High energy description of dark energy in an approximate 3-brane Brans-Dicke cosmology

We consider a Brans-Dicke cosmology in five-dimensional space-time. Neglecting the quadratic and the mixed Brans-Dicke terms in the Einstein equation, we derive a modified wave equation of the Brans-Dicke field. We show that, at high energy limit, the 3-brane Brans-Dicke cosmology could be described as the standard one by changing the equation of state. Finally as an illustration of the purpose, we show that the dark energy component of the universe agrees with the observations data.


Introduction:
Theories, in which scalar fields are coupled directly to the curvature, are termed scalar-tensor gravity, such as the low-energy effective string theory [1] which couples a dilaton field to the Ricci curvature tensor. However, the simplest and the best known is the Brans-Dicke (BD) theory [2]. The BD theory, which is a generalization of the general relativity, must recover the latter as the BD parameter ω goes to infinity [3]. From timing experiments using the Viking space probe [4], ω must exceed 500. This constraint ruled out many of extended inflation [5,6] and provides a succession of improved models of extended inflation [7,8,9,10].
Superstring theory suggests that the space-time of our universe might be of higher dimension [11,12], in which the extra dimensions are compactified and only the 4-dimensional (4D) is observed experimentally. Recently, a great deal of interest has been done in cosmological scenario in which matter field are confined to a 3-brane world imbedded in a 5-dimensional (5D) bulk space-time [13,14,15].
Dark energy, distributed homogeneously in the Universe, is a component of the critical density of our universe as showed by the cosmic microwave background (CMB) and type Ia supernovae (SNe) observations [16,17,18]. Using type Ia SNe [17] as standard candles to gauge the expansion of the universe shows that the dark energy causes the expansion of the universe to speed up. This two experiments are in agreement with Ω Λ ≃ 0.7 and Ω m ≃ 0.3 .
In this paper, the dynamical system and dark energy of the universe are studied using the BD field in the 3-brane world. In brane cosmology [13,14,15], at the very early time, i.e. at high energy limit, dynamical evolution of the universe is modified by the extra terms in the Einstein equations, otherwise by the square of the matter density on the brane. In this way, we generalize the 4D BD theory to the 5D one by considering that BD field is sensitive only to the physical 3-brane. So it is described by the same 4D action and must recover the standard BD cosmology at low energy. To this aim, we add simply a BD stress energy tensor to the modified Einstein equations, 4 G µν , in 5D bulk space-time by neglecting all quadratic and mixed terms of this stress-energy tensor. To illustrate our consideration, we follow first Kolitch's work [19] in order to show, first, that at high energy limit the 5D space-time could be described by 4D space-time cosmology, i.e. the information contained in the extra dimensions are now involved in the equation of state. The γ factor characterizing the matter content of the universe in standard BD cosmology is equal to twice of the one in the 3-brane. Second, following Ç alik's work [20], we show the contribution of the dark energy in the dynamical evolution of the universe. The paper is organized as follows: a short review of the standard BD cosmology with vacuum cosmological constant as well as the 3-brane world with BD field is presented in section 2. Section 3 is devoted to resolve the dynamical system of the universe, while in section 4 we relate the cosmological parameters to the dark energy. A conclusion is given in section 5.

2
BD cosmology with cosmological constant:

4-dimensional BD cosmology:
Brans-Dicke cosmology with a nonzero cosmological constant, Λ 4 , was studied by many authors [19,21,22,23]. In this section, we follow the notations and the work of the author [19] where the action has the form: By varying this action with respect to the metric and BD field φ, the homogeneous and isotropic Friedmann-Robertson-Walker equation with scale factor a(t) and spatial curvature index k and the wave equation of the BD field are: and where ω is the BD parameter, the dot denotes the derivative with respect to the time and T µν = diag(ρ, p, p, p) is the stress-energy tensor of a perfect fluid in an orthonormal frame. From the conservation equation of T µν we have: Equations (2), (3) and (4) could be rewritten, as in [19,21]: where the new variables are defined as: The equation of state is given by The solutions of this planar dynamical system have previously been examined in [19,20,21] In the next subsection we will be interested in this dynamical system but in an extra dimension particularly in the 3-brane world.

5-dimensional BD cosmology:
The modified Einstein equations on the 3-brane, derived from 5-dimensional bulk space-time, have the form [15] where and k 2 5 are the 4D and the 5D gravitational constant respectively (G N is the Newton's constant of gravity) and the quadratic tensor Π µν is given by: E µν is a part of the 5D Weyl tensor which we take equal to zero. Λ 5 is the cosmological constant of the bulk space-time. λ, τ µν and q µν are the tension, the energy momentum tensor and the metric, respectively, confined on the brane world.
Our proposal to generalize the gravitational equation (10) in 3-brane world is done as follows. First, to obtain the equivalent 4D BD of the field equations in the 5-dimensions universe, we consider that the behaviour of BD field is sensitive only to physical 3-brane, so it is described by the same action as in 4-dimension, Eq. (1). Second, to recover the BD cosmology at low energy, we propose to add simply a BD stress energy tensor to the Einstein equation (10) in 5D bulk space-time, where all quadratic and mixed terms of this stress-energy tensor are neglected. So the modified Einstein equations are then written as: The BD field equations obtained by varying the action S, Eq. (1), with E µν = 0 and G N = 1 φ , are: The wave equation of the BD field, (14), in the 3-brane world differs from the one, Eq. (3), in the standard BD cosmology. This means that the extra dimensions affect not only the Einstein equations but also the wave equation and the equation of state as will be stated later. So at low energy limit, ρ ≫ ρ 2 , the field equations are the same as (2) and (3); or equivalently in terms of the variables X and Y, we recover the equations (6) and (5). At high energy limit, ρ ≪ ρ 2 , the field equations become: In term of the variables X and Y, we recover the following dynamical system: which are the same as (6) and (5) with γ in the standard case is equal to twice of the one in 5D bulk space-time. Therefore the 3-brane BD theory at high energies limit, could be described by the 4D one with the following equation of state p = (2γ − 1)ρ (21) 3 Dynamical system of the universe: To show how the dark energy contributes to the dynamical system of the universe, and how changes appear from 4D to 5D; we linearize the dynamical system about the stable cosmological non vacuum solution with flat space and show how the Hubble parameter varies with the scale factor a(t).

Equilibrium solutions for a flat space:
To study the dark energy in four and five dimensional BD cosmology using the Hubble parameter, we follow the work of [20] by introducing the variables, H =ȧ a and F =φ φ rather than X and Y . The field equations (2) and (3) become: where the prime denotes the derivative with respect to the scale factor. Since the term k/a 2 decreases dramatically as a(t) increases with the expansion of the universe, we drop it in the next by taking k = 0.
Let (H ∞ , F ∞ ) be the equilibrium points for a flat space (k = 0), which are obtained by setting H' and F' equal to zero in equations (22) and (23). The solutions of the resulting equations are: and the (∞) index means that the equilibrium is taken for a late time expansion.
The first solution is unstable while the second one is stable. In the next we will be interested only in the second solution and we must have ω > −4/3 or ω < −3/2, and in the limit ω −→ +∞ we have:

Linearized dynamical system:
To solve the dynamical system (22) and (23), we linearize the solution as in [20]: where h(a) and f (a) are linearized perturbation functions to be determined later.
Putting (27) and (28) into the field equations (22) and (23) and neglecting higher terms in h(a) and f(a), one obtains the following system: (29) This system becomes and Therefore, the linearized solutions (27) and (28), for ω ⇀ ∞, in the form as: where the subscript '0' indicates the present value. C 1 , C 2 and C ′ 2 = C 2 ω are dimensionless integration constants.
The equation (33) shows that the Hubble parameter varies with the scale factor as in [20] but with an extra term whose exponent depends on the γ-parameter.

The cosmological parameters and dark energy:
In this section we show that the Brans-Dicke theory is successful in explaining the dark energy which we relate to the most important cosmological parameters.
The Hubble parameter today, H 0 =ȧ/a(t 0 ), is used to estimate the order of magnitude for the present size and the age of the universe, has the value: In particular, we can define the individual ratios Ω i ≡ ρ i /ρ c , for matter, radiation, cosmological constant and even curvature, today. And from the standard Friedmann equations we have [24,25]: with Now inserting the solution (33) in equation (36) one gets, in 4D space-time, the expression of the constants C 1 and C 2 by comparing respectively the expressions of Ω i in 4D with the BD ones in (33) for ω −→ ∞. First, we mention that all forms of matter/energy are possible. However, we are interested in the γ-parameter of the equation of state in order to recover the different exponents of the equation (36). From (26) we obtain the following result: The integration constants C 1 and C 2 for different values of the γ-parameter are summarized as follows: γ −1/3, 0, 1/3 1/2 2/3 1 4/3 2 we notice that in the cases γ = 1 2 (extended inflation) and γ = 1 (dust universe) the integration constants C 1 and C 2 depend on each other. The symbol ∀ means that all values of C 2 are possible.
In the 3-brane, one obtains the same results by replacing the preceding γ by γ 2 . Hence, C 2 and C 1 are related to each other for γ = 1 4 and γ = 1 2 .
To compare ours results with the experimental data, we use the deceleration parameter q 0 = Ω R + 1 2 Ω M − Ω Λ [3,24,25]. Neglecting Ω R , one can parameterize the matter/energy content of the universe with just two components: the matter, characterized by Ω M , and the vacuum energy by Ω Λ . Except the cases γ = 1 2 , 1, where C 1 and C 2 are not independent,