Evolution of density perturbations in fðrþ theories of gravity

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1 PHYSICAL REVIEW D 77, 2355 (28) Evolution of density perturbations in fðrþ theories of gravity A. de la Cruz-Dombriz,* A. Dobado, + and A. L. Maroto Departamento de Física Teórica I, Universidad Complutense de Madrid, E-284 Madrid, Spain (Received 29 February 28; published 2 June 28) In the context of fðrþ theories of gravity, we study the evolution of scalar cosmological perturbations in the metric formalism. Using a completely general procedure, we find the exact fourth-order differential equation for the matter density perturbations in the longitudinal gauge. In the case of sub-hubble modes, the expression reduces to a second-order equation which is compared with the standard (quasistatic) equation used in the literature. We show that for general fðrþ functions the quasistatic approximation is not justified. However, for those functions adequately describing the present phase of accelerated expansion and satisfying local gravity tests, it provides a correct description for the evolution of perturbations. DOI:.3/PhysRevD PACS numbers: k, 4.5.Kd, x *dombriz@fis.ucm.es + dobado@fis.ucm.es maroto@fis.ucm.es I. INTRODUCTION The present phase of accelerated expansion of the universe [] poses one of the most important problems of modern cosmology. It is well-known that ordinary Einstein s equations in either a matter or radiation dominated universe give rise to decelerated periods of expansion. In order to have acceleration, the total energymomentum tensor appearing on the right-hand side of the equations should be dominated at late times by a hypothetical negative pressure fluid usually called dark energy (see [2] and references therein). However, there are other possibilities to generate a period of acceleration in which no new sources are included on the right-hand side of the equations, but instead Einstein s gravity itself is modified [3]. In one of such possibilities, new functions of the curvature scalar [fðrþ terms] are included in the gravitational action, which amounts to modifiying the left-hand side of the equations of motion. Although such theories are able to describe the accelerated expansion on cosmological scales correctly, they typically give rise to strong effects on smaller scales. In any case viable models can be constructed to be compatible with local gravity tests and other cosmological constraints [4]. The important question that arises is therefore how to discriminate dark energy models from modified gravities using present or future observations. It is known that by choosing particular fðrþ functions, one can mimic any background evolution (expansion history), and, in particular, that of CDM. Accordingly, the exclusive use of observations such as high-redshift Hubble diagrams from supernovae type Ia [], baryon acoustic oscillations [5] or CMB shift factor [6], based on different distance measurements which are sensitive only to the expansion history, cannot settle the question of the nature of dark energy [7]. However, there exist observations of a different type which are sensitive, not only to the expansion history, but also to the evolution of matter density perturbations. The fact that the evolution of perturbations depends on the specific gravity model, i.e., it differs in general from that of Einstein s gravity even though the background evolution is the same, means that observations of this kind will help distinguish between different models for acceleration. In this work we study the problem of determining the exact equation for the evolution of matter density perturbations for arbitrary fðrþ theories. Such a problem had been previously considered in the literature ([8 3]) and approximated equations have been widely used. They are typically based on the so-called quasistatic approximation in which all the time derivative terms for the gravitational potentials are discarded, and only those including density perturbations are kept [4]. From our exact result, we will be able to determine under which conditions such an approximation can be justified. The paper is organized as follows: In Sec. II, we briefly review the perturbations equations for the standard CDM model. In Sec. III we obtain the perturbed equations for general fðrþ theories. In Sec. IV we describe the procedure to obtain the general equation for the density perturbation. In Sec. V we summarize the main viability condition for fðrþ theories. Section VI is devoted to the study of the validity of the quasistatic approximation. In Sec. VII we apply our results to some particular models, and finally in Sec. VIII we include the main conclusions. In Appendixes A and B we have also included complete expressions for the relevant coefficients of the perturbation equation. II. DENSITY PERTURBATIONS IN CDM Let us start by considering the simplest model for dark energy described by a cosmological constant. The cor =28=77(2)=2355(9) Ó 28 The American Physical Society

2 A. DE LA CRUZ-DOMBRIZ, A. DOBADO, AND A. L. MAROTO PHYSICAL REVIEW D 77, 2355 (28) responding Einstein s equations read G ¼ 8GT ; () where G is the Einstein s tensor and T is the energymomentum tensor for matter. In the metric formalism for the CDM model it is possible to obtain a second-order differential equation for the growth of matter density perturbation =. Let us consider the scalar perturbations of a flat Friedmann- Robertson-Walker metric in the longitudinal gauge and in conformal time: ds 2 ¼ a 2 ðþ½ð þ 2Þd 2 ð 2 Þðdr 2 þ r 2 d 2 2 ÞŠ; (2) where ð; ~xþ and ð; ~xþ are the scalar perturbations. From this metric, we obtain the first-order perturbed Einstein s equation: G ¼ 8GT ; (3) where the perturbed energy-momentum tensor reads T ¼ ¼ ; T i j ¼ Pi j ¼ c2 s i j T i ¼ Ti ¼ ðþc2 i v; (4) with the unperturbed energy density and v the potential for velocity perturbations. We assume that the perturbed and unperturbed matter have the same equation of state, i.e., P= c 2 s P =, where c s ¼ for matter perturbations. The resulting differential equation for in Fourier space is written as þ H k4 6~k 2 8~ 2 k 4 ~ð3k 2 þ 9H 2 Þ ~ k4 þ 9~ð2~ 3H 2 Þ k 2 ð9~ 3H 2 Þ k 4 ~ð3k 2 þ 9H 2 ¼ ; (5) Þ where ~ 4G a 2 ¼ H þ H 2 and H a =a with prime denoting derivative with respect to time. We point out that it is not necessary to explicitly calculate potentials and to obtain Eq. (5), but algebraic manipulations in the field equations are enough to get this result. In the extreme sub-hubble limit, i.e., k or equivalently k H,(5) is reduced to the well-known expression þ H 4G a 2 ¼ : (6) In this regime and at early times, the matter energy density dominates over the cosmological constant, and it is easy to show that solutions for (6) grow as aðþ. At late times (near today) the cosmological constant contribution is not negligible and power-law solutions for (6) no longer exist. It is necessary in this case to assume an ansatz for. One which works very well is the one proposed in [7,5]: ðaþ a ¼ e R a a i ½ m ðaþ Šdlna : (7) This expression fits with high precision the numerical solution for with a constant parameter ¼ 6=. III. PERTURBATIONS IN fðrþ THEORIES Let us consider the modified gravitational action S ¼ Z d 4 p x ffiffiffiffiffiffiffi g ðr þ fðrþþ; (8) 6G where R is the scalar curvature. The corresponding equations of motion read G 2 g fðrþþr f R ðrþ g hf R ðrþþf R ðrþ ; ¼ 8GT ; (9) where f R ðrþ ¼dfðRÞ=dR. For the background flat Robertson-Walker metric they read and 3H a 2 ð þ f R Þ 2 ðr þ f Þ 3H a 2 f R ¼ 8G a 2ðH þ 2H 2 Þð þ f R Þ 2 ðr þ f Þ a 2 ðh f R þ f R Þ () ¼ 8Gc 2 s ; () where R denotes the scalar curvature corresponding to the unperturbed metric, f fðr Þ, f R dfðr Þ=dR, and prime means derivative with respect to time. A very useful equation to use in the following calculations is the () and () combination 2ð þ f R Þð H þ H 2 Þþ2H f R f R ¼ 8G ð þ c 2 sþa 2 : (2) Finally we have the conservation equation þ 3ð þ c2 sþh ¼ : (3) Using the perturbed metric (2) and the perturbed energymomentum tensor (4), the first-order perturbed equations, assuming that the background equations hold, may be written as ð þ f R ÞG þðr þr r hþf RR R þ½ðg Þr r ðg Þr r Šf R ½g ð Þ g ð ÞŠ@ f R ¼ 8GT ; (4) The Riemann tensor definition is þ, which has an opposite sign to the one proposed in [6]

3 EVOLUTION OF DENSITY PERTURBATIONS IN fðrþ... PHYSICAL REVIEW D 77, 2355 (28) where f RR ¼ d 2 fðr Þ=dR 2, h r r, and r is the usual covariant derivative with respect to the unperturbed Friedmann-Robertson-Walker metric (see [6] for perturbed metric, connection symbols, and other useful perturbed quantities). Notice that unlike the ordinary Einstein- Hilbert (EH) case, with second-order equations, this is a set of fourth-order differential equations. By computing the covariant derivative with respect to the perturbed metric r ~ of the perturbed energy-momentum tensor ~T, we find the conservation equations ~r ~T ¼ (5) which do not depend on fðrþ. For the linearized Einstein s equations, the components (), ðiiþ, ðiþ ðiþ, and (ij), where i, j ¼, 2,3,i Þ j, in Fourier space, read, respectively, ð þ f R Þ½ k 2 ð þ Þ 3Hð þ Þ þð3h 6H 2 Þ 3H ŠþfR ð 9H þ 3H 3 Þ ¼ 2~; (6) ð þ f R Þ½ þ þ 3H ð þ Þþ3H þðh þ 2H 2 Þ ŠþfR ð3h H þ 3 Þ þ fr ð3 Þ ¼ 2c 2 s ~; (7) ð þ f R Þ½ þ þ H ð þ ÞŠ þ fr ð2 Þ ¼ 2~ð þ c 2 SÞv; (8) þ H þ k 2 3 3H ¼ ; (23) which will be very useful in future calculations. IV. EVOLUTION OF DENSITY PERTURBATIONS Our porpose is to derive a fourth-order differential equation for matter density perturbation alone. This can be performed by means of the following process. Let us consider Eqs. (6) and (8) for a matter dominated universe i.e., c 2 s ¼, and combine them to express the potentials and in terms of f ; ;; g by means of algebraic manipulations. The resulting expressions are the following: ¼ ½3ð þ f R ÞH ð þ ÞþfR þ 2~Š and ¼ where DðH ;kþ ð þ f R ÞðH fr Þþ ð þ f R Þð þ Þþ 2~ k 2 ð 3 Þ ½ð þ f R Þð k 2 3H Þþ3f R H Š ½ 3ð þ f R ÞH ð þ Þ 3f R DðH ;kþ 2~Š½ð þ f R ÞH þ 2fR ð Š þ f R Þð þ Þþ 2~ k 2 ð 3 Þ ½ð þ f R Þð k 2 þ 3H (24) 6H 2 Þ 9H f R Š ; (25) ¼ f RR R; þ f R (9) where R is given by R ¼ 2 a 2 ½3 þ 6ðH þ H 2 Þ þ 3H ð þ 3 Þ k 2 ð 2 ÞŠ: (2) Finally, from the energy-momentum tensor conservation (5), we get to first order: and 3 ð þ c 2 sþ þ k 2 ð þ c 2 sþv ¼ (2) c2 s þ þ c 2 þ v þ H vð 3c 2 sþ¼ (22) s for the temporal and spatial components, respectively. In a dust matter dominated universe, i.e., c 2 s ¼, (2) and (22) can be combined to give DðH ;kþ 6ð þ f R Þ 2 H 3 þ 3H ½f 2 R þ 2ð þ f R Þ 2 H Šþ3ð þ f R Þf R ð 2H 2 þ k 2 þ H Þ: (26) The second step will be to derive Eqs. (24) and (25) with respect to and obtain and algebraically in terms of f ; ; ; ; g. These last results can be substituted in Eqs. (6) and (8) to obtain potentials and just in terms of f ; ;; ; g. So at this stage we are able to express, but we do not do here explicitly, the following: ¼ ð ; ; ; ; Þ; ¼ ð ; ; ; ; Þ; ¼ ð ; ; ; ; Þ; ¼ ð ; ; ; ; Þ; (27) where we mean that the functions on the left-hand side are algebraically dependent on the functions inside the parenthesis on the right-hand side

4 A. DE LA CRUZ-DOMBRIZ, A. DOBADO, AND A. L. MAROTO PHYSICAL REVIEW D 77, 2355 (28) The natural reasoning at this point would be to try to It is very useful to define the parameter H =k since obtain the potentials second derivatives f ; g in terms it will allow us to perform a perturbative expansion of of f; ; g by an algebraic process. The chosen equations to do so will be (9) and (23), the first derivative with Other dimensionless parameters which will be used are the previous and coefficients in the sub-hubble limit. respect to. In(23) it is necessary to substitute and the following: i H ðiþ =H iþ (i ¼, 2, 3) and f i by the expressions obtained in (27), whereas in (9) the f ðjþ R first derivative may be sketched as follows: =ðh j f R Þ (j ¼, 2). Expressing the and coefficients with those dimensionless quantities we may write ¼ f RR R frr fr þ f RR ð þ f RÞ þ f R ð þ f R Þ 2 R: i;eh ¼ X3 ðjþ i;eh i ¼ ; ; 2; (28) j¼ Before deriving, we are going to substitute the that appears in (9) by lower derivatives potentials f; ; ; g,, and its derivatives. To do so we consider in (6) and (8) the first derivatives with respect to where the quantity v has been previously substituted by its expression in (2). Following this process we may express as follows: ¼ ð; ; ; ; ; ; Þ; (29) and now substituting in (9) we can derive that equation with respect to. Solving a two algebraic equations system with Eqs. (23) and (28) and introducing (27) we are able to express f ; g in terms of f; ; ; g: ¼ ð; ; ; Þ; ¼ ð; ; ; Þ: (3) We substitute the results obtained in (3) straightforwardly in (27) in order to express f; ; ; g in terms of f; ; ; g. With the two potentials and their first derivatives as algebraic functions of f; ; ; g,we perform the last step: we consider ð; ; ; Þ and derive it with respect to. The result should be equal to ð; ; ; Þ, so we only need to express together these two results obtaining a fourth-order differential equation for. Note that this procedure is completely general to first order for scalar perturbations in the metric formalism for fðrþ gravities. Once this fourth-order differential equation has been solved, we may go backward, and by using the results for we obtain f ; g from (3) as functions of time. Analogously from (27) the behavior of the potentials f; g and their first derivatives could be determined. The resulting equation for can be written as follows: 4;f iv þ 3;f þð 2;EH þ 2;f Þ þð ;EH þ ;f Þ þð ;EH þ ;f Þ ¼ ; (3) where the coefficients i;f (i ¼ ;...; 4) involve terms with fr and f R, i.e., terms disappearing if we take f R constant. Equivalently, i;eh (i ¼,, 2) contain terms coming from the linear part of f in R. i;f ¼ X7 j¼ i;f ¼ X8 j¼ ðjþ i;f i ¼ 3; 4; ðjþ i;f i ¼ ; ; 2; (32) where two consecutive terms in each series differ in the 2 factor. The expressions for the coefficients are too long to be written explicitly. Instead, in the following sections we will show different approximated formulas useful in certain limits. V. VIABLE fðrþ THEORIES Results obtained so far are valid for any fðrþ theory. However, as mentioned in the introduction, this kind of model is severely constrained in order to provide consistent theories of gravity. In this section we review the main conditions [9]: () f RR > for high curvatures [7]. This is the requirement for a classically stable high-curvature regime and the existence of a matter dominated phase in the cosmological evolution. (2) þ f R > for all R. This condition ensures the effective Newton s constant to be positive at all times and the graviton energy to be positive. (3) f R < ensures ordinary general relativity behavior is recovered at early times. Together with the condition f RR >, it implies that f R should be a negative and monotonically growing function of R in the range <f R <. (4) jf R jat recent epochs. This is imposed by local gravity tests [7], although it is still not clear what is the actual limit on this parameter. This condition also implies that the cosmological evolution at late times resembles that of CDM. In any case, this constraint is not required if we are only interested in building models for cosmic acceleration. VI. EVOLUTION OF SUB-HUBBLE MODES AND THE QUASISTATIC APPROXIMATION We are interested in the possible effects on the growth of density perturbations once they enter the Hubble radius in the matter dominated era. In the sub-hubble limit,it

5 EVOLUTION OF DENSITY PERTURBATIONS IN fðrþ... PHYSICAL REVIEW D 77, 2355 (28) can be seen that the 4;f and 3;f coefficients are suppressed by 2 with respect to 2;f, ;f, and ;f, i.e., in this limit the equation for perturbations reduces to the following second-order expression: þ H þ ð þ f RÞ 5 H 2 ð þ Þð2 2 Þ 6 a 8 f 4 RR ð 2 2Þk 8 8G a 2 ð þ f R Þ 5 ð þ Þþ 24 a 8 f 4 RR ð þ f RÞð 2 2Þk 8 ¼ ; (33) where we have taken only the leading terms in the expansion for the and coefficients. This expression can be compared with that usually considered in literature, obtained after performing strong simplifications in the perturbed equations (6) (9), (2), and (22) by neglecting time derivatives of and potentials (see [4]). Thus in [,8] the authors obtain þ H þ 4 k2 a 2 þ 3 k2 a 2 f RR þf R f RR þf R ~ þ f R ¼ : (34) This approximation has been considered as too aggressive in [] since neglecting time derivatives can remove important information about the evolution. Note also that there exists a difference in a power k 8 between those terms coming from the f part and those coming from the EH part in (33). This result differs from that in the quasistatic approximation where difference is in a power k 2 according to (34). In order to compare the evolution for both equations, we have considered a specific function f test ðrþ ¼ 4R :63, where H 2 units have been used, which gives rise to a matter era followed by a late-time accelerated phase with the correct deceleration parameter today. Initial conditions in the matter era were given at redshift z ¼ 485 where the EH part was dominant. Results for k ¼ 6H are presented in Fig.. We see that, as expected, both expressions give rise to the same evolutions at early times (large redshifts) where they also agree with the standard CDM evolution. However, at late times the quasistatic approximation fails to correctly describe the evolution of perturbations. Notice that the model example satisfies all the viability conditions described in the previous section except for the local gravity tests. As we will show in the following, it is precisely this last condition jf R j that will ensure the validity of the quasistatic approximation. We will now restrict ourselves to models satisfying all the viability conditions, including jf R j. In Appendix A we have reproduced all the and the first four coefficients for each term in (3). These are the dominant ones for sub-hubble modes (i.e., ) once the condition jf R j has been imposed. Thus, keeping only P 4 j¼ ðjþ i¼;...;4;f and ðþ i¼;;2;eh as the relevant contributions for the general coefficients, the full differential equation (3) can be simplified as c 4 iv þ c 3 þ c 2 þ c þ c ¼ ; (35) where the c coefficients are written in Appendix B. We see that indeed in the sub-hubble limit the c 4 and c 3 coefficients are negligible and the equation can be reduced to a second-order expression. As a consistency check, we find that both in a matter dominated universe and in CDM all coefficients vanish identically since f, f 2. For these cases, Eq. (3) becomes Eq. (6) as expected. For instance, in the pure matter dominated case, the coefficients are constant, and they take the following values: ¼ =2, 2 ¼ =2, 3 ¼ 3=4, and 4 ¼ 3=2. Another important feature from our results is that, in general, without imposing jf R j, the quotient ð ;EH þ ;f Þ=ð 2;EH þ 2;f Þ is not always equal to H. In fact only the quotients ðþ ;EH =ðþ 2;EH and ðþ ;f =ðþ 2;f are identically equal to H, which is in agreement with the coefficient in (6). However, for our approximated expressions it is true that c =c 2 H. From expressions in Appendix B, the second-order equation for becomes þ H 4 3 which can also be written as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 9 a 2 4 ð þ 2 ÞŠ½ 6f RRk 2 þ 9 a 2 4 ð þ þ 2 ÞŠ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð ÞH 2 ¼ ; (36) þ 5 a 2 2 ð 24 þ 25 ÞŠ½ 6f RRk 2 þ 5 a 2 2 ð þ 24 þ 25 ÞŠ ½ 6f RRk 2 ½ 6f RRk 2 þ H 4 3 2þ 2 2þ 2 ð 6f RRk 2 þ 9 a 2 4 Þ2 8 6 þ þ 2 ð 6f RRk 2 þ 5 a 2 2 Þ þ 6 ð þ ÞH 2 ¼ : (37) 2þ

6 A. DE LA CRUZ-DOMBRIZ, A. DOBADO, AND A. L. MAROTO PHYSICAL REVIEW D 77, 2355 (28) k = h Mpc - quasistatic ΛCDM k =.67 h Mpc - quasistatic ΛCDM k k z FIG.. k with k ¼ :2 h Mpc for f test ðrþ model and CDM. Both standard quasistatic evolution and Eq. (33) have been plotted in the redshift range from to. Note that the quasistatic expression (34) is only recovered in the matter era (i.e., for H ¼ 2=) or for a pure CDM evolution for the background dynamics. Nevertheless in the considered limit j f R j it can be proven using the background equations of motion that þ 2 (38) and therefore þ 2 þ, which allows simplifying expression (37) to approximately become (34). This is nothing but the fact that for viable models the background evolution resembles that of CDM [9]. In other words, although for general fðrþ functions the quasistatic approximation is not justified, for those viable functions describing the present phase of accelerated expansion and satisfying local gravity tests, it gives a correct description for the evolution of perturbations. VII. SOME PROPOSED MODELS In order to check the results obtained in the previous section, we propose two particular fðrþ theories which allow us to determine at least numerically all the quantities involved in the calculations and therefore to obtain solutions for (3). As commented before, for viable models the background evolution resembles that of CDM at low redshifts and that of a matter dominated universe at high redshifts, i.e., the quantity ðr þ fðrþþ=r tends to one in the high-curvature regime. Nevertheless the fðrþ contribution gives the dominant contribution to the gravitational action for small curvatures, and therefore it may explain the cosmological acceleration. For the sake of concreteness we will fix the model parameters imposing a deceleration parameter today q :6. Thus, our first model (A) will be fðrþ ¼c R p. According to the results presented in [2,9] viable models of this type include both a matter dominated and late-time accelerated universe provided the parameters satisfy c < and <p<. We have chosen c ¼ 4:3 and p ¼ : in H 2 units. This choice does verify all the viability conditions, including jf R j today. For the second model (B), fðrþ ¼ c R e þc 2, we have chosen c ¼ 2:5 4, e ¼ :3, and c 2 ¼ :22 also in the same units. For each model, we compare our result (36) with the standard CDM and the quasistatic approximation (34) (see Figs. 2 and 3). In both cases, the initial conditions are given at redshift z ¼ where is assumed to behave as in a matter dominated universe, i.e., k ðþ /aðþ with no k dependence. We see that for both models the quasistatic k... k =.67 h Mpc - +z +z quasistatic ΛCDM FIG. 2. k with k ¼ :67 h Mpc for fðrþ model A evolving according to (36), CDM and quasistatic approximation given by Eq. (34) in the redshift range from to. The quasistatic evolution is indistinguishable from that coming from (36), but diverges from CDM behavior as z decreases.. FIG. 3. k with k ¼ :67 h Mpc for fðrþ model B evolving according to (36), CDM and quasistatic evolution given by Eq. (34) in the redshift range from to. The quasistatic evolution is indistinguishable from that coming from (36), but diverges from CDM behavior as z decreases

7 EVOLUTION OF DENSITY PERTURBATIONS IN fðrþ... PHYSICAL REVIEW D 77, 2355 (28) today (k) k/h today (k) f(r) model A today (k) f(r) model B FIG. 4. Scale dependence of k evaluated today (z ¼ ) for k=h in the range from to 4. approximation gives a correct description for the evolution which clearly deviates from the CDM case. In Fig. 4 the density contrast evaluated today was plotted as a function of k for both models. The growing dependence of with respect to k is verified. This modified k dependence with respect to the standard matter dominated universe could give rise to observable consequences in the matter power spectrum, as shown in [3], and could be used to constrain or even discard fðrþ theories for cosmic acceleration. VIII. CONCLUSIONS In this work we have studied the evolution of matter density perturbations in fðrþ theories of gravity. We have presented a completely general procedure to obtain the exact fourth-order differential equation for the evolution of perturbations. We have shown that for sub-hubble modes the expression reduces to a second-order equation..9 We have compared this result with that obtained within the quasistatic approximation used in the literature and found that for arbitrary fðrþ functions such an approximation is not justified. However, if we limit ourselves to theories for which jf R j today, then the perturbative calculation for sub- Hubble modes requires taking into account, not only the first terms, but also higher-order terms in ¼ H =k. In that case, the resummation of such terms modifies the equation which can be seen to be equivalent to the quasistatic case but only if the universe expands as in a matter dominated phase or in a CDM model. Finally, the fact that for models with jf R j the background behaves today precisely as that of CDM makes the quasistatic approximation correct in those cases. ACKNOWLEDGMENTS We would like to thank A. Starobinsky for useful comments and J. A. R. Cembranos and J. Beltrán for their continuous encouragement. This work has been partially supported by the DGICYT (Spain) under projects FPA , FPA , and CAM/UCM 939 and by UCM-Santander PR34/ APPENDIX A: AND COEFFICIENTS Coefficients for the iv term: ðþ 4;f 8f4 R ð þ f RÞ 6 f 42 ; 4;f 72f3 R f3 4 ð 2 þ 2 Þ; 4;f 26f2 R f2 6 ð 2 þ 2 Þ 2 ; 4;f 26f Rf 8 ð 2 þ 2 Þ 3 : Coefficients for the term: (A) ðþ 3;f 8f4 R ð þ f RÞ 5 f 4H 2 ½3 þ f R ð3 þ f ÞŠ; 3;f 6f3 R f2 H 4 f8f 2 ð 2 þ 2 Þþ4f ½2 þ 9 2 2ð9 þ 3 ÞŠg; 3;f 72f2 R f H 6 ð 2 þ 2 Þ½ 4f 2 ð 2 þ 2 Þþf ð þ 4 3 ÞŠ; 3;f 26f RH 8 ð 2 þ 2 Þ 2 ½ 2f 2 ð 2 þ 2 Þþf ð7 4 2 þ 2 3 ÞŠ: (A2) Coefficients for the term: ðþ 2;EH ¼ 432ð þ f RÞ H 2 8 ð þ Þð 2 þ 2 Þ 3 ; 2;EH ¼ 296ð þ f RÞ H 2 ð þ Þ 2 ð 2 þ 2 Þ 3 ; 2;EH ¼ 3888ð þ f RÞ H 2 2 ð þ Þ 2 ð 2 þ 2 Þ 3 ; ðþ 2;f 8f4 R ð þ f RÞ 6 f 4H 2 ; 2;f 88f3 R f3 H 2 2 ð 2 þ 2 Þ; 2;f 24f2 R f2 H 2 4 ð 2 þ 2 Þð 28 þ 2 þ 3 2 Þ; 2;f 72f Rf H 2 6 ð 2 þ 2 Þ 2 ð 4 þ 4 þ 5 2 Þ: Coefficients for the term: (A3)

8 A. DE LA CRUZ-DOMBRIZ, A. DOBADO, AND A. L. MAROTO PHYSICAL REVIEW D 77, 2355 (28) ðþ ;EH ¼ 432ð þ f RÞ H 3 8 ð þ Þð 2 þ 2 Þ 3 ; ;EH ¼ 2592ð þ f RÞ H 3 ð þ Þ 2 ð 2 þ 2 Þ 3 ; ;EH ¼ 7776ð þ f RÞ H 3 2 ð þ Þ 3 ð 2 þ 2 Þ 3 ; ðþ ;f 8f4 R ð þ f RÞ 6 f 4H 3 ; ;f 88f3 R f3 H 3 2 ð 2 þ 2 Þ; ;f 24f2 R f2 H 3 4 ð 2 þ 2 Þð 28 þ 2 þ 3 2 Þ; ;f 72f Rf H 3 6 ð 2 þ 2 Þ 2 ð 4 þ 4 þ 5 2 Þ: Coefficients for the term: ðþ ;EH ¼ 432ð þ f RÞ H 4 8 ð þ Þð2 2 Þð 2 þ 2 Þ 3 ; ;EH ¼ 296ð þ f RÞ H 4 ð þ Þ 2 ð þ 4 2 Þð 2 þ 2 Þ 3 ; ;EH ¼ 3888ð þ f RÞ H 4 2 ð þ Þ 2 ð2 2 2Þð 2 þ 2 Þ 3 ; (A4) ðþ ;f 6 3 f4 R ð þ f RÞ 5 f 4H 4 ½2 þ f R ð2 þ 2f f 2 2 Þ 2 Š; ;f 2f3 R f3 H 4 2 ð þ Þð 2 þ 2 Þ; ;f 48f2 R f2 H 4 4 ð þ Þð 2 þ 2 Þð 6 þ 2 þ 7 2 Þ; ;f 44f Rf H 4 6 ð þ Þð 2 þ 2 Þ 2 ð 6 þ 4 þ 2 Þ: (A5) APPENDIX B: c COEFFICIENTS c 4 ¼ f R f ½ f R f k 2 3H 2 ð 2 þ 2 ÞŠ 3 ; c 3 ¼ 3f R H ½ f R f k 2 3H 2 ð 2 þ 2 ÞŠffR 2 f3 k4 þ 6f 2 H 4 ð 2 þ 2 Þ 2 þ f H 2 ð 2 þ 2 Þ½2f R f 2 k 2 þ 3H 2 ð 7 þ þ ÞŠ þ 2f R f H 2 k 2 ð 6 þ 6 þ Þg; c 2 ¼½ f R f k 2 3H 2 ð 2 þ 2 ÞŠ 2 ½fR 2 f2 k4 þ 5f R f H 2 k 2 ð 2 þ 2 Þþ6H 4 ð þ Þð 2 þ 2 ÞŠ; c ¼ H c 2 ; c ¼ 2 3 H 2 ð þ Þ½ f R f k 2 3H 2 ð 2 þ 2 ÞŠ 2 ½2fR 2 f2 k4 þ 9f R f H 2 k 2 ð 2 þ 2 Þþ9H 4 ð2 2 Þð 2 þ 2 ÞŠ: (B) [] A. G. Riess et al. (Supernova Search Team Collaboration), Astron. J. 6, 9 (998); S. Perlmutter et al. (Supernova Cosmology Project Collaboration), Astrophys. J. 57, 565 (999). [2] E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys. D 5, 753 (26). [3] S. M. Carroll, V. Duvvuri, M. Trodden, and M. S. Turner, Phys. Rev. D 7, (24); S. M. Carroll et al., Phys. Rev. D 7, 6353 (25); A. Dobado and A. L. Maroto, Phys. Rev. D 52, 895 (995); G. Dvali, G. Gabadadze, and M. Porrati, Phys. Lett. B 485, 28 (2); A. de la Cruz-Dombriz and A. Dobado, Phys. Rev. D 74, 875 (26); J. A. R. Cembranos, Phys. Rev. D 73, 6429 (26); S. Nojiri and S. D. Odintsov, Int. J. Geom. Methods Mod. Phys. 4, 5 (27). [4] T. P. Sotiriou, Gen. Relativ. Gravit. 38, 47 (26); O. Mena, J. Santiago, and J. Weller, Phys. Rev. Lett. 96, 43 (26); V. Faraoni, Phys. Rev. D 74, (26); S. Nojiri and S. D. Odintsov, Phys. Rev. D 74, 865 (26); I. Sawicki and W. Hu, Phys. Rev. D 75, 2752 (27). [5] D. J. Eisenstein et al., Astrophys. J. 633, 56 (25). [6] D. N. Spergel et al. (WMAP Collaboration), Astrophys. J. Suppl. Ser. 7, 377 (27). [7] E. Linder, Phys. Rev. D 72, (25). [8] J. M. Bardeen, Phys. Rev. D 22, 882 (98); J. C. Hwang and H. Noh, Phys. Rev. D 54, 46 (996); S. M. Carroll, I. Sawicki, A. Silvestri, and M. Trodden, New J. Phys. 8, 323 (26); Y. S. Song, W. Hu, and I. Sawicki, Phys. Rev. D 75, 444 (27); S. Carloni, P. K. S. Dunsby, and A. Troisi, Phys. Rev. D 77, 2424 (28); S. Tsujikawa, Phys. Rev. D 77, 2357 (28); S. Tsujikawa, K. Uddin, and R. Tavakol, Phys. Rev. D 77, 437 (28). [9] L. Pogosian and A. Silvestri, Phys. Rev. D 77, 2353 (28). [] P. Zhang, Phys. Rev. D 73, 2354 (26). [] R. Bean, D. Bernat, L. Pogosian, A. Silvestri, and M. Trodden, Phys. Rev. D 75, 642 (27)

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