, via F. Marzolo, 8I-35131 Padova, Italy b INFN Sezione di Padova, via F. Marzolo, 8 I-35131 Padova, Italy c Institute of Cosmology & Gravitation, University of Portsmouth, Dennis SciamaBuilding, Portsmouth, PO1 3FX, United KingdomE-mails: [email protected] , [email protected] , [email protected] Abstract:
We propose an analytic procedure that allows to determine quantitatively thedeviation in the behavior of cosmological perturbations between a given f ( R ) modifiedgravity model and a ΛCDM reference model. Our method allows to study structure for-mation in these models from the largest scales, of the order of the Hubble horizon, down toscales deeply inside the Hubble radius, without employing the so-called “quasi-static” ap-proximation. Although we restrict our analysis here to linear perturbations, our techniqueis completely general and can be extended to any perturbative order. Keywords: alternative theory of gravity; cosmological perturbations; analyticalmethod. ontents
1. Introduction 12. Preliminary relations 33. Description of the analytical approach 5
4. From the synchronous to the Poisson gauge 115. Comparison with the “quasi-static” approximation 136. Matter Power Spectrum 157. Conclusions 16A. Evolution of first-order perturbations of Λ CDM model in the synchronousgauge 17B. Evolution of first-order perturbations of Λ CDM model in the Poissongauge 19
1. Introduction
During the last decade, independent observational data such as type-Ia Supernovae (SNIa)[1, 2, 3, 4] and Cosmic Microwave Background (CMB) [5, 6] and Baryonic Acoustic Oscilla-tions [7, 8, 9] suggest that two dark components govern the dynamics of the Universe. Theyare the Dark Matter (DM), thought to be the main responsible for structure formation,and a non-zero cosmological constant Λ (see, e.g. ref. [10]) or a dynamical dark energy(DE) component, that is supposed to drive the observed cosmic acceleration [11, 12].The standard cosmological model ΛCDM provides a good fit to observations, but itdescribes the Universe by means of two unknown components, which represent 96% of thetotal energy density. However, it should be recognised that, while some form of Cold DarkMatter (CDM) is independently expected to exist within any modification of the StandardModel of high energy physics, the really compelling reason to postulate DE has been thediscovery that the Universe is experiencing a phase of accelerated expansion. This couldbe theoretically unsatisfactory and, recently, alternative models have been proposed withrespect to the class of DE models. In particular, many of them are based on modifications– 1 –f gravity at large distances. For example, scalar-tensor theories [13, 14, 15, 16, 17, 18, 19],Brane-World models (see e.g the review [20] and refs. therein), Galileon models [21, 22,23, 24, 25, 26, 27], Gauss-Bonnet gravity [28, 29, 30] and other scenarios (see, for example,the review [11] and refs. therein).An interesting class of modified gravity models is represented by f ( R ) theories (seethe reviews [31, 32, 33, 34, 35, 36, 37] and refs. therein ), whose Lagrangian density issimply defined by an arbitrary function of the Ricci scalar R . These Lagrangians wereproposed for the first time in connection with Inflation in the early Universe [61] and, onlyrecently, they have been used in the context of DE models, to explain the present-daycosmic acceleration (see e.g. [62, 63, 64]).In general, there are two ways to study these models [65]: 1) specifying directly thetype of Lagrangian that satisfies cosmological and local gravity constraints (e.g. [66, 67]and see also [34]) 2) through a parametrized post-Newtonian framework or similar (see e.g.Refs. [68, 69, 70, 71, 72, 73, 65, 74, 75]), to provide a scale-dependent parameterization ofcosmological perturbations (see also [76]).Recently, it was found that, in viable f ( R ) models (see e.g. [77, 78, 66, 67, 79, 80, 81,82, 83, 56, 34]) when the superhorizon long-wavelength limit is taken (i.e. when spatialgradients may be neglected in the equations), the evolution of metric and density perturba-tions of a Robertson-Walker background can be described by the Friedmann equation andenergy-momentum conservation [68, 69, 66, 65, 74]. In addition, the viable f ( R ) modelsusually need to be close to the ΛCDM model during the matter dominated epoch in orderto satisfy cosmological constraints, local gravity constraints (see [67, 34, 84, 86, 87]) andgalactic constraints [67]. Motivated by these results, in this paper, we propose an ana-lytical method which allows to describe both the background evolution and cosmologicalperturbations in f ( R ) modified gravity models. This approach is completely different fromprevious ones. Indeed, through this analytic technique we can determine quantitatively thedeviation in the behavior of cosmological perturbations between a given f ( R ) model and aΛCDM reference model. Moreover, our treatment is general in that all the results dependonly on the initial conditions that characterize the type of f ( R ) model we are studying.Finally, our approach allows to study structure formation in these models from the largestscales, of the order of the Hubble horizon, down to scales deeply inside the Hubble radius,without employing the so-called “quasi-static” approximation (see also [88, 89]). Althoughwe restrict our analysis here to linear perturbations, the technique can be applied to lin-ear and weakly non-linear scales and can be extended to any perturbative order (see also[90, 91]). Of course, in order to study these models at non linear scales another approachmust be adopted (for example, see [34]).The rest of the paper is organised as follows. In Section 2 we present the basic equationsdescribing the background and the perturbative evolution of a generic f ( R ) gravity model.In Section 3 we describe our approach, analyzing both the background evolution and thefirst-order perturbation equations in the synchronous gauge and, in Section 4, we introducethe evolution of cosmological perturbations in the Poisson gauge. In Section 5 we recover See also [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60] for otherrecent contributions to this area. – 2 –he “quasi-static” approximation and Section 6 we compute the matter power spectrumwith our approach. Section 7 is devoted to our conclusions. Appendix A and Appendix Bare devoted to the evolution of linear perturbations in the reference ΛCDM model, in thesynchronous and Poisson gauge, respectively.Throughout the paper we use G = c = 1 units and the ( − , + , + , +) signature for themetric. Greek indices run over { , , , } , denoting space-time coordinates, whereas Latinindices run over { , , } , labelling spatial coordinates.
2. Preliminary relations
Let us consider the following action in f ( R ) gravity (see e. g. [61, 62, 63, 64]; see also thereviews [31, 33, 34, 35] and refs. therein): S = S (GR) + S (m) = 116 π Z d x √− g f ( R ) + Z d x √− g L m [Ψ m , g µν ] . (2.1)In this case f ( R ) is a general function of the Ricci scalar, R , and L m is the matter La-grangian and Ψ m are the matter fields. Defining φ ≡ ∂f /∂R , S (GR) can be cast in the formof Brans-Dicke (DB) theory [13] with a potential for the scalar field φ [31, 33, 34, 35].In particular, S (GR) = 116 π Z d x √− g [ φR − V ( φ )] , (2.2)where V ( φ ) = Rφ − f ( R ). The field equations obtained from varying the action withrespect to g µν are φG µν − πT φµν = 8 πT m µν , (2.3)where G µν = R µν − (1 / Rg µν ,8 πT φµν = ∇ µ ∇ ν φ − (cid:3) φg µν − V ( φ ) g µν , (2.4)and T m µν is the energy-momentum tensor of a perfect fluid, i.e. T m µν = − √− g δS (m) δg µν = ( ρ m + p m ) u µ u ν + p m g µν . (2.5)The vector u µ is the fluid rest-frame four-velocity, ρ is the energy density and p the isotropicpressure. In this case the equation of motion of the scalar field is simply R = d V d φ . (2.6)Finally by taking the trace of Eq. (2.3) and using Eq. (2.6), we obtain the dynamics of thescalar field for a given matter source (cid:3) φ + 13 (cid:18) V − φ d V d φ (cid:19) = 8 π T m , (2.7)where T m = g µν T m µν . – 3 –hen considering the background cosmological evolution, we take the metric to be ofthe flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) form, ds = a ( η )( − dη + d x ),where η is the comoving time and a ( η ) the scale factor. Then the gravitational fieldequations, the scalar field equation of motion and the continuity equation become A H = 8 πa ρ (0)m + ( A − ϕ ) H + ( A − ϕ ) ′ H + a V ( ϕ ) , (2.8) A (cid:0) H ′ + H (cid:1) = − πa p (0)m + ( A − ϕ ) (cid:0) H ′ + H (cid:1) + ( A − ϕ ) ′′ + ( A − ϕ ) ′ H + a V ( ϕ ) , (2.9) H ′ + H = a V d ϕ , (2.10) ρ (0)m ′ + 3 H ( p (0)m + ρ (0)m ) = 0 , (2.11)where A is a suitable constant (for its physical meaning see Ref. [34]), H = a ′ /a andprimes indicate differentiation w.r.t. η . Note that φ (0) = ϕ , p (0)m and ρ (0)m are respectivelythe background scalar field, matter pressure and matter energy density. From Section 3.1onwards we will set A = 1.Now taking into account only the non-relativistic matter, i.e. p m = 0, let us considerthe metric of a flat FLRW Universe with small perturbations. In particular we want tostudy the evolution of the scalar perturbations in the synchronous gauge (see also Ref.[78]). In this case the line-element is written in the form ds = a {− dη + [(1 − ψ (1) ) δ ij + D ij χ (1) )] dx i dx j } , where D ij = ∂ i ∂ j − (1 / δ ij ∇ is a trace-free operator. Let us allow forsmall inhomogeneities of the scalar field, φ ( η, x ) = ϕ ( η ) + ϕ (1) ( η, x ). Perturbing Eq. (2.3),we get 3 (cid:18) H + ϕ ′ ϕ (cid:19) ψ (1) ′ − (cid:18) ∇ ψ (1) + 13 ∇ ∇ χ (1) (cid:19) = 8 πa ϕ ρ (0)m ϕ (1) ϕ − δ (1) ! +3 H ϕ (1) ′ ϕ − H ϕ ′ ϕ ϕ (1) ϕ − ∇ ϕ (1) ϕ + a (cid:18) Vϕ − d V d ϕ (cid:19) ϕ (1) ϕ , (2.12)2 ψ (1) ′ + 13 ∇ χ (1) ′ = ϕ (1) ′ ϕ − H ϕ (1) ϕ , (2.13) χ (1) ′′ + (cid:18) H + ϕ ′ ϕ (cid:19) χ (1) ′ + 2 ψ (1) + 13 ∇ χ (1) = 2 ϕ (1) ϕ , (2.14)where δ (1) = ( ρ (1)m − ρ (0)m ) /ρ (0)m . From Eq. (2.7), we obtain ϕ (1) ′′ + 2 H ϕ (1) ′ − ∇ ϕ (1) − ϕ ′ ψ (1) ′ = 8 πa ρ (0)m δ (1) − a (cid:18) ϕ d V d ϕ − d V d ϕ (cid:19) ϕ (1) . (2.15)– 4 –inally, at the linear order, Eqs. (2.6) and (2.11) become − ψ (1) ′′ − H ψ (1) ′ + 4 ∇ ψ (1) + 23 ∇ ∇ χ (1) = a d V d ϕ ϕ (1) , (2.16) δ (1) ′ = 3 ψ (1) ′ . (2.17)
3. Description of the analytical approach
Motivated by the fact that, in viable f ( R ) models [34], 1) the cosmic scale-factor a ( η ) canbe described by the Friedmann equation with background expansion of the Universe closeto the ΛCDM model and 2) gravity can be described by a classical four-dimensional metrictheory having a well-defined infrared limit [77, 68, 78, 66, 79, 65] (i.e. in the modifiedgravity models, the long-wavelength perturbations obey the same constraints as they do ingeneral relativity [68, 69, 66, 65]), let us make the following ans¨atz: i) at the background level, the scalar field can be described in the following way ϕ = ¯ ϕ + ǫAξ + ... , (3.1)where we introduced a suitable perturbative parameter | ǫ | ≪ ii) at the first order level ψ (1) = ψ (1 , + ǫA ψ (1 , + ... , (3.2) χ (1) = χ (1 , + ǫA χ (1 , + ... , (3.3) δ (1) = δ (1 , + ǫA δ (1 , + ... , (3.4)and ϕ (1) = ξ (1 , + ǫAξ (1 , + ǫ Aξ (1 , + ... , (3.5)where in the double superscript ( i, j ) the left index i refers to the order in standard pertur-bation theory (e.g. i = 1 denotes linear theory), while the right index j refers to the order ǫ j of the deviation of our modified gravity model w.r.t. ΛCDM. In particular, setting ¯ ϕ = 1, A = 1 and ξ (1 , = 0 we recover the ΛCDM model when ǫ →
0. In other words,all the equations previously obtained will be also iteratively perturbed by the parameter ǫ . This approach allows us to understand the relevance of the various terms that comefrom modified gravity in the description of the large scale structure of the Universe. Let usstress that this prescription is completely general and can be used to describe any viable f ( R ) theory on cosmological scales.In the next two subsections we will analyze in detail with this technique the backgroundand the cosmological perturbations. Specifically we will expand all the equations in section2 up to order ǫ . If we consider ¯ ϕ non-constant then the background could change completely. Indeed, as we will seefrom Eqs. (3.6) and (3.8), when ¯ ϕ = 1 and A = 1, the value of H does not depend on ϕ and, obviously, on ǫξ .Therefore the background is the same of the ΛCDM model for ǫξ ≪ – 5 – .1 FLRW background equations In this case, by construction, V ( ¯ ϕ ) = V is a constant (specifically, it is of the order of thecosmological constant), ¯ ϕ = 1, A = 1 and ρ (0)m is unperturbed with respect to ǫ . In thiscase from Eq. (2.8) we get H = 8 πa ρ (0)m + a V , (3.6) H ξ ′ + H ξ = a V , (3.7)where we have defined the expansion of V ( ϕ ) with respect to ǫ as V ( ϕ ) = V + ǫV ( ξ ( η )).Instead, from Eq. (2.9) we obtain (cid:0) H ′ + H (cid:1) = a V , (3.8) ξ ′′ + H ξ ′ + (cid:0) H ′ + H (cid:1) ξ = a V . (3.9)We note immediately that we retrieve the field equations of general relativity at the back-ground level. Now from Eqs. (3.7) and (3.9) we find the equation of motion for ξξ ′′ − H ξ ′ + 2 (cid:0) H ′ − H (cid:1) ξ = 0 . (3.10)Notice that this field equation describes the dynamics of ξ as a free field, without anysource term proportional to the matter fields.At this point, knowing that the Cauchy problem is well-formulated for metric f ( R )gravity in the presence of matter [94], the initial conditions for ξ and ξ ′ are crucial becausethey characterize the type of f ( R ) model that we are studying and the validity of theapproach that we are using . Specifically, assuming that for η < η rec , where η rec is someepoch when the Universe is matter dominated and radiation is negligible (usually at therecombination epoch), f ( R ) gravity should be the same as general relativity. Thus we canchoose at recombination ξ ( η rec ) = 0 and | ξ ( η rec ) ′ | ≪
1. In other words, we have to setboth ǫ and | ξ ( η rec ) ′ | “small enough” to allow only at late times a different solution of theperturbation evolution w.r.t. a ΛCDM model.Let us make another comment. For the stability at high curvature of perturbations ofthe scalar field the sign of ǫ and ξ ′ is crucial. Indeed, from Eq. (2.15) we must assume that( a / (cid:2) ϕ (d V / d ϕ ) − (d V / d ϕ ) (cid:3) > V d ϕ = 6 /a ǫξ ′ (cid:0) H ′′ − H (cid:1) = 8 π ρ (0)m ′ ǫξ ′ = − π ρ (0)m ǫ ( a d ξ/ d a ) , (3.11)for ǫ →
0, we have to impose that ǫ ( a d ξ/ d a ) <
0. Moreover, analyzing Eq. (3.9) we notethat if ξ ′ < >
0) then ξ < > ǫ > a d ξ/ d a ) < This translates to imposing the conditions on ∂f/∂R and ∂ f/∂R (or B defined by [77]) at a specifictime (for example, at recombination or today). In particular, let us look at Eq. (3) of Ref. [46] (see also[67]). In this paper the authors need simply to set two parameters, namely f R and n in order to study oneclass of viable f ( R ) models (see Section 6). – 6 –ow, in order to better understand the behaviour of ξ , let us rewrite Eq. (3.10) ina slightly different way. Indeed, using Eqs. (3.6) and (3.8), defining V = 16 πρ Λ and ν = ρ (0)0 m /ρ Λ = (Ω / Ω ), where ρ (0)m = ρ (0)0 m a − and a = 1, we obtaind ξ d a − ν/aν + a d ξ d a − ν/a ν + a ξ = 0 . (3.12) - - -Ξ Figure 1:
Illustrative plot of − ξ , as a function of a , with (Ω / Ω ) = 3 /
7. The lines, fromshort to long dashes, correspond to (d ξ/ d a ) ( a rec ) = − − , − − respectively; the black solid linecorresponds to (d ξ/ d a ) ( a rec ) = − − . In Figs. 1 and 2 we show the evolution of ( − ξ ) and ( − d ξ/ d a ) as a function of thescale-factor a . In particular we note that the value of | ξ | grows rapidly after recombinationand if the initial conditions are not sufficiently small, in future this effect could lead toan “explosive phenomenon” for these models [95]. It is obvious that in that regime ourperturbative technique cannot be used.Finally, it is useful to compare our formalism to the existing literature by consideringthe dimensionless quantity that quantifies the deviation from the ΛCDM reference modelintroduced in Ref. [77] (see also [80, 34]) B = m R ′ R H ( H ′ − H ) , (3.13)where [96, 34] m = R ( ∂ f /∂R )( ∂f /∂R ) = (d V / d ϕ ) ϕ (d V / d ϕ ) . (3.14)In our case it becomes B = H ( H ′ − H ) (cid:20) a d ln(1 + ǫξ )d a (cid:21) . (3.15) a the value of the scalar factor today. – 7 – .0 0.2 0.4 0.6 0.8 1.010 - - d Ξ (cid:144) da Figure 2:
Illustrative plot of ( − d ξ/ d a ), as a function of a , with (Ω / Ω ) = 3 /
7. The lines, fromshort to long dashes, correspond to (d ξ/ d a ) ( a rec ) = − − , − − respectively; the black solid linecorresponds to (d ξ/ d a ) ( a rec ) = − − . We see immediately that, imposing ξ, ξ ′ < ǫ >
0, we get the corresponding stabilitycondition
B >
In this section we analyse in detail the field perturbation equations through this iterativetechnique. From Eq. (2.12) we get6 H ψ (1 , ′ − (cid:18) ∇ ψ (1 , + 13 ∇ ∇ χ (1 , (cid:19) + 8 πa ρ (0)m δ (1 , = 0 , (3.16)6 H ψ (1 , ′ − (cid:18) ∇ ψ (1 , + 13 ∇ ∇ χ (1 , (cid:19) + 8 πa ρ (0)m δ (1 , = − ξ ′ ψ (1 , ′ + 3 H ξ (1 , ′ − ∇ ξ (1 , + a (cid:20) V − a (cid:0) H ′ + H (cid:1)(cid:21) ξ (1 , + 8 πa ρ (0)m (cid:16) ξ (1 , + ξδ (1 , (cid:17) . (3.17)From Eq. (2.13) we obtain2 ψ (1 , ′ + 13 ∇ χ (1 , ′ = 0 , (3.18)2 ψ (1 , ′ + 13 ∇ χ (1 , ′ = ξ (1 , ′ − H ξ (1 , ; (3.19)– 8 –hile from Eq. (2.14) χ (1 , ′′ + 2 H χ (1 , ′ + 2 ψ (1 , + 13 ∇ χ (1 , = 0 , (3.20) χ (1 , ′′ + 2 H χ (1 , ′ + 2 ψ (1 , + 13 ∇ χ (1 , = 2 ξ (1 , − ξ ′ χ (1 , ′ . (3.21)Moreover, from Eq. (2.15), we get8 πa ρ (0)m δ (1 , = 2 ξ ′ (cid:0) H ′′ − H (cid:1) ξ (1 , , (3.22) ξ (1 , ′′ + 2 H ξ (1 , ′ − ∇ ξ (1 , − ξ ′ ψ (1 , ′ + 2 ξξ ′ (cid:0) H ′′ − H (cid:1) ξ (1 , − (cid:0) H ′ + H (cid:1) ξ (1 , = 8 πa ρ (0)m δ (1 , ; (3.23)while, from Eq. (2.16), − ψ (1 , ′′ − H ψ (1 , ′ + 4 ∇ ψ (1 , + 23 ∇ ∇ χ (1 , = 6 ξ ′ (cid:0) H ′′ − H (cid:1) ξ (1 , , (3.24) − ψ (1 , ′′ − H ψ (1 , ′ + 4 ∇ ψ (1 , + 23 ∇ ∇ χ (1 , = 3 ξ ′ (cid:0) H ′′ − H (cid:1) ξ (1 , . (3.25)Let us note that in the LHS of Eqs. (3.22) and (3.24) there is a term proportional to ξ (1 , .This might appear as a mismatch of the 0-th order and of the first order in ǫ . This is notthe case, however, since, in these equations, ξ (1 , behaves as an auxiliary field which allowsto connect these two equations. Indeed, replacing the RHS of Eq. (3.22) in Eq. (3.24) onerecovers the trace of the gravitational field equations of ΛCDM.Finally, from Eq. (2.17), we find δ (1 , ′ = 3 ψ (1 , ′ , (3.26) δ (1 , ′ = 3 ψ (1 , ′ . (3.27)At this point, from Eqs. (3.16), (3.18), (3.20), (3.24) [substituting the RHS of Eq.(3.22) in the RHS of Eq. (3.24)] and (3.26), we obtain the perturbation equations in thesynchronous gauge in general relativity (see Appendix A and Refs. [92, 93]). Instead,from the other equations we are able to get their correction terms, i.e. δ (1 , , 2 ∇ ψ (1 , +(1 / ∇ ∇ χ (1 , . In particular, knowing that H ′′ = H ′ H + H , and after some tedious– 9 –alculations, we derive the following relations πa ρ (0)m δ (1 , = 13 (cid:20)(cid:18) H − H ′ + 2 H ′ H + 4 πa ρ (0)m (cid:19) (cid:18) − a d ξ d a (cid:19) + 2 (cid:18) H ′ − H − H ′ H + 4 πa ρ (0)m (cid:19) ξ (cid:21) δ (1 , + 13 (cid:20) (cid:18) H − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 4 (cid:18) −H + H ′ H (cid:19) ξ (cid:21) δ (1 , ′ − (cid:18) − a d ξ d a (cid:19) ∇ δ (1 , , (3.28)2 ∇ ψ (1 , + 13 ∇ ∇ χ (1 , = ( H πa ρ (0)m (cid:20)(cid:18) H − H ′ + 23 H ′ H − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 2 (cid:18) H ′ − H − H ′ H + 12 H ′ H (cid:19) ξ (cid:21) + (cid:18) H − H ′ + 2 H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) − (cid:18) H − H ′ + 2 H ′ H (cid:19) ξ + 4 πa ρ (0)m (cid:18) − a d ξ d a (cid:19)(cid:27) δ (1 , + ( H πa ρ (0)m (cid:20)(cid:18) H − H ′ + 6 H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 3 (cid:18) H ′ − H − H ′ H (cid:19) ξ (cid:21) + (cid:18) H − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 2 (cid:18) −H + 2 H ′ H (cid:19) ξ (cid:27) δ (1 , ′ + ( πa ρ (0)m (cid:20)(cid:0) − H + H ′ (cid:1) (cid:18) − a d ξ d a (cid:19) + 2 (cid:0) H − H ′ (cid:1) ξ (cid:21) − (cid:18) − a d ξ d a (cid:19)) ∇ δ (1 , − H πa ρ (0)m (cid:18) − a d ξ d a (cid:19) ∇ δ (1 , ′ . (3.29)Moreover, by linearizing the solution of the continuity equation (3.27) we obtain ψ (1 , ( η, x ) = ψ (1 , ( x ) + 13 (cid:16) δ (1 , ( η, x ) − δ (1 , ( x ) (cid:17) . (3.30) In order to illustrate better that the background is the same of the ΛCDM model for ǫ ξ ≪
1, let usconsider for example Eq. (3.28). In the RHS of this equation most of the coefficients are proportional to H or H ′ / H . In this case, they describe the same background as the ΛCDM model because the correctionsare of order ǫ . In other words, they are negligible at first order in ǫ . – 10 –e denote by a subscript 0 the condition at the present time of the referred quantity. Then13 ∇ ∇ χ (1 , = ( H πa ρ (0)m (cid:20)(cid:18) H − H ′ + 23 H ′ H − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 2 (cid:18) H ′ − H − H ′ H + 12 H ′ H (cid:19) ξ (cid:21) + (cid:18) H − H ′ + 2 H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) − (cid:18) H − H ′ + 2 H ′ H (cid:19) ξ + 4 πa ρ (0)m (cid:18) − a d ξ d a (cid:19)(cid:27) δ (1 , + ( H πa ρ (0)m (cid:20)(cid:18) H − H ′ + 6 H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 3 (cid:18) H ′ − H − H ′ H (cid:19) ξ (cid:21) + (cid:18) H − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 2 (cid:18) −H + 2 H ′ H (cid:19) ξ (cid:27) δ (1 , ′ + ( πa ρ (0)m (cid:20) (cid:18) − H + 6 H ′ − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 4 (cid:18) H − H ′ + H ′ H (cid:19) ξ (cid:21) − (cid:18) − a d ξ d a (cid:19) − ξ (cid:27) ∇ δ (1 , + 112 πa ρ (0)m (cid:20)(cid:18) − H + 2 H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 4 (cid:18) H − H ′ H (cid:19) ξ (cid:21) ∇ δ (1 , ′ + 112 πa ρ (0)m (cid:18) − a d ξ d a (cid:19) ∇ ∇ δ (1 , − ∇ (cid:18) ψ (1 , − δ (1 , (cid:19) . (3.31)At this point, one can remove the residual gauge ambiguity of the synchronous coordinatesby imposing that δ (1 , = − (1 / ∇ χ (1 , . Therefore, evaluating Eq. (3.31) to presenttime, we can determine immediately the value of ψ (1 , .Eqs. (3.28), (3.29) and (3.31), which govern the lowest-order modified gravity correc-tions w.r.t. ΛCDM in the behavior of scalar perturbations, represent the main result of thispaper. Let us stress two important facts: 1) all our expressions are completely determinedby using the well-known results of the ΛCDM model (see Appendix A) and the dynamicalsolution of ξ ; 2) through this method it is possible to obtain analytically a more preciseresult if we consider the next orders in ǫ .In the next section we will analyze in detail the evolution of perturbations in thePoisson gauge.
4. From the synchronous to the Poisson gauge
In this section we obtain the evolution of cosmological perturbations in the Poisson gauge (also known as the conformal Newtonian gauge or the longitudinal gauge) through a gaugetransformation of the results obtained in the synchronous gauge in the previous section. Inparticular we will follow the approach used in Ref. [93] (e.g. see also [92]). Setting linear In addition, we can conclude that δ (1)0 = − (1 / ∇ χ (1)0 , see Appendix A. In the Poisson gauge one scalar degree of freedom is eliminated from the g i component of the metric,and one scalar and two vector degrees of freedom are eliminated from g ij . – 11 –ector and tensor modes to zero, the flat linear metric becomes ds = a [ − (1 + 2Φ (1)p ) dη +(1 − (1)p ) δ ij dx i dx j ]. Instead, perturbing the mass-density and fluid four-velocity we get ρ m = ρ (0)m (1 + δ (1)p ) and u µ = ( δ µ + v (1) µ p ) /a , where v (1) 0 = − Φ (1)p and v (1) i p = ∂ i v (1)p .Considering the perturbations at the same space-time coordinate values, the syn-chronous gauge and the conformal Newtonian gauge can be related by the following rela-tions [93, 92] − (1)p = χ (1) ′′ + H χ (1) ′ , (4.1)2Ψ (1)p = 2 ψ (1) + 13 ∇ χ (1) + H χ (1) ′ , (4.2) δ (1)p = δ (1) + 32 H χ (1) ′ , (4.3) v (1)p = 12 χ (1) ′ . (4.4)At this point, as we have already done in section 3, we can split Φ (1)p , Ψ (1)p , δ (1)p and v (1)p in the following way Φ (1)p = Φ (1 , + ǫ Φ (1 , + ... , (4.5)Ψ (1)p = Ψ (1 , + ǫ Ψ (1 , + ... , (4.6) δ (1)p = δ (1 , + ǫ δ (1 , + ... , (4.7) v (1)p = v (1 , + ǫ v (1 , + ... . (4.8)Also in this case the terms Φ (1 , , Ψ (1 , , δ (1 , and v (1 , are the perturbation termsthat one obtains in a ΛCDM model. Now, we want to study and determine the terms toorder ǫ . Immediately we note that using Eqs. (3.19), and (3.27) we can obtain v (1 , and,consequently, δ (1 , . Indeed v (1 , = 18 πa ρ (0)m (cid:26)(cid:20)(cid:18) − H + 35 H ′ H − H ′ H + 6 H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) − (cid:18) H ′ H − H − H ′ H + 6 H ′ H (cid:19) ξ − πa ρ (0)m (cid:18) H − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 16 πa ρ (0)m (cid:18) H − H ′ H (cid:19) ξ (cid:21) ∇ − δ (1 , − (cid:20)(cid:18) H − H ′ + 6 H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 6 (cid:18) H ′ − H − H ′ H (cid:19) ξ + 8 πa ρ (0)m ξ (cid:21) ∇ − δ (1 , ′ + (cid:20)(cid:18) H − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 2 (cid:18) −H + H ′ H (cid:19) ξ (cid:21) δ (1 , + (cid:18) − a d ξ d a (cid:19) δ (1 , ′ (cid:27) ; (4.9)where ∇ − stands for the inverse of the Laplacian operator; while from Eqs. (4.3) and– 12 –3.28) we get δ (1 , = 14 πa ρ (0)m (cid:20)(cid:18) H − H ′ + H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + (cid:18) H ′ − H − H ′ H (cid:19) ξ + 2 πa ρ (0)m (cid:18) − a d ξ d a (cid:19) + 4 πa ρ (0)m ξ (cid:21) δ (1 , + 18 πa ρ (0)m (cid:20)(cid:18) H − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 4 (cid:18) −H + H ′ H (cid:19) ξ (cid:21) δ (1 , ′ − πa ρ (0)m (cid:18) − a d ξ d a (cid:19) ∇ δ (1 , + 3 H πa ρ (0)m (cid:20)(cid:18) − H + 35 H ′ − H ′ H + 6 H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) − (cid:18) H ′ H − H − H ′ H + 6 H ′ H (cid:19) ξ − πa ρ (0)m (cid:18) H − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 16 πa ρ (0)m (cid:18) H − H ′ H (cid:19) ξ (cid:21) ∇ − δ (1 , − H πa ρ (0)m (cid:20)(cid:18) H − H ′ + 6 H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 3 (cid:18) H ′ − H − H ′ H (cid:19) ξ + 8 πa ρ (0)m ξ (cid:21) ∇ − δ (1 , ′ . (4.10)Instead from Eqs. (4.1), (4.4), (4.10), (3.21) and (3.29) we findΦ (1 , = (cid:20)(cid:18) H − H ′ + H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) − (cid:18) H − H ′ + 2 H ′ H (cid:19) ξ + 2 πa ρ (0)m (cid:18) − a d ξ d a (cid:19)(cid:21) ∇ − δ (1 , + 12 (cid:20)(cid:18) H − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 4 (cid:18) −H + H ′ H (cid:19) ξ (cid:21) ∇ − δ (1 , ′ − (cid:18) − a d ξ d a (cid:19) δ (1 , . (4.11)Moreover from Eqs. (4.2), (4.4), (4.10) and (3.29) we obtainΨ (1 , = (cid:20)(cid:18) H − H ′ + H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) − (cid:18) H − H ′ + 2 H ′ H (cid:19) ξ + 2 πa ρ (0)m (cid:18) − a d ξ d a (cid:19)(cid:21) ∇ − δ (1 , + 12 (cid:20)(cid:18) H − H ′ H (cid:19) (cid:18) − a d ξ d a (cid:19) + 4 (cid:18) −H + H ′ H (cid:19) ξ (cid:21) ∇ − δ (1 , ′ − (cid:18) − a d ξ d a (cid:19) δ (1 , . (4.12)Finally, another useful quantity is the anisotropic contribution Π (1) which is one ofthe parameters that allow to quantify the departure of f ( R ) gravity from the standardΛCDM model [79]. Indeed in our formalism Π (1) = Ψ (1)p − Φ (1)p = ǫ (Ψ (1 , − Φ (1 , ) becauseΨ (1 , = Φ (1 , (see appendix B). ThenΠ (1) /ǫ = 2 (cid:18) − a d ξ d a (cid:19) (cid:18) δ (1 , − H∇ − δ (1 , ′ (cid:19) . (4.13)
5. Comparison with the “quasi-static” approximation
As it is well known, when we consider scales deep inside the Hubble radius, in order toderive the equation of matter perturbations approximately, one uses the quasi-static ap-– 13 –roximation (for details, for example, see [34]). This section is devoted to recover this ap-proximation, as an important example of how our formalism confront with and recover someknown results in the literature. In particular we want to calculate the Poisson equationfor these models with the approach studied in this work. Defining Φ (1)eff = − (Φ (1)p + Ψ (1)p ) / (1)eff = − (cid:18) ψ (1) + 13 ∇ χ (1) − ψ (1) ′′ (cid:19) . (5.1)Then, knowing that Φ (1)eff = Φ (1 , + ǫ Φ (1 , and using Eqs. (3.21), (3.17) and (3.19), weobtain ∇ Φ (1 , = − h H ′ − H ) ξ (1 , + 8 πa ρ (0)m (cid:16) δ (1 , − ξδ (1 , (cid:17)i . (5.2)At this point, let us consider in detail Eq. (3.23). In that scales we can drop the terms withtemporal derivatives when we compare them with spatial gradients term in ξ (1 , . Thenwe find (cid:20) ξξ ′ (cid:0) H ′′ − H (cid:1) − (cid:0) H ′ + H (cid:1) − ∇ (cid:21) ξ (1 , = 13 8 πa ρ (0)m δ (1 , − ξ ′ δ (1 , ′ . (5.3)As we see from Figs. 1 and 2 , [ ξ/ (d ξ/ d a )] is less of 1 /
10. Moreover, considering scaleswhere the square of wavenumber k is larger than H ′ and H , we conclude that −∇ ξ (1 , ≃ [(8 πa / ρ (0)m δ (1 , − ξ ′ δ (1 , ′ ]. Consequently, in this case, the additive term proportionalto ξ (1 , becomes negligible with respect to the other terms in Eq. (5.2). Then ∇ Φ (1 , ≃ − πa ρ (0)m (cid:16) δ (1 , − ξδ (1 , (cid:17) . (5.4)On the other hand, starting from the literature (for example, see Refs. [77, 34]) andassuming the quasi-static approximation (i.e. when ∇ | X | ≫ H | X | and | X ′ | < H| X | ,where X = Φ (1)p , Ψ (1)p , ϕ, ϕ ′ , φ (1) , φ (1) ′ ), we find ∇ Φ (1)eff = − πa ρ (0)m δ (1) /ϕ . Then we canquickly obtain the same result. Indeed ∇ Φ (1)eff ≃ − πa ρ (0)m h δ (1 , + ǫ (cid:16) δ (1 , − ξδ (1 , (cid:17)i . (5.5)However, let us stress that our formalism is designed in order to easily account for effectsthat go beyond the quasi-static approximation, in particular those taking place on scalescomparable to the horizon size, where time derivatives cannot be neglected. Indeed theycould be important, for example, for accurate calculation of the Integrated Sachs-Wolfe(ISW) effect and the large-scale matter power spectrum (for example, see [103]). In general, for these models, one also adds the approximation that corrisponds to | ϕ (d V / d ϕ ) | ≫| (d V / d ϕ ) | . Let us stress that this condition is automatically satisfied through our approach (see section3.1). – 14 – . Matter Power Spectrum In this section we compute the matter power spectrum P ( k, a ): h δ (1) ( k , a ) δ (1) ( k ′ , a ) i = (2 π ) δ D ( k + k ′ ) P ( k, a ) , (6.1)where k = | k | . Within our approach, it becomes P ( k, a ) = P (0) ( k, a ) + ǫP (1) ( k, a ) (6.2)where P (0) ( k, a ) is the matter power spectrum in the ΛCDM model. In terms of theprimordial power spectrum and the transfer function, we have P (0) ( k, a ) = P prim ( k ) T ( k ) (cid:18) D ( a ) D (0) (cid:19) , (6.3)where D is the growing mode of δ (1 , , see Appendix A. From Eq. (3.28) we get P (1) ( k ) = 2 A ( k, a ) P (0) ( k, a ) (6.4)with3Ω m ( a ) A ( k, a ) = (cid:20) −
152 Ω m ( a ) + 92 Ω Λ ( a ) (cid:21) + 2 (cid:20) Ω m ( a ) −
12 Ω Λ ( a ) (cid:21) (cid:18) − a d ξ d a (cid:19) + 2 ( − m ( a ) −
52 Ω Λ ( a ) − (cid:20) Ω m ( a ) −
12 Ω Λ ( a ) (cid:21) ) ξ ( a )+ f ( a ) (cid:26) (cid:20) − Ω m ( a ) + 12 Ω Λ ( a ) (cid:21) (cid:18) − a d ξ d a (cid:19) − (cid:20) − Ω m ( a ) + 12 Ω Λ ( a ) (cid:21) ξ ( a ) (cid:27) + (cid:18) − a d ξ d a (cid:19) (cid:18) k H (cid:19) , (6.5)where f is the growth rate, Ω m ( a ) is the density parameter of non-relativistic matterand Ω Λ ( a ) is the density parameter of the cosmological constant in a ΛCDM model, seeAppendixes A and B.Then, defining the initial solutions as in Section 3.1, we can finally derive P ( k, a = 1).For simplicity, we fix ξ ( a rec ) = 0 and (d ξ/ d a ) ( a rec ) = − . − in order to get ξ ( a = 1) ≃− − and (d ξ/ d a ) ( a = 1) ≃ − . · − .In Fig 3 we show P ( k, a = 1) and (cid:18) ∆ PP (cid:19) ( k, a = 1) = [ P ( k, a = 1) − P (0) ( k, a = 1)] P (0) ( k, a = 1) = 2 ǫ A ( k, a = 1) , (6.6)which measures the relative deviation from the matter power spectrum in a ΛCDM model.At this point, let us stress that, knowing ∂f∂R = 1 + ǫξ (6.7)– 15 – .001 0.002 0.005 0.010 0.020 0.050 0.100 0.200100010000500020002000030001500150007000 k h H Mpc - L P - H Mpc - L D P (cid:144) P Figure 3:
Illustrative plot of P and (∆ P/P ), as a function of k and for a = 1. The lines, from shortto long dashes, correspond to ǫ = 10 − , − , − respectively; the black solid line corresponds to ǫ = 10 − . and defining Eq. (3.15) today, i.e. B = 1(1 − Ω + Ω / (cid:18) − ǫ d ξ d a (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) a =1 , (6.8)we can compare a generic model in the literature characterized by B and f R = ( ∂f /∂R ) with our approach (see the discussion in Section 3.1). In particular we make a comparisonof our results with the Hu-Sawicki model, see for example [67]. In this case, using theirdefinitions, we get R + ˜ f ( R ) = − − ˜ f R n R n +10 R n , (6.9)where ˜ f ( R ) = f ( R ) − R and ˜ f R = ( ∂ ˜ f /∂R ) in the notation of [67]. Here we have used anapproximate definition of their Lagrangian, i.e. when this model describes a backgroundsimilar to ΛCDM. Then ǫξ ( a = 1) = ˜ f R and findd ξ d a (cid:12)(cid:12)(cid:12)(cid:12) a =1 = ( n + 1) (1 − Ω + Ω / − Ω / ξ | a =1 . (6.10)Using the initial solutions as in Section 3.1 we see that we are considering a Hu-Sawickimodel with n = 0 . | ˜ f R | < − . Obviously, in order to get a more precise result we have to add ǫ -ordercontributions.
7. Conclusions
In this paper we have introduced a novel method which allows to describe the backgroundevolution as well as cosmological perturbations in f ( R ) modified gravity models, under themain assumption that the background evolution is close to ΛCDM. Here we restricted ouranalysis to linear perturbations, although the method is completely general and can beextended to any perturbative order. – 16 –et us conclude by adding some comments. As we stressed before, this approach iscompletely different with respect to previous ones. Indeed, through this analytic techniquewe can determine quantitatively the deviation in the behavior of cosmological perturbationsbetween a given f ( R ) model and ΛCDM reference model. Moreover, our treatment isgeneral in that all the results depend only on the initial conditions that characterize the typeof f ( R ) model we are studying. Specifically all our expressions are completely determinedby using the well-known results of the ΛCDM model (see Appendix A) and the dynamicalsolution of ξ .Our method allows to study structure formation in these models from the largest scales,of the order of the Hubble horizon, down to scales deeply inside the Hubble radius, withoutemploying the so-called “quasi-static” approximation [97] (see also the review [34] and refs.therein). This can be very useful as: 1) one can use this approach including all horizon-scale corrections to correctly interpret data on very large scales [98, 99, 100, 101, 102]; 2) atthe non-linear level, it can be used to get the expressions for the effect of primordial non-Gaussianity on the matter density perturbation in an f ( R ) cosmology, fully accounting forthe corrections arising on scales comparable with the Hubble radius [104] (e.g. in ΛCDMcosmology see [105, 106, 107, 108, 109, 110]).We believe that this technique can be extended to other classes of modified theoriesof gravity to providing information about the evolution of large-scale structures in theuniverse. Acknowledgments
DB would like to acknowledge the ICG at the University of Portsmouth for hospitalityduring the development of this project. DB research has been partly supported by ASIcontract I/016/07/0 “COFIS”. Support was given by the Italian Space Agency throughthe ASI contracts Euclid-IC (I/031/10/0). We thank V. Acquaviva, D. Bacon, B. Hu,K. Koyama, L. Pogosian, A. Raccanelli, F. Schmidt, G. Zhao for helpful discussions.
A. Evolution of first-order perturbations of Λ CDM model in the syn-chronous gauge
In this section let us consider briefly the evolution, at the linear level, of relativistic per-turbations of a ΛCDM model in the synchronous gauge. Starting from the line-elementdefined in section 2, in a perfectly homogeneous and isotropic universe, the momentum– 17 –onstraint, the continuity equation and the energy constraint, respectively, give ψ (1) + 16 ∇ χ (1) = ψ (1)0 + 16 ∇ χ (1)0 = const . , (A.1) δ (1) = δ (1)0 − ∇ (cid:16) χ (1) − χ (1)0 (cid:17) , (A.2) ∇ (cid:20) H χ (1) ′ + 4 πa ρ (0)m (cid:16) χ (1) − χ (1)0 (cid:17) + 2 ψ (1)0 + 13 ∇ χ (1)0 (cid:21) = 8 πa ρ (0)m δ (1)0 . (A.3)The evolution equation becomes χ (1) ′′ + 2 H χ (1) ′ + 13 ∇ χ (1) = − ψ (1) . (A.4)An equation only for the scalar mode χ (1) can be obtained by combining together theevolution equation and the energy constraint, ∇ h χ (1) ′′ + H χ (1) ′ − πa ρ (0)m (cid:16) χ (1) − χ (1)0 (cid:17)i = − πa ρ (0)m δ (1)0 , (A.5)and, consequently, we can get the equation for the linear density fluctuation δ (1) ′′ + H δ (1) ′ − πa ρ (0)m δ (1) = 0 . (A.6)The equations above have been obtained in whole generality; one could have used in-stead the well-known residual gauge ambiguity of the synchronous coordinates (see, e.g.,Refs.[111, 112, 93]) to simplify their form. For instance, one could fix χ (1)0 so that ∇ χ (1)0 = − δ (1)0 , and thus the χ (1) evolution equation takes the same form as that for δ (1) , i.e. δ (1) = − (1 / ∇ χ (1) . Now, this gauge-condition replaced in Eqs. (A.4) and (A.5) yields H χ (1) ′ + 4 πa ρ (0)m χ (1) + 13 ∇ χ (1) = − ψ (1) (A.7)Moreover, from the momentum constraint we get ψ (1) = ψ (1)0 − (1 / ∇ ( χ (1) − χ (1)0 ). Finallywith such a gauge fixing one obtains δ (1) ∝ D ± , where D ± represent the the growing (+)and decaying ( − ) solution of the equation D ± ′′ + HD ± ′ − πa ρ (0)m D ± = 0 . (A.8)In what follows, we shall restrict ourselves to the growing mode. Then χ (1) = D + ( η ) χ (1)0 ; and ψ (1) = ψ (1)0 −
16 ( D + ( η ) − ∇ χ (1)0 , (A.9)where we have defined D + ( η ) = a = 1. Replacing (A.7) in (A.9) we find H D + (cid:20) f (Ω m ) + 32 Ω m (cid:21) χ (1)0 = − (cid:18) ψ (1)0 + 16 ∇ χ (1)0 (cid:19) = const . (A.10) For simplicity, in these appendices, we have substituted the double superscript (1 ,
0) with (1). – 18 –here Ω m = 8 πa ρ (0)m / (3 H ) and f (Ω m ) = d ln D + / d ln a . Then ψ (1) = − H (cid:20) f (Ω ) + 32 Ω (cid:21) χ (1)0 − D + ( η ) ∇ χ (1)0 . (A.11)It may be convenient to define the gravitational potential today Φ (see Appendix B)through the relation ∇ Φ = 4 πa ρ (0)0 m δ (1)0 = − πa ρ (0)0 m ∇ χ (1)0 . Then χ (1)0 = − / (3 H Ω ).Finally we obtain ψ (1) = (cid:20) f (Ω )Ω (cid:21) Φ + 29 ∇ Φ H Ω D + . (A.12) B. Evolution of first-order perturbations of Λ CDM model in the Poissongauge
The goal of this Appendix is to briefly recall the results for the linear perturbations in thecase of a non-vanishing cosmological Λ term in the Poisson gauge [116, 117].Starting from the line-element defined in section 4, we note immediately that, at linearorder, the traceless part of the ( i - j )-components of Einstein’s equations gives Φ (1) = Ψ (1) ≡ Φ. Its trace gives the evolution equation for the linear scalar potential ΦΦ ′′ + 3 H Φ ′ + (2 H ′ + H )Φ = 0 . (B.1)Selecting only the growing mode solution one can writeΦ( x , η ) = g ( η ) Φ ( x ) , (B.2)where Φ is the peculiar gravitational potential, linearly extrapolated to the present time η and g ( η ) = D + ( η ) /a ( η ) is the so called growth-suppression factor. The exact form of g can be found in Refs. [113, 114, 115]. In the Λ = 0 case, g = 1. A very good approximationfor g as a function of redshift z is given in Refs. [113, 114] g ∝ Ω m h Ω / − Ω Λ + (1 + Ω m /
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