Can structure formation distinguish\Lambda CDM from non-minimal f ...

Can structure formation distinguish ΛCDM from Non-minimal f (R) Gravity? Shruti Thakur1, ∗ and Anjan A Sen2, † 1

Department of Physics and Astrophysics, University of Delhi, Delhi-110007, India Center for Theoretical Physics, Jamia Millia Islamia, New Delhi-110025, India (Dated: August 13, 2013)

arXiv:1305.6447v3 [astro-ph.CO] 12 Aug 2013

2

Non-minimally coupled f (R) gravity model is an interesting approach to explain the late time acceleration of the Universe without introducing any exotic matter component in the energy budget of the Universe. But distinguishing such model with the concordance ΛCDM model using present observational data is a serious challenge. In this paper, we address this issue using the observations related to the growth of matter over density g(z) as measured by different galaxy surveys. As background cosmology is not sufficient to distinguish different dark energy models, we first find out the functional form for f (R) ( which is non-minimally coupled to the matter lagrangian) that produces the similar background cosmology as in ΛCDM. Subsequently we calculate the growth for the matter over density g(z) for such non minimally coupled f (R) models and compare them with the ΛCDM Universe. We also use the measurements of g(z)σ8 (z) by different galaxy surveys to reconstruct the behavior for g(z) and σ8 (z) for both the non-minimally coupled f (R) gravity models as well as for the ΛCDM. Our results show that there is a small but finite window where one can distinguish the non-minimally coupled f (R) models with the concordance ΛCDM. PACS numbers:

I.

INTRODUCTION

The quest to explore gravity theory beyond Einstein’s general theory relativity (GTR) on large cosmological scales, in recent years, has been aroused by the prospect of explaining the late time acceleration of the Universe. It has now been established beyond any doubt that our Universe is currently going through an accelerated expanding phase which has been started in recent past [1]. The theoretical approach to explain such an acceleration can be broadly divided into two categories. In one approach, one has to add some exotic component with negative pressure (known as dark energy) in the energy budget of the Universe as normal matter or radiation component can not initiate accelerated expansion. Simplest of such component, known as cosmological constant ( with equation of state w = −1) can explain all the available cosmological data but at the same time is plagued by the embarrassing problem of fine tuning and cosmic coincidence [2]. One can also consider more exotic scalar fields [3–20] to mimic the required negative pressure which may solve at least the cosmic coincidence problem and can have far more striking cosmological consequences. Unfortunately the presently available cosmological data can not distinguish these two models decisively. The other approach is to give up the idea of adding extra component in energy budget of the Universe but to modify the gravity theory on large cosmological scales. The idea is to look for departure from the GTR that can effectively mimic a cosmological constant and result the necessary acceleration. f (R) gravity theory [21] is

∗ Electronic † Electronic

address: [email protected] address: [email protected]

one such example where instead of linear dependence of the Einstein-Hilbert (EH) action on the Ricci scalar, one includes nonlinear dependence. It is particularly interesting and has been studied extensively. Recently a generalization of such theory has been proposed which involves a non-minimal coupling between curvature and matter [22]. In this model, one extends the presence of non-linear functions of the scalar curvature in the EH action by incorporating an additional term involving the coupling between the Ricci scalar and the matter. Astrophysical and cosmological signatures of such non-minimally coupled f (R) gravity models have been studied by various authors [22] (see also [23] for similar investigation) . However describing background cosmological evolution for any particular model is not sufficient to remove the observed degeneracies between different modified gravity models and ΛCDM. Growth of matter perturbation is extremely important in this regard as the evolution equations for growth of matter perturbation are completely different in two scenarios. This can result characteristic signatures in CMB as well as in matter power spectrum from galaxy clusters which in turn can help to remove the degeneracies between the two models. In a recent paper [25] the evolution of cosmological perturbations in the presence of non-minimal coupling between curvature and matter has been studied. In this work, we extend the work done in [25] to the observational front. We study how far the present cluster data can distinguish the model with non-minimal coupling between curvature and matter from the standard ΛCDM model. For this we proceed in a two stage manner. We first look for the possible form for the f (R) that can give rise to the same background evolution as in ΛCDM Universe. After reconstructing the suitable form for the f (R), we calculate the matter power spectrum for such a model and see whether the currently available

2 observational data can distinguish this model from the ΛCDM. II.

THE NON-MINIMAL COUPLING IN MODIFIED GRAVITY THEORIES

where κ2 = 8πG, f (R) is a dimensionless function of the scalar curvature R and Lm is the matter lagrangian. Due to the presence of coupling between the curvature and the matter, there will be an energy transfer between them which can be seen in the corresponding conservation equation given by df /dR (gµν Lm − Tµν )∇µ R. 1 + f (R)

(2)

The non vanishing term in the r.h.s of the above equation shows the energy transfer between curvature and matter. This exchange of energy between the curvature and matter is a special feature in non minimally coupled f (R) gravity models. But in a homogenous and isotropic Universe, the energy momentum tensor for the matter should have a perfect fluid form and once we assume the form of the matter lagrangian as Lm = −ρm , [24](ρm being the energy density for the matter), it is easy to show that ∇µ Tµν = 0 is satisfied for the background evolution. Varying the action (1) with respect to the metric tensor gµν , one obtains the Einstein’s equation as 1 R(1 − χ)gµν + χ,µ;ν − gµν χ + κ2(1 + f (R))Tµν 2 (3) df where χ = 1 + 2κ2 Lm dR . Assuming a spatially flat Friedmann-Robertson-Walker spacetime with scale factor a(t) χGµν =

ds2 = −dt2 + a(t)2 δij dxi dxj

   Ωm0 Ωm0 −af . f − Ω − Ω ,a Λ0 Λ0 a3 2a3 (7) Here f,a represents the first derivative of f with respect to the scale factor. One has to solve this equation to find the form for f which can mimic the similar background evolution as in ΛCDM Universe. Equation (7) is a second order differential equation and needs two initial conditions, f (initial) and f,a (initial) , to solve. We solve the equation from the era of decoupling (a ∼ 10−3 ) till present day (a = 1). We observe that the behavior does not vary with f,a (initial) but is very sensitive to f (initial). In Figure 1, we show the different behaviors for f (a) and f ′ (a) with varying f (initial) that give rise to the same background evolution as in ΛCDM model. Given these forms for f (R) that produce identical background evolution as in ΛCDM Universe, the next question is how different is the growth history of the Universe for these non-minimally coupled f (R) gravity models from the concordance ΛCDM model? In the next section, we try to address this issue. a2 h2 (

The action for non minimally coupled f (R) gravity models is given by  Z  √ R + (1 + f (R))Lm S= −gd4 x, (1) 2κ2

∇µ Tµν =

flat Universe, Ωm0 + ΩΛ0 = 1. With such a background evolution, the equation (5) takes the form f,a 3 +f,a,a ) = a 2

III.



COSMOLOGICAL PERTURBATIONS

Perturbed FRW metric in longitudinal gauge takes the form: ds2 = −(1 + 2φ)dt2 + a2 (1 − 2ψ)δij dxi dxj ,

(8)

where φ and ψ are the two gravitational potentials. Using equations (3) and (8), one can obtain the equations for growth in the linear regime: k2 1 k2 ψ + 3H(ψ˙ + Hφ) = − ((3H˙ + 3H 2 − 2 )δχ 2 a 2χ a ˙ ˙ −3H δχ + 3H χφ ˙ + 3χ(Hφ ˙ + ψ)

+κ2 (1 + f )δρm ), (9)

(4)

the time component of above equation is H χ˙ = (H˙ + 2H 2 )χ −

R κ2 − H 2 χ + ρm (1 + f (R)). (5) 6 3

Our first goal is to look for those models that mimic the background evolution for the ΛCDM Universe. We know the form of the Hubble parameter for the ΛCDM Universe which is given by H2 Ωm0 h2 = 2 = 3 + ΩΛ0 , H0 a

(6)

where H0 , Ωm0 and ΩΛ0 are the Hubble parameter, matter density parameter and density parameter for cosmological constant at present respectively. For a spatially

2 ¨ + 3H δχ ˙ + [ 1 ( χ + κ2 ρm f,R ) − R + k ]δχ = δχ 2 χ,R 3 3 a

˙ + (2χ χ(3Hφ ˙ + 3ψ˙ + φ) ¨ + 3H χ)φ ˙ +

ψ−φ=

δχ . χ

κ2 (1 + f )δρm ,(10) 3 (11)

Here δρm (t, ~x) = ρm (t, ~x)−ρm (t) and δχ(t, ~x) = χ(t, ~x)− χ(t). Next we assume the velocity perturbation as uµ = uµ(0) + δuµ .

(12)

3

2500 2500

2000 2000

âf âa

1500

f

1500

1000 1000

500 500

0

0 -7

-6

-5

-4

-3

-2

-1

-7

0

-6

-5

-4

-3

-2

-1

0

lna

lna

df FIG. 1: Left figure shows behavior of f with log of scale factor. Right figure corresponds to the behavior of da with log of scale factor. In each figure different curves correspond to different values of parameter finitial . From bottom to top finitial varies as 0.02, 0.2, 2.0, 4.0

In the FRW Universe, uµ(0) = (−1, 0, 0, 0). Also with uµ uµ = −1, one can write δu0 = −δu0 = φ. The spatial part of the δuµ is the peculiar velocity. Writing the spatial part δui as gradient of a scalar: δui = δ ij vm,j ,

(13)

one can now get the following equations on perturbing equation (2): δ˙m + ▽2 vm − 3ψ˙ = 0,

inside the Hubble radius [26] : k2 k2 k2 |φ|, 2 |ψ|, 2 |δχ|} >> {H 2 |φ|, H 2 |ψ|, H 2 |δχ|}. 2 a a a (17) This actually implies |Y˙ | ≤ |HY |, where Y = ˙ φ, ψ, χ, χ, ˙ δχ, δχ Under this approximation, equations (9), (11) and (10) take the form {

(14)

   f,R 1 f,R ˙ R vm + 2 φ + δR = 0 v˙ m + 2H + 1+f a 1+f (15) m . Using these two equations, one finally where δm = δρ ρm arrives at     2 f,R ˙ f,R ˙ +k δ¨m + 2H + φ + R δm δR = 1+f a2 1+f   f,R ˙ 3ψ¨ + 3ψ˙ 2H + R .(16) 1+f

k2 1 ψ= 2 a 2χ



A.

Evolution of Matter Overdensities



k2 1 φ=− a2 2χ

k2 δχ − κ2 (1 + f )δρm a2





k2 δχ + κ2 (1 + f )δρm a2



(18)

(19)

 κ2 R k2 1 χ ( + κ2 ρm f,R ) − + 2 δχ = (1 + f )δρm . χ,R 3 3 a 3 (20) Using equations (18), (19) and (20), and under the approximation (17), equation (16) finally becomes ! f,x R˙ ¨ δm + 2H + δ˙m − 4πGef f ρm δm = 0, (21) 1 + f R0 

Equation (16) governs the cosmological evolution of the growth of matter over density in a non minimally where coupled f (R) gravity model which has identical backh i βf −1 3 f,x 4 k2 k2 f,x ground evolution as in the ΛCDM model. We are in- G (1 + f ) ( βf,x,x,x )(1 − αa2 2 H − x + 2 1+f ) − 2 f 2 H2 αa ,x,x ef f 0 0 terested in the perturbations which are deep inside the = βf,x −1 3Ω0m 3 k2 G ( )(1 + )f + x(βf − 1) − (βf − 1) ,x ,x ,x Hubble radius for which k >> aH. We apply the followβf,x,x αa3 αa2 a2 (22) ing approximations to calculate the perturbations deep

4 where W (kR) is the window function defined as   Sin(kR) Cos(kR) . − W (kR) = 3 (kR)3 (kR)2

2

gL HzL

100* HgNM HzL - gL HzLL

4

0

-2

-4

-6 0.0

0.5

1.0

1.5

2.0

z FIG. 2: Percentage deviation for the growth g(z) from the corresponding ΛCDM value for different f (initial). The solid, dashed, dot-dashed and dotted lines are for f (initial) = 0.002, 0.006, 0.02, 0.2 respectively. ΩΛ0 = 0.7.

R0 m0 Where β = 6Ω αa3 and α = H02 . To solve the equation (21), we use the initial conditions δ(ainitial ) = ainitial and ′ δ (ainitial ) = 1 where we choose ainitial ∼ 10−3 , which is the decoupling era. This is to ensure that the growth evolution reproduces the matter-like behavior during the early time.

IV.

OBSERVATIONAL CONSTRAINTS

Growth of matter overdensity is an important independent probe for different cosmological models that explain the late time acceleration of the Universe. The growth rate for matter over density is defined as g≡

dlnδm . dlna

(23)

In Figure 2, we show the percentage deviation of the growth factor g for the non-minimally coupled f (R) gravity model from the corresponding ΛCDM model. One can see that there can be four to six percent deviation one may expect for different values of f (initial). The question is whether such deviation can be probed by the current observational data. Another important quantity related to the growth is the rms fluctuation of the linear density field at the scale 8h−1 Mpc, σ8 (z). This defined as

σ82 (R, z)

=

Z

∞

∆2 (k, z)W (kR) 0

dk , k

(24)

(25)

The quantity ∆2 (k, z) as defined in [27]  ns −1  4 4 1 k k ∆2 (k, z) = As D(z)2 T (k)2 , 25 Ω2m k0 H0 (26) Where D(z) is the normalized density contrast defined as δm (z) δm (z=0) , As is the amplitude of the primordial curvature perturbation produced during inflation, k0 = 0.05 M pc−1 is the pivot scale at which the primordial fluctuation is calculated. T (k) is the transfer function which incorporates the effects of evolution of perturbations through horizon crossing and matter/radiation transition. We use analytical form for T (k) as proposed in [28]: ln(1 + 2.34q) 2.34q  −0.25 2 3 (27) 1 + 3.89q + (16.2q) + (5.47q) + (6.71q)4 T (k) =

−Ω

(1+

q

2h

)

b0 k 1 Ωm0 Where q ≡ Γh . Mpc−1 and Γ = Ωm0 h exp Galaxy surveys are directly sensitive to the combination g(z)σ8 (z). This combination is almost a model independent estimator for the observed growth history of the Universe and that is why most of the surveys e.g. the 2dF, VVDS, SDSS, 6dF, BOSS, as well as the Wiggle-Z galaxy survey provide the measurement for this estimator. In a recent paper [29] compilation of the current measurements for g(z)σ8 (z) has been given and we use those measurements for our purpose. Using these measurements, we calculate the posterior for our case, which is given by

P osterior = P (θ)L(θ)

(28)

where θ corresponds to parameters in our theory. P (θ) is the prior probability distribution for different parameters. In our calculations, we fix the value of the spectral index ns = 0.9616 which is the best fit value obtained by the Planck [30]. This is because, posterior is not very sensitive to the value of ns . We use Gaussian prior for the parameters ΩΛ0 and As , with their valuse as ΩΛ0 = 0.686 ± 0.020 and (109 As ) = 2.23 ± 0.16 from recent measurement by Planck [30]. For the model parameter f (initial), we use a flat prior between 0.002 to 4. The likelihood function is defined as L(θ) = e−

χ(θ)2 2

.

(29)

with χ2 = Σ



gσ8 obs (zi ) − gσ8 th (zi , ΩΛ , fi , As ) σgσ8

2

(30)

Covariance matrices for f (R) with non minimal coupling and for ΛCDM are constructed from the posterior.

5 0.9 0.8 0.8

0.7

gHzL

Σ8HzL

0.6 0.5

0.7

0.6

0.4 0.0

0.2

0.4

0.6

0.8

1.0

z

0.0

0.2

0.4

0.6

0.8

1.0

z

FIG. 3: Left figure corresponds to the growth rate of matter densities with respect to redshift. Right figure corresponds to the σ8 . The inner, outer shaded region corresponds to 1σ and 2σ deviation from the best fit value of parameters corresponding to non minimally coupled f (R). The inner, outer dashed curves corresponds to 1σ and 2σ deviation from the best fit value of parameters corresponding to ΛCDM

For non minimally coupled f(R) with parameters as ΩΛ , fi and As it becomes

CN M

ΩΛ0 finitial As 0.00167062  0.000135109 −0.000183621 = −0.000183621 1.17092 7.28979 ∗ 10−12 −12 0.00167062 7.28979 ∗ 10 0.0285568 

Covariance matrix for ΛCDM with parameters as ΩΛ and As becomes   ΩΛ0 As CΛCDM =  0.000134686 0.0016725  0.0016725 0.0285589

Using these covariance matrices of non minimally coupled f (R) as well as for ΛCDM, we have reconstructed evolution of the growth factor g(z) and σ8 (z) at 1σ and 2σ confidence level. This has been shown in Figure 3. One can see clearly that it is possible to distinguish the ΛCDM and the non-minimal f (R) model through the g(z) evolution both at 1σ and 2σ confidence level, whereas for the σ8 (z) evolution, one can distinguish them at 2σ confidence level. We should stress the fact that two models have identical background evolution and can not be distinguished by any observation related to background cosmology. V.

CONCLUSION

Explaining late time acceleration is one of the most significant challenges for cosmologists today. Modifying the Einstein gravity at large cosmological scales like f (R) gravity models, is one interesting approach to explain such late time acceleration. Coupling curvature with matter in f (R) gravity has been recently proposed which has the advantage, that the singularity that occurs in the standard f (R) gravity models may not occur

in non minimally coupled f (R). The background cosmology in such non-minimally coupled f (R) gravity has been studied. Recently its implication on inhomogeneous Universe has also been explored in [25]. Our goal is to extend further the work done in [25] to investigate the possibility to distinguish such a non  minimally coupled f (R) gravity model from the concordance ΛCDM model using the current measurements by different galaxy surveys. For this we do not assume any particular form for the function f (R). We take a different approach in this investigation. The growth of matter overdensity in any modified gravity model depends both on the background expansion as well as the extra effects that arise solely due to the modification in the gravity action ( in this case it is the non minimal coupling between the curvature and matter). To study the effect of this non minimal coupling on the growth history and the subsequent deviation from the ΛCDM behavior, we fix the background evolution same as the that of the ΛCDM model thereby eliminating the effect due to the background evolution. Now departure from the ΛCDM evolution is solely due to the modified equation for the growth evolution in the non-minimally coupled case. This is one crucial difference in approach from the investigation done in [25] where a specific form for f (R) was assumed. Subsequently we calculate the allowed behaviors for f (R) modification, that can give rise to the same background cosmology as in ΛCDM Universe. Reconstructing this form, we subsequently study the growth history and the σ8 normalization for such f (R) forms and compare them with ΛCDM Universe. We use the measurements for g(z)σ8 (z) ( which is mostly model independent) from different galaxy surveys to reconstruct the form for growth factor g(z) and σ8 (z) for both the non minimally coupled f (R) model as well as for ΛCDM Universe. The results show that it is possible to distinguish these two models using the growth of matter over density, even if they have same background evolutions. This result is

6 the important extension of the work done in [25] to observationally distinguish the non minimally coupled f (R) gravity model from the concordance ΛCDM model. We should mention that we have considered the simple form for the action as given in equation (1). One can consider more general form like L = f1 (R) + (1 + f2(R))Lm . In that case, if one considers the same background evolution as in ΛCDM, there will be effects on the growth history from the modified part of the pure gravity action (f1 (R)) as well as the effect due to the non minimal coupling between the matter and the curvature. One can then address two questions. Firstly whether such model can be distinguished from the concordance ΛCDM using the current observational data. And also given a specific form for the pure gravity lagrangian f1 (R), whether the non minimal f (R) model can be distinguished from the

standard minimal f (R) model using the observational data. We shall address these issues in future.

[1] S.Perlmutter et al., Astrophys. J.517 (1999) 565 [arXiv:astro-ph/9812133] .A. G. Riess et al., Astron.J.116 (1998) 1009 [arXiv:astro-ph/9805201] D.N.Spergel et al.[WMAP collaboration], Astrophys.J.Suppl. 170 (2007) 377 [arXiv:astro-ph/0603449]; U.Seljak et al [SDSS collboration],Phys.Rev. D 71 (2005) 103515 [arXiv:astro-ph/0407372]; D.J.Eisenstein et al.[SDSS collboration], Astrophys.J.633 (2005) 560 [arXiv:astro-ph/0501171]; B.Jain and A.Taylor, Phys.Rev Lett. 91 (2003) 141302 [arXiv:astro-ph/0306046] [2] E.J.Copeland, M.Sami and S.Tsujikawa,Int.J.Mod.Phys.D 15, 1753 (2006): M.Sami, arXiv:0904.3445; V.Sahni and A.A.Starobinsky, Int.J.Mod.Phys.D 9, 373 (2000); T.Padmanabhan,Phys.Rep. 380, 235 (2003);E.V.Linder, asrto-ph/0704.2064; J.Frieman, M.Turner and Huterer, arXiv:0803.0982; R.Caldwell and M.Kamionkowski, arXiv:0903.0866; A.Silvestri and M.Trodden, arXiv:0904.0024 [3] B. Ratra and P.J.E. Peebles, Phys. Rev. D 37 3406 (1988) ; R.R. Caldwell, R. Dave and P.J. Steinhardt, Phys. Rev. Lett. 80 1582 (1998) ; A.R. Liddle and R.J. Scherrer, Phys. Rev. D 59 (1999) 023509; P.J. Steinhardt, L. Wang and I. Zlatev, Phys. Rev. D 59 (1999) 123504 [4] C.Armendariz-Picon, T.Damour, and V.Mukhanov, Phys.Lett.B 458, 209 (1999) [5] J.Garriga and V.F.Mukhanov, Phys.Lett.B 458, 219 (1999) [6] T.Chiba, T.Okabe, M.Yamaguchi, Phys.Rev.D 62,023511 [7] C.Armendariz-Picon, V.Mukhnov, and P.J.Steinhardt, Phys.Rev.Lett 85, 4438(2000) [8] C.Armendariz-Picon, V.Mukhnov, and P.J.Steinhardt, Phys.Rev.D 63,103510 (2001) [9] T.Chiba, Phys.Rev.D 66, 063514 (2002) [10] L.P.Chimento and A.Feinstein, Mod.Phys.Lett.A 19, 761 (2004) [11] L.P.Chimento, Phys.Rev.D 69,123517 (2004) [12] R.J.Scherrer, Phys.Rev.Lett. 93, 011301 (2004) [13] A.Y.Kamenshchik, U.Moschella, and V.Pasquier, Phys.Lett.B 511, 265 (2001)

[14] N.Bilic, G.B.Tupper, and R.D.Viollier, Phys.Lett.B 535, 17 (2002) [15] M.C.Bento, O.Bertolami, and A.A.Sen, Phys.Rev.D 66, 043507 (2002) [16] A.Dev, J.S.Alcaniz, and D.Jain, Phys.Rev.D 67, 023515(2003) [17] V.Gorini, A.Kamenshchik and U.Moschella, Phys.Rev.D 67, 063509 (2003) [18] R.Bean and O.Dore, Phys.Rev.D 68,23515 (2003) [19] T.Multamaki, M.Manera and E.Gaztanaga, Phys.Rev.D bf 69, 023004 (2004) [20] A.A,Sen and R.J.Scherrer, Phys.Rev.D 72, 063511 (2005) [21] T.P.Sotiriou and V.Faraoni, arXiv:0805.1726[gr-qc]; S.Nojiri and S.D.Odinstov, arXiv:0807.0685; L. Amendola, R. Gannouji, D. Polarski and S. Tsujikawa, Phys. Rev. D 75 083504, (2007) ; W. Hu and I. Sawicki, Phys. Rev. D 76 064004, (2007); A. A. Starobinsky, JETP Lett. 86 157, (2007); S. Tsujikawa, Phys. Rev. D 77 023507, (2008) ; G. J. Olmo, Phys. Rev. D 72 083505,(2005); A. L. Erickcek, T. L. Smith and M. Kamionkowski, Phys. Rev. D 74 121501,(2006) ; V. Faraoni, Phys. Rev. D 74 023529, (2006); T. Chiba, T. L. Smith and A. L. Erickcek, Phys. Rev. D 75 124014,(2007); P. Brax, C. van de Bruck, A. C. Davis and D. J. Shaw, Phys. Rev. D 78 104021,(2008); I. Thongkool, M. Sami, R. Gannouji and S. Jhingan, Phys. Rev. D 80 043523, (2009); R. Gannouji and D. Polarski, JCAP 0805 018, (2008); S Nojiri, S D Odintsov and D saez-Gomez, Phys. Lett. B 681, 74,(2009); G Cognola, E Elizalde, S D Odintsov, P.Tretyakov and S Zerbini, Phys.Rev.D 79, 044001, (2009); S Nojiri and S D Odintsov, arXiv:0706.1378; F. G. Alvarenga et al., arXiv:1302.1866 [astro-ph.CO]; A. Abebe, A. de la Cruz-Dombriz and P. K. S. Dunsby, arXiv:1304.3462 [astro-ph.CO]; A. Abebe, M. Abdelwahab, A. de la Cruz-Dombriz and P. K. S. Dunsby, Class. Quant. Grav., 29, 135011 (2012); A. de la CruzDombriz, A. Dobado and A. L.Maroto, Phys. Rev. D, 77, 123515 (2008). [22] O.Bertolami and J.Paramos arXiv:0805.1241v2[gr-qc]; O.Bertolami, F. Lobo and J. Paramos, Phys. Rev. D, 78, 064036, (2008); O. Bertolami, J. Paramos,

VI.

ACKNOWLEDGEMENT

The authors thank T.R. Seshadri for useful discussions and comments. ST acknowledges the computational facilities provided by the CTP, JMI, New Delhi, India. ST thank C.S.I.R, Govt. of India for financial support through Senior Research Fellowship. AAS. acknowledges the partial financial support provided by C.S.I.R. Govt. of India through the research grant (Grant No:03(1187)/11/EMR-II).

7 T. harko and F. Lobo, arXiv:0811.2876; O. Bertoami and M.C.Sequeira, Phys. Rev.D, 79, 104010, (2009); O. Bertoami, C.G.Boehmer, T.Harko and F.Lobo Phys. Rev. D, 75, 104016, (2007); V. Faraoni, Phys.Rev. D. 80, 124040 (2009); S. Nojiri and S.D. Odintsov, Phys.Lett.B 599, 137, (2004); G. Allemandi, A.Borowiec, M.Francaviglia, S.D.Odintsov, Phys.Rev.D 72 063505,(2005); S. Nojiri, S.D. Odintsov and P.Tretyakov, arXiv:0710.5232; O. Bertolami and J. Paramos, arXiv:0906.4757; O. Bertolami, P.Frazao and J. Paramos, arXiv:1003.0850 [23] T. Koivisto, Class. Quant. Grav., 23, 4289 (2006). [24] J.D.Brown, Class Quantum Grav. 10 1579 (1993)

[25] O.Bertolami, Pedro Frazao and Jorge Paramos, JCAP, 1305, 029 (2003). [26] S. Tsujikawa, Phys. Rev .D 76, 023514, (2007); S. Tsujikawa, R. Gannouji, B. Moraes, D. Polarski, Phys.Rev. D 80, 084044, (2009) [27] D. Huterer, arXiv:1010.1162 [astro-ph.CO]. [28] M. Bardeen, J.R. Bond, N. Kaiser, K.S. Szalay, Astrophysical Journal 304, 15(1986). [29] S. Basilakos, S. Nesseris and L. Perivolaropoulos, arXiv:1302.6051 [astro-ph.CO]. [30] Planck Collaboration(P. A. R. Ade et al.), arXiv:1303.5076 [astro-ph.CO]