aa r X i v : . [ g r- q c ] F e b Modified gravity and Space-Time-Matter theory
F. Darabi ∗ Department of Physics, Azarbaijan University of Tarbiat Moallem, Tabriz 53741-161, Iran
October 22, 2018
Abstract
The correspondence between f ( R ) theories of gravity and model theories explaining induceddark energy in a 5 D Ricci-flat universe, known as the Space-Time-Matter theory (STM), is studied.It is shown that such correspondence may be used to interpret the four dimensional expressions,induced from geometry in 5 D STM theories, in terms of the extra terms appearing in f ( R ) theoriesof gravity. The method is demonstrated by providing an explicit example in which a given f ( R )is used to predict the properties of the corresponding 5 D Ricci-flat universe. The acceleratedexpansion and the induced dark energy in a 5D Ricci-flat universe characterized by a big bounceis studied and it is shown that an arbitrary function µ ( t ) in the 5D solutions can be rewritten, interms of the redshift z , as a new arbitrary function F ( z ) which corresponds to the 4D curvaturequintessence models. The recent distance measurements from the light-curves of several hundred type Ia supernovae [1, 2]and independently from observations of the cosmic microwave background (CMB) by the WMAPsatellite [3] and other CMB experiments [4, 5] suggest strongly that our universe is currently under-going a period of acceleration. This accelerating expansion is generally believed to be driven by anenergy source called dark energy. The question of dark energy and the accelerating universe has beentherefore the focus of a large amount of activities in recent years. Dark energy and the acceleratinguniverse have been discussed extensively from various point of views over the past few years [6, 7, 8].In principle, a natural candidate for dark energy could be a small positive cosmological constant. Oneapproach in this direction is to employ what is known as modified gravity where an arbitrary functionof the Ricci scalar is added to the Einstein-Hilbert action. It has been shown that such a modificationmay account for the late time acceleration and the initial inflationary period in the evolution of theuniverse [9, 10]. Alternative approaches have also been pursued, a few example of which can be foundin [11, 12, 13]. These schemes aim to improve the quintessence approach overcoming the problemof scalar field potential, generating a dynamical source for dark energy as an intrinsic feature. Thegoal would be to obtain a comprehensive model capable of linking the picture of the early universeto the one observed today, that is, a model derived from some effective theory of quantum gravitywhich, through an inflationary period would result in today accelerated Friedmann expansion drivenby some Ω Λ -term. However, the mechanism responsible for this acceleration is not well understoodand many authors introduce a mysterious cosmic fluid, the so called dark energy, to explain this ∗ email: [email protected] D is addressed can be found in[28]. One of the first proposals in this regard was suggested in [9] where a term of the form R − wasadded to the usual Einstein-Hilbert action. In f ( R ) gravity, Einstein equations posses extra termsinduced from geometry which, when moved to the right hand side, may be interpreted as a mattersource represented by the energy-momentum tensor T Curv , see equation (5).In a similar fashion, the Space-Time-Matter (STM) theory, discussed below, results in Einsteinequations in 4 D with some extra geometrical terms which may be interpret as induced matter. Ittherefore seems plausible to make a correspondence between the geometrical terms in STM and T Curv resulting in f ( R ) gravity. We shall explore this idea to show that different choices of the parameter µ ( t ) in STM may correspond to different choices of f ( R ) in curvature quintessence models in modifiedgravity.The correspondence discussed above is based on the idea of extra dimensions. The idea that ourworld may have more than four dimensions is due to Kaluza [17], who unified Einstein’s theory ofGeneral Relativity with Maxwell’s theory of Electromagnetism in a 5 D manifold. Since then, higherdimensional or Kaluza-Klein theories of gravity have been studied extensively [18] from differentangles. Notable amongst them is the STM theory mentioned above, proposed by Wesson and hiscollaborators, which is designed to explain the origin of matter in terms of the geometry of the bulkspace in which our 4 D world is embedded, for reviews see [20]. More precisely, in STM theory, ourworld is a hypersurface embedded in a five-dimensional Ricci-flat ( R AB = 0) manifold where all thematter in our world can be thought of as being manifestations of the geometrical properties of thehigher dimensional space. The fact that such an embedding can be done is supported by Campbell’stheorem [21] which states that any analytical solution of the Einstein field equations in N dimensionscan be locally embedded in a Ricci-flat manifold in ( N + 1) dimensions. Since the matter is inducedfrom the extra dimension, this theory is also called the induced matter theory. Applications of theidea of induced matter or induced geometry can also be found in other situations [22]. The STMtheory allows for the metric components to be dependent on the extra dimension and does not requirethe extra dimension to be compact. The sort of cosmologies stemming from STM theory is studiedin [23, 24, 26].In this paper we consider the correspondence between f ( R ) gravity and STM theory. In section2 we present a short review of 4 D dark energy models in the framework of f ( R ) gravity. In section 3the field equations are solved in STM theory by fixing a suitable metric and the resulting geometricterms are interpreted as dark energy. The cosmological evolution in STM are considered in section4. Section 5 deals with an example for a special form of f ( R ). Conclusions are drawn in the lastsection. f ( R ) gravity General coordinate invariance in the gravitational action, without the assumption of linearity, allowsinfinitely many additive terms to the Einstein-Hilbert action [25] S = Z d x √− g [ c R + c R + c R µν R µν + c R µνλδ R µνλδ + · · · ] + S m , (1)where R, R µν and R µνλδ are Ricci scalar, Ricci tensor and Reimann tensor, respectively and S m is theaction for the matter fields. The fourth order term R µνλδ R µνλδ may be neglected as a consequence ofthe Gauss-Bonnet theorem. The action (1) is not canonical because the Lagrangian function containsderivatives of the canonical variables of order higher than one. This means that, not only do we expecthigher order field equations, but also the validity of the Euler-Lagrange equations is compromised.This problem is particularly difficult in the general case, but can be solved for specific metrics. Inhomogeneous and isotropic spacetimes, the Lagrangian in (1) can be further simplified. Specifically,2he variation of the term R µν R µν can always be rewritten in terms of the variation of R . Thus,the effective fourth order Lagrangian in cosmology contains only powers of R and we can suppose,without loss of generality, that the general form for a non-linear Lagrangian is given by S = Z d x √− gf ( R ) + S m , (2)where f ( R ) is a generic function of the Ricci scalar . Variation with respect to the metric g µν leadsto the field equations f ′ ( R ) R µν − f ( R ) g µν = f ′ ( R ) ; αβ ( g αµ g βν − g αβ g µν ) + ˜ T mµν , (3)where ˜ T mµν = 2 √− g δ S m δg µν (4)and the prime denotes a derivative with respect to R . It is easy to check that standard Einsteinequations are immediately recovered if f ( R ) = R . When f ′ ( R ) = 0 the equation (3) can be recast inthe more expressive form G µν = R µν − g µν R = T Curv µν + T mµν , (5)where an stress-energy tensor has been defined for the curvature contribution T Curv µν = 1 f ′ ( R ) (cid:26) g µν (cid:2) f ( R ) − Rf ′ ( R ) (cid:3) + f ′ ( R ) ; αβ ( g αµ g βν − g αβ g µν ) (cid:27) , (6)and T mµν = 1 f ′ ( R ) ˜ T mµν , (7)is an effective stress-energy tensor for standard matter. This step is conceptually very important sincea gravity model with a complicated structure converts to a model in which the gravitational field hasthe standard GR form with a source made up of two fluids: perfect fluid matter and an effective fluid(curvature fluid) that represents the non-Einsteinian part of the gravitational interaction.We now consider the Robertson-Walker metric for the evolution of the universe ds = dt − a ( t ) (cid:18) dr − kr + r d Ω (cid:19) , (8)where k is the curvature of the space, namely, k = 0 , , − k = 0 in equation (5) we obtain the 4 D , spatiallyflat Friedmann equations as follows H = 13 ( ρ m + ρ Curv ) , (9)and ˙ H = −
12 [( ρ m + p m ) + ρ Curv + p Curv ] , (10)where a dot represents derivation with respect to time. Such a universe is dominated by a barotropicperfect fluid with the equation of state (EOS) given by p m = w m ρ m ( w m = 0 for pressureless colddark matter and w m = 1 / p Curv = 1 f ′ ( R ) (cid:26) (cid:18) ˙ aa (cid:19) ˙ Rf ′′ ( R ) + ¨ Rf ′′ ( R ) + ˙ R f ′′′ ( R ) − (cid:2) f ( R ) − Rf ′ ( R ) (cid:3)(cid:27) , (11) We use units such that 8 πG N = c = ~ = 1. ρ Curv = 1 f ′ ( R ) (cid:26) (cid:2) f ( R ) − Rf ′ ( R ) (cid:3) − (cid:18) ˙ aa (cid:19) ˙ Rf ′′ ( R ) (cid:27) , (12)respectively. The equation of state of the curvature quintessence is w Curv = p Curv ρ Curv . (13)Recently, cosmological observations have indicated that our universe is undergoing an acceleratedexpanding phase. This could be due to the vacuum energy or dark energy which dominates ouruniverse against other forms of matter such as dark matter and Baryonic matter. We thus concentrateon the vacuum sector i.e. ρ m = p m = 0, from which the evolution equation of curvature quintessencebecomes ˙ ρ Curv + 3 H ( ρ Curv + p Curv ) = 0 , (14)which yields ρ Curv ( z ) = ρ Curv exp (cid:20) Z z (1 + w Curv ) d ln(1 + z ) (cid:21) ≡ ρ Curv E ( z ) , (15)where, 1 + z = a a is the redshift and the subscript 0 denotes the current value. In terms of theredshift, the first Friedmann equation can be written as H ( z ) = H Ω Curv E ( z ) , (16)where Ω Curv and H are the current values of the dimensionless density parameter and Hubbleparameter, respectively. Equation (16) is the Friedmann equation in terms of redshift, z , which issuitable for cosmological observations. In fact, equations (16) and (28), obtained in section 4, are thecosmological connections between f ( R ) gravity and STM theory. According to the old suggestion of Kaluza and Klein the 5 D vacuum Kaluza-Klein equations canbe reduced under certain conditions to the 4 D vacuum Einstein equations plus the 4 D Maxwellequations. Recently, the idea that our four-dimensional universe might have emerged from a higherdimensional spacetime is receiving much attention [19]. One current interest is to find out in amore general way how the 5 D field equations relate to the 4 D ones. In this regard, a proposal wasmade recently by Wesson [20] in that the 5 D Einstein equations without sources R AB = 0 (theRicci flat assumption) may be reduced to the 4 D ones with sources G ab = 8 πGT ab , provided anappropriate definition is made for the energy-momentum tensor of matter in terms of the extra partof the geometry. Physically, the picture behind this interpretation is that curvature in (4 + 1) spaceinduces effective properties of matter in (3 + 1) spacetime. This idea is known as space time matter (STM) or modern KaluzaKlein theory.In this popular non-compact approach to Kaluza-Klein gravity, the gravitational field is unifiedwith its source through a new type of 5 D manifold in which space and time are augmented by an extranon-compact dimension which induces 4 D matter within four dimensional universe. Unlike the usualKaluza-Klein theory in which a cyclic symmetry associated with the extra dimension is assumed, thenew approach removes the cyclic condition and derivatives of the metric with respect to the extracoordinate are retained. This induces non-trivial matter on the hypersurface of l = constant . Thistheory basically is guaranteed by an old theorem of differential geometry due to Campbell [21].In the context of STM theory, a class of exact 5 D cosmological solutions has been investigatedand discussed in [27]. This solution was further pursued in [23] where it was shown to describe a4osmological model with a big bounce as opposed to the ubiquitous big bang. The 5 D metric of thissolution reads dS = B dt − A (cid:18) dr − kr + r d Ω (cid:19) − dy , (17)where d Ω ≡ (cid:0) dθ + sin θdφ (cid:1) and A = (cid:0) µ + k (cid:1) y + 2 νy + ν + Kµ + k ,B = 1 µ ∂A∂t ≡ ˙ Aµ . (18)Here µ = µ ( t ) and ν = ν ( t ) are two arbitrary functions of t , k is the 3 D curvature index ( k = ± , K is a constant. This solution satisfies the 5 D vacuum equation R AB = 0. The Kretschmanncurvature scalar I = R ABCD R ABCD = 72 K A , (19)shows that K determines the curvature of the 5 D manifold. Such a solution was considered in [27]with a different notation.Using the 4 D part of the 5 D metric (17) to calculate the 4 D Einstein tensor, we obtain (4) G = 3 (cid:0) µ + k (cid:1) A , (4) G = (4) G = (4) G = 2 µ ˙ µA ˙ A + µ + kA . (20)As was mentioned earlier, since the recent observations show that the universe is currently goingthrough an accelerated expanding phase, we assume that the induced matter contains only darkenergy with ρ DE , i.e. ρ m = 0. We then have3 (cid:0) µ + k (cid:1) A = ρ DE , (21)2 µ ˙ µA ˙ A + µ + kA = − p DE . (22)From equations (21) and (22), one obtains the EOS of dark energy w DE = p DE ρ DE = − µ ˙ µ / A ˙ A + (cid:0) µ + k (cid:1)(cid:14) A µ + k )/ A . (23)The Hubble and deceleration parameters are given in [23, 26] and can be written as H ≡ ˙ AAB = µA , (24)and q ( t, y ) ≡ − A d Adt (cid:30) (cid:18) dAdt (cid:19) = − A ˙ µµ ˙ A , (25)from which we see that ˙ µ / µ > µ / µ < µ ( t ) therefore plays a crucial role in defining the properties of theuniverse at late times. 5 Correspondence between modified f ( R ) gravity and STM theory In this section we will concentrate on the predictions of the cosmological evolution in the spatiallyflat case ( k = 0). To avoid having to specify the form of the function ν ( t ), we change the parameter t to z and use A / A = 1 + z and define µ (cid:14) µ = F ( z ), noting that F (0) ≡
1. We then find thatequations (23)-(25) reduce to w DE ( z ) = − z ) d ln F ( z ) / dz , (26)and q DE ( z ) = 1 + 3Ω DE w DE − (1 + z )2 d ln F ( z ) dz . (27)There is an arbitrary function µ ( t ) in the present 5 D model. Different choices of µ ( t ) may correspondto different choices of f ( R ) in curvature quintessence models in modified gravity. Various choicesof µ ( t ) correspond to the choices of F ( z ). This enables us to look for the desired properties of theuniverse via equations (26) and (27). Using these definitions, the Friedmann equation becomes H = H (1 + z ) F ( z ) − . (28)This would allow us to use the supernovae observational data to constrain the parameters containedin the model or the function F ( z ). By comparing equation (28) with equation (16), we find that thereexists a correspondence between the functions f ( R ) and F ( z ). We thus take F ( z ) as F ( z ) = (1 + z ) [Ω Curv E ( z )] − . (29)According to (15), it is easy to see that the function E ( z ) is determined by the particular choicefor f ( R ) which, in turn, determines the function F ( z ) through equation (29). The evolution of thedensity components and the EOS of dark energy may now be derived. To this end, we must determinethe functional form of f ( R ). Thus, for example, we choose f ( R ) as a generic power law of the scalarcurvature and assume for the scale factor a power law solution in 4 D , investigated in [28]. Therefore f ( R ) = f R n , a ( t ) = a (cid:18) tt (cid:19) α . (30)The interesting cases are for the values of α satisfying α > ρ m = 0. Inserting equation (30) intothe dynamical system (9) and (10), for a spatially flat space-time we obtain an algebraic system forparameters n and α α (cid:2) α ( n −
2) + 2 n − n + 1 (cid:3) = 0 ,α (cid:2) n − n + 1 + α ( n − (cid:3) = n ( n − n − , (31)from which the allowed solutions are α = 0 → n = 0 , / , ,α = 2 n − n + 12 − n , ∀ n, n = 2 . (32)The solutions with α = 0 are not interesting since they provide static cosmologies with a non-evolvingscale factor. Note that this result matches the standard General Relativity result n = 1 in the absence6f matter. On the other hand, the cases with generic α and n furnish an entire family of significantcosmological models. Using equations (11) and (12) we can also deduce the equation of state for thefamily of solutions α = 2 n − n + 12 − n as w Curv ( n ) = − (cid:18) n − n − n − n + 3 (cid:19) , (33)where w Curv → − n → ∞ . This shows that an infinite n is compatible with recovering an infinite cosmological constant. Thus, using equation (33), E ( z ) and F ( z ) are given by E ( z ) = (1 + z ) h − n +46 n − n +3 i , (34) F ( z ) = (1 + z ) (cid:20) Ω Curv (1 + z ) h − n +46 n − n +3 i (cid:21) − . (35)Now, using the above equations, equations (26) and (27) can be written as w DE ( n ) = − (cid:18) n − n − n − n + 3 (cid:19) , (36)and q DE ( n ) ≡ − A ˙ µµ ˙ A = − n + 2 n + 12 n − n + 1 . (37)Therefore, within the context of the present investigation, the accelerating, dark energy dominateduniverse, can be obtained by using the correspondence between F ( z ) and f ( R ) in modified gravitytheories. We observe that in STM theory, 5 D dark energy cosmological models correspond to 4 D curvature quintessence models. This result is consistent with the correspondence between exactsolutions in Kaluza-Klein gravity and scalar tensor theory [29]. Note that, as is well known, with asuitable conformal transformation, f ( R ) gravity reduces to the scalar tensor theory.From equations (34) and (35), we can rewrite equation (28) as h ( z, n ) = Ω Curv (1 + z ) h − n +46 n − n +3 i , (38)where h ( z, n ) ≡ H ( z ) H and the contribution of ordinary matter has been neglected. Figure 1 shows thebehavior of h ( n ) as a function of n for z ∼ . Curv ≃ .
70. As can be seen, for n −→ ±∞ and z −→ h ( z, n ) −→ Ω Curv , that is, the universe finally approaches the curvature dominantstate, thus undergoing an accelerated expanding phase. Figure 2 shows the behavior of h ( z ) as afunction of z for n = 2 , , −
10 and Ω Curv ≃ .
70. We see that for small z , h ( z ) −→ .
70. Thus,we have obtained late-time accelerating solutions only by using the correspondence between f ( R )gravity and STM theory. Here, we have interpreted the properties of 5 D Ricci-flat cosmologies bydark energy models in modified gravity.
In this paper we have studied the correspondence between modified f ( R ) gravity and Space-Time-Matter theory by investigation of the present accelerated expanding phase of the universe using ageneral class of 5 D cosmological models, characterized by a big bounce as opposed to a big bang, whichis the standard prediction in 4 D cosmological models. Such an exact solution contains two arbitraryfunctions, µ ( t ) and ν ( t ), which are analogous to different forms of f ( R ) in curvature quintessence7 H n L Figure 1:
Behavior of h ( n ) as a function of n for z ∼ . Curv ≃ .
70. An accelerating universe occurs for n . − n & H z L Figure 2:
Behavior of h ( z ) as a function of z for n = 2 (solid line), n = 10 (dashed line), n = −
10 (dot-dashed line)and Ω Curv ≃ .
70. Note that for n = 2 , , − z → h ( z ) → . F ( z ) and f ( R ) plays a crucial role and defines the form of the function F ( z ). Finally,by taking a specific form for f ( R ) we obtained solutions that describe the late-time acceleration ofthe universe. Explicitly, the induced dark energy and the resulting accelerated expansion in a 5DRicci-flat universe is studied and it is shown that an arbitrary function µ ( t ) in the 5D solutions canbe rewritten as a new arbitrary function F ( z ) which corresponds to the 4D curvature quintessencemodels. References [1] A. G. Riess et. al. [Supernova Search Team Collaboration], Astron. J. 116 (1998) 1006, [astro-ph/9805201].[2] S. Perlmutter et. al. , Astron. J. 517 (1999) 565, [astro-ph/9812133];D. N. Spergel et. al. , Astrophys. J. Suppl. 148 (2003) 175, [astro-ph/0302209].[3] C. L. Bennett et. al. , Astrophys. J. Suppl. 148 (2003) 1, [astro-ph/0302207].[4] C. B. Netterfiled et. al. , Astrophys. J. 571 (2002) 604, [astro-ph/0104460].[5] N. W. Halverson et. al. , Astrophys. J. 568 (2002) 38, [astro-ph/0104489].[6] I. Zlatev, L. Wang and P. J. Steinhardt , Phys. Rev. Lett. 82 (1999) 896, [astro-ph/9807002];P. J. Steinhardt, L. Wang, I. Zlatev, Phys. Rev. D 59 (1999) 123504, [astro-ph/9812313];M. S. Turner, Int. J. Mod. Phys. A 17S1 (2002) 180, [astro-ph/0202008];V. Sahni, Class. Quant. Grav. 19 (2002) 3435, [astro-ph/0202076].[7] R. R. Caldwell, M. Kamionkowski, N. N. Weinberg, Phys. Rev. Lett. 91 (2003) 071301, [astro-ph/0302506];R. R. Caldwell, Phys. Lett. B 545 (2002) 23, [astro-ph/9908168];P. Singh, M. Sami, N. Dadhich, Phys. Rev. D 68 (2003) 023522, [hep-th/0305110];J.G. Hao, X.Z. Li, Phys. Rev. D 67 (2003) 107303, [gr-qc/0302100].[8] Armend´ariz-Pic´on, T. Damour, V. Mukhanov, Phy. Lett. B 458 (1999) 209 ;M. Malquarti, E.J. Copeland , A. R. Liddle, M. Trodden, Phys. Rev. D 67 (2003) 123503;T. Chiba , Phys. Rev. D 66 (2002) 063514, [astro-ph/0206298].[9] S. M. Carroll, V. Duvvuri, M. Trodden, and M. S. Turner, Phys. Rev. D 70 (2004) 043528.[10] S. Capozziello, S. Nojiri, S. D. Odintsov and A. Troisi, Phys. Lett. B 639 (2006) 135;S. Nojiri and S. D. Odintsov, J. Phys. Conf. Ser. 66 (2007) 012005;S. Nojiri and S. D. Odintsov, J. Phys. A 40 (2007) 6725;F. Briscese, E. Elizalde, S. Nojiri, and S. D. Odintsov, Phys. Lett. B 646 (2007) 105;S. Nojiri, S.D. Odintsov and H. Stefan´cic, Phys. Rev. D 74, (2006) 086009;M. E. Soussa and R. P. Woodard, Gen. Rel. Grav. 36 (2004) 855;R. Dick, Gen. Rel. Grav. 36 (2004) 217;A. E. Dominguez and D. E. Barraco, Phys. Rev. D 70, (2004) 043505;V. Faraoni, Phys. Rev. D 75 (2007) 067302;J. C. C. de Souza and V. Faraoni, Class. Quantum Grav. 24 (2007) 3637;D. A. Easson, Int. J. Mod. Phys. A 19 (2004) 5343;G. J. Olmo, Phys. Rev. Lett. 95 (2005) 261102;G. Allemandi, M. Francaviglia, M. L. Ruggiero and A. Tartaglia, Gen. Rel. Grav. 37 (2005) 1891;S. Capozziello and A. Troisi, Phys. Rev. D 72 (2005) 044022 ;9. Clifton and J.D. Barrow, Phys. Rev. D 72 (2005) 103005;T. P. Sotiriou, Gen. Rel. Grav. 38 (2006) 1407;S. Capozziello, A. Stabile and A. Troisi, Mod. Phys. Lett. A 21 (2006) 2291;A. Dolgov and D. N. Pelliccia, Nucl. Phys. B 734 (2006) 208 ;K. Atazadeh and H.R. Sepangi, Int. J. Mod. Phys. D 16 (2007) 687, [gr-qc/0602028];S. Nojiri and S. D. Odintsov, Int. J. Geom. Meth. Mod. Phys. 4 (2007) 115, [hep-th/ 0601213].[11] A. Ujjaini, V. Sahni, [astro-ph/0209443];V. Sahni, Y. Shtanov, JCAP 0311 (2003) 014, [astro-ph/0202346].[12] K. Freese, M. Lewis, Phys. Lett. B 540 (2002) 1, [astro-ph/0201229];Y. Wang, K. Freese, P. Gondolo, M. Lewis, Astrophys. J. 594 (2003) 25, [astro-ph/0302064].[13] N. Bilic, G. B. Tupper and R. D. Viollier, Phys. Lett. B 535 (2002) 17;J. C. Fabris, S. V. B. Goncalves and P. E. de Souza, [astro-ph/0207430];A. Dev, J. S. Alcanitz and D. Jain, D 67, (2003) 023515.[14] S. M. Carroll, Living Rev. Rel. 4 (2001) 1, [astro-ph/0004075].[15] C. Deffayet, Phys. Lett. B 502 (2001) 199 [hep-th/0010186];C. Deffayet, G. R. Dvali and G. Gabadadze, Phys. Rev. D 65 (2002) 044023, [astro-ph/0105068];C. Deffayet and S. J. Landau, J. Raux, M. Zaldarriaga and P. Astier, Phys. Rev. D 66 (2002)024019, [astro-ph/0201164];J. S. Alcaniz, Phys. Rev. D 65 (2002) 123514, [astro-ph/0202492];D. Jain, A. Dev and J. S. Alcaniz, Phys. Rev. D 66 (2002) 083511, [astro-ph/0206224];A. Lue, R. Scoccimarro, G. Starkman, Phys. Rev. D 69 (2004) 044005, [astro-ph/0307034].[16] S. Capozziello, Int. J. Mod. Phys. D 11 (2002) 483, [astro-ph/0201033];S. Capozziello, Int. J. Mod. Phys. D 12 (2003) 1969, [astro-ph/0307018].[17] T. Kaluza,
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