Cosmological expansion governed by a scalar field from a 5D vacuum

We consider a single field governed expansion of the universe from a five dimensional (5D) vacuum state. Under an appropiate change of variables the universe can be viewed in a effective manner as expanding in 4D with an effective equation of state which describes different epochs of its evolution. In the example here worked the universe fistly describes an inflationary phase, followed by a decelerated expansion. Thereafter, the universe is accelerated and describes a quintessential expansion to finally, in the future, be vacuum dominated.


I. INTRODUCTION AND OVERVIEW OF THE 5D FORMALISM
In the last years has been an uprising interest in finding exact solutions of the Kaluza-Klein field equations in 5D, where the fifth coordinate is considered as noncompact [1]. Unlike the usual Kaluza-Klein theory in which a cyclic symmetry associated with the extra dimension is assumed, the new approach removes the cyclic condition on the extra dimension and derivatives of the metric with respect to the extra coordinate are retained. This induces non-trivial matter on the hypersurfaces with ψ = constant and other nontrivial frames. This theory reproduces and extends known solutions of the Einstein field equations in 4D. Particular interest revolves around solutions which are not only Ricci flat [2], but also Riemann flat [3]: R A BCD = 0 ( A, B, C, D can take the values 0, 1,2,3,4). This is because it is possible to have a flat 5D manifold which contains a curved 4D submanifold, as implied by the Campbell theorem [4]. So, the universe may be "empty" and simple in 5D, but contain matter of complicated forms in 4D [5].
One of the greatest challenges of modern cosmology is understanding the nature of the observed late-time acceleration of the universe. Recent measurements of type Ia Supernovae (SNIa) [6] at redshifts z ∼ 1 and also the observational results coming from the Cosmic Microwave Background Radiation (CMBR) along with the Maxima [7] and Boomerang data [8] indicate that the expansion of the present universe is accelerated. In fact the present day results show that supernovae are moving faster than expected from the luminosity redshift relationship in a decelerating universe. A possible explanation is that in the universe there exists an important matter component which, in its most simple description, has the characteristic of a cosmological constant as vacuum energy density which contributes to a large component of negative pressure [9,10]. The idea that the expansion of the universe could be governed by a scalar field has been developed in the quintessential models, where the dynamics of the scalar field is governed by an appropiate potential V (ϕ) [11]. This paper is devoted to study the evolution of the universe which firstly suffers an inflationary expansion followed by a decelerated expansion (matter and radiation dominated) to finally be monotonically accelerated until a quasi-de Sitter expansion from a 5D vacuum state, where the (space-like) fifth dimension is considered as noncompact. We shall suppose that the universe is governed by a scalar field, which is minimally coupled to gravity. For the system, we shall consider that the action is where ϕ is a scalar (neutral) quantum field, G = M −2 p is the gravitational constant [being M p = 1.2 10 19 GeV the Planckian mass], and (5) R is the Ricci scalar. Furtheremore (5) g is the determinant of the covariant tensor metric g AB (A, B can take the values 0, 1,2,3,4). We are interested to describe a manifold in apparent vacuum, so that the Lagrangian density L in (1) should be only kinetic (5) To describe a 3D spatially isotropic and homogeneous universe, which is Ricci flat: R A BCD = 0 and describes a 5D vacuum: G AB = 0, we shall consider the background metric [12] where ψ is the space-like fifth coordinate, r(x, y, z) = √ x 2 + y 2 + z 2 and N, x, y, z are dimensionless coordinates. Furthermore, the metric (3) is 3D spatially isotropic, homogeneous and flat. For the metric (3), the determinant of the covariant metric tensor g AB is | (5) g| = ψ 8 e 6N and | (5) g 0 | = ψ 8 0 e 6N 0 is a constant of dimensionalization, for the constants ψ = ψ 0 and N = N 0 . On the other hand, the energy-momentum tensor is given by which is symmetric because the symmetry of g AB . The dynamics for ϕ is described by the Lagrange equation In absence of 3D spatially isotropic field fluctuations, the background field ϕ b (N, ψ) corresponding to the vacuum equation (3), is a constant of N and ψ: ϕ b (N, ψ) = const.. The commutator between ϕ and ⋆ ϕ will be In order to simplify the structure of the equation (5) we can make the transformation and the commutator between χ and ⋆ χ is We can make a Fourier expansion for χ where the asterisk denotes the complex conjugate and (a krk ψ , a † krk ψ ) are respectively the annihilation and creation operators. They satisfy the following commutation expressions The expression (8) complies if the modes are normalized by the following condition: This equation provides the normalization for the complete set of solutions on all the spectrum (k r , k ψ ). As was demonstrated in a previous work [13], such that the normalization condition forξ kr (N) becomes where the overstar denotes the derivative with respect to N. Hence, the field χ in eq. (9) can be rewritten as χ(N, r) = 1 (2π) 3/2 d 3 k r dk ψ a krk ψ e i kr. rξ kr (N) + a † krk ψ e −i kr. rξ * kr (N) .
Furthermore, the functionsξ kr (N) are given by [13] ξ kr (N) = i √ π 2 H Finally, the field ϕ is given by with χ(N, r) given by eq. (15). It is very important to notate that exponentials e ±i k ψ . ψ disappear in χ(N, r) and there is not dependence on the fifth coordinate ψ in this field. This fact is an evidence of that the field ϕ(N, r, ψ) propagates only on the 3D spatially isotropic space r(x, y, z), but not on the additional space-like coordinate ψ.

II. AN EFFECTIVE 4D MODEL OF EXPANSION FOR THE UNIVERSE
In order to develope an effective 4D model for an universe which is governed by the scalar field ϕ, we shall use the following change of variables on the metric (3) dt = ψ dN, dR = ψ dr, ψ = ψ so that the resulting metric becomes On hypersurfaces ψ = 1/H(t) the metric (19) becomes which also can be rewritten as an effective 4D metric which is well defined when 1 −  (3)], to the particular frame u R = 0 of the effective 4D metric (21). Here, U A = dx A dS are the penta velocities of the original metric (3) and u µ = dX µ ds are the tetra velocities of the effective 4D metric (21), such that g AB U A U B = 1 and g µν u µ u ν = 1, respectively. Furthermore, the density Lagrangian L of eq. (2) can be expanded as so that we can make the following identification for the scalar potential on the effective 4D metric (21): Using the change of variables (18), the equation (5) can be rewritten on the 4D effective submanifold (21), asφ with A. The effective 4D equation of state Using the effective 4D metric (21), we obtain the following Einstein's equations where ρ and p are, respectively, the energy density and the pressure. The effective 4D equation of state for the universe becomes p ρ = ω ef f (t) (28) with which, for a Hubble parameter H = p/t (with constant p) agrees exactly with that of a spatially flat 4D Friedmann-Robertson-Walker (FRW) metric ds 2 = dt 2 − e 2 Hdt dR 2 : ω ef f (t)| H=p/t = 2−3p 3p = ω F RW (t). Some particular cases p → ∞, p = 2/3 and p = 1/2, give us respectively ω ef f → −1, ω ef f = 0 and ω ef f = 1/3, which describes respectively expansions dominated by vacuum, matter and radiation. Note that for p > 1 the effective 4D metric (21) is always Lorentzian in nature. However for p = 1 this metric is not well defined. On the other hand, for p(t) < 1 the metric (21) is Euclidean and hence losses its relativistic nature. Hence, a well defined model for the expansion of the universe must be developed using p > 1 in the metric (21). For a more realistic model with a time dependent p(t), one obtains ω ef f (t)| H=p(t)/t = ω F RW (t). In the section (III) we shall study with more detail this last case.

B. Comoving frame
In order to describe the evolution of the universe on a comoving frame, which is the relevant for cosmological models, we can make use of the hyperbolic condition g µν u µ u ν = 1 on the 4D effective metric (21). Here, u µ = dx µ dS(t) (µ can take the values 0, 1, 2, 3), are the tetravelocities being u R = 0 in the comoving frame. In this frame the velocity u t is Note that when we obtain u t ≃ 1 and the metric (21) describes an asymptotic 4D FRW metric, which usually is used to describe the universe in cosmological models.

C. Evolution of the background field ϕ b
The effective 4D evolution of the background field ϕ b (t) is given by the equation (24) which has the general solution where ϕ b (0) = ϕ b (t = 0). It is evident that ϕ b (t) decreases monotonically with the time and rolls down the minimum of the potential, so that ϕ b (t → ∞) → 0.

III. MODEL OF EXPANSION OF THE UNIVERSE
In order to describe all the evolution of the universe we can propose the following expresion for the Hubble: H(t) = p(t)/t, such that where a = 1/6 10 30n G n/2 , b = 8/7 10 15n G n/4 , c = 2.0 10 −61 G −1/2 and n = 0.352. The general solution (33) in the example we are worked assume the explicit expression In the figure (1) is ploted the parameter p[x(t)] [where x(t) = log 10 (t)], which always remains with values p > 1. In the figure (2) is shown the evolution for the cosmological parameter w ef f [x(t)]. Note that for x(t) < 20 the universe is governed by vacuum and describes an inflationary expansion, but later w ef f [x(t)] increases to thereafter describe a phase with positive pressure: ω ef f > 0 (in the range 30 < x(t) < 60). After it, the pressure decreases and for x(t) > 60 the universe expands with negative pressure until the present day, when ω ef f [x ≃ 60.652] ≃ −0.7. This result agrees with the experimental data [15]. In the figure . It is very clear that the discrepance between ω ef f and ω F RW becomes more notorious in epochs where the universe expands with positive pressure. However, in both epochs (in the very early universe and the present day universe), the equation of state for the universe agrees with that predicted by a 4D spatially flat FRW metric: ds 2 = dt 2 − e 2 Hdt dR 2 . More exactly, the condition (31) holds for

IV. FINAL COMMENTS
In this letter we have studied the evolution of the universe which is considered as governed by a single scalar field from a 5D vacuum state, where the fifth dimension is considered as noncompact [see the metric (3)]. However when we make the change of variables (18) the universe describes an effective 4D evolution which can be described by the metric (21). This metric is well defined for Ḣ /H 2 2 = 1, which is the case we are considered in this letter. If we consider a Hubble paramter H(t) = p(t)/t, this metric remains Lorentzian for p > 1. In the example here considered, with a power law expansion (34), the universe initially describes an equation of state with ω ef f ≃ −1, which increases until take values of the order of ω ef f ≃ 0.4. Thereafter, this cosmological parameter begins to descrease to take values which agree very good with the present day experimental data ω ef f ≃ −0.7. This result is consistent with a quintessential expansion. The model predicts that, in the future, the universe will be more and more accelerated to finally describe an effective 4D vacuum dominated expansion ω ef f ≃ −1.