D-dimensional f ( R , φ ) gravity

In this work, we explore a the different forms of a new type of modified gravity, namely 1 f (φ) gravity. We construct the Big Rip type for the energy density and the curvature of the uni2 verse. We show that dark energy is a result of the transformation of the field φ mass (dark matter) 3 to energy. In addition, we provide that Ωm ≈ 0, 050, ΩDM ≈ 0, 2, ΩDE ≈ 0, 746, is in excellent 4 agreement with observation data. We explore a generalized formalism of braneworld modified 5 gravity. We also construct a new field equations, which generalize the Einstein field equations. We 6 provide a relation between the extra dimension in 3-brane with the vacuum energy density. We 7 show that the energy density of matter depends directly on the number of dimensions. We manage 8 to find the value of the Gauss-Bonnet coupling α = 1/4 which is a good agreement with the results 9 in the literature, this correspondence creates a passage between f (R) gravity and Gauss-Bonnet 10 gravity, this comparison leads to a number of bulk dimensions equal to D = 10121 + 4. 11 12 PACS numbers: 98.70.Vc, 04.30. w, 98.80.Cq 13


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In a homogeneous and isotropic spacetime, the Einstein field equations give rise 17 to the Friedmann equations that describe the evolution of the Universe. In fact, the 18 standard big-bang cosmology based on radiation and matter dominated era can be well from 4d to 5d in f (φ) gravity, we will simultaneously study matter and dark matter 48 in the framework of f (φ) gravity. In Section 3; we study dark energy and we find 49 equivalent results with the model of holographic dark energy, then we obtain the values 50 of the density paramete. We study generally the holographic f (φ) gravity. Section 4 51 focus on in the general framework of D-dimonsional f (R, φ) gravity. In section 5 we will 52 study a passage between Gauss-Bonnet gravity and f (R) gravity. Section 6 contains an 53 discussion.
we can write the action (1) with the geometrical form by with a(t) is a scale factor. In this context we write g MN which leads to In the following, we will consider a some specific forms of the eqution (6). To describe the dark matter in f (φ) gravity, we assume that the term of DE in Eq.(6) 92 is zero: which leads to or equivalently where q is the deceleration parameter (a ∝ t q ). The field φ describes the acceleration of 109 the expansion of the Universe (dark energy). So that the equation (6)   Hubble parameter is given as follows a(t) ∝ t q =⇒ H = q t , we find We can also give the parameter: -In the radiation-dominated era: 143 We take the small values of the angle φR in this era. From Eq.(26) we obtain

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-In the matter-dominated era : This equation shows us that the gravity in the far future directly depends on DE density.
since we have chosen φR as the angle of the transition rotation between 4d and 5d. 162 We propose that the 5th dimension is perpendicular on 3-brane, in this case we take 163 φR = π 2 .We suppose that the change of Ω DM relative to Ω m is minimal, then using 164 This result is in excellent agreement with new observation data [16,17]. In our previous 166 paper [10], we just found the values of Ω m and Ω DM but in this paper we also obtained 167 Ω DE . Next, we will study the holographic f (φ) gravity.
show that the gravity on the 3-brane is a conformal transformation of the gravity on the  Hence, one can find a justification for the problem of the transition of the ω DE : The we can see that the fields (φ, R) describe a exceptional dimensions, are D = (4, 5).

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Additionally, Ricci scalar R describe the gravity at 4D, and the field φ describe gravity at 218 5D [10]. According to the action (37) we can write I 4 and I 5 as Only D = (4, 5) which describe the fields (φ, R) without a coupling. For all the dimen- We define a metric of a flat space, that is written as a direct sum of Friedmann-Lemaître- logical. In that case, therefore, density represents hidden dimensions in the Universe. 261 We can solve the equation (43), one can obtain with C is an integration constant.

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This result shows us that the density of matter always depends on the number of  In what follows is to study the densities ρ 5 and ρ D within the framework of Gauss-

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Bonnet gravity. Eq.(46) can be expressed in the vacuum of 3-brane as The term 1/(D − 4) has already been proposed by [26], they multiplied the GB term by showing in this case the properties similar to EGB gravity; this expression is equivalent to To past from f (R) gravity to GB gravity, we take G φ equivalent to the form proposed by [10], but with an additional factor. We take 296 a maximally Symmetric Space-time where we compare the last value of G with the value of G φ , we get consequently, the relationship between f (R) gravity and GB gravity implies that the 306 density ρ 5 is not the density of the dark matter but it is the density of the dark energy.

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The density ρ D zero in our 3-brane but on the bulk this density will take place. For D = 4 308 we can find the relation between the scalar field and the Ricci scalar φ ≈ 0, 35R −1 . Then, 309 we consider ρ vac ≈ 10 74 Gev 2 is the quantum vacuum density found by the quantum 310 mechanics. And ρ Λ ≈ 10 −47 Gev 2 is the vacuum density found by the ΛCDM model.