Quintessential inflation from a variable cosmological constant in a 5D vacuum

We explore an effective 4D cosmological model for the universe where the variable cosmological constant governs its evolution and the pressure remains negative along all the expansion. This model is introduced from a 5D vacuum state where the (space-like) extra coordinate is considered as noncompact. The expansion is produced by the inflaton field, which is considered as nonminimally coupled to gravity. We conclude from experiental data that the coupling of the inflaton with gravity should be weak, but variable in different epochs of the evolution of the universe.


I. INTRODUCTION
Recent years have witnessed a large amount of interest in higher-dimensional cosmologies where the extra dimensions are noncompact. A popular example is the so-called Randall-Sundrum Brane World (BW) scenario [1]. Particular interest revolves around solutions which are not only Ricci flat, but also Riemann flat (R A BCD = 0), where the vanishing of the Riemann tensor means that we are considering the analog of the Minkowsky metric in 5D. An achievement of this theory is that all the matter fields in 4D can arise from a higher-dimensional vacuum. One starts with the vacuum Einstein field equations in 5D and dimensional reduction of the Ricci tensor leads to an effective 4D energy-momentum tensor [2]. For this reason the Space-Time-Matter (STM) theory is also called Induced-Matter (IM) theory. BW and IM theories may appear different, but their equivalence has been recently shown by Ponce de Leon [3].
The potential energy of the scalar field and/or the presence of a variable cosmological term could drive inflation, resolving puzzles such as the monopole, horizon and flatness problems [4]. The variable cosmological term has also been mentioned as a possible solution to the cosmological "constant" problem [5] and, most recently, as a candidate for the dark matter (or quintessence) making up most of the Universe [6]. A mechanism for obtaining the decay of the cosmological parameter consists in relax Λ to its small present day value [7,8,9].
In this letter we are aimed to study the evolution of the universe which is governed by a variable cosmological constant (Λ < 0) in a 5D vacuum state, such that the expansion of the universe is due to a scalar field (the inflaton field) coupled to gravity. However, on an effective 4D metric the universe evolves with an equation of state with negative pressure p (but with p ≥ −ρ). This kind of expansion is the well known quintessential inflation [10].
To describe a scalar field ϕ, which is nonminimally coupled to gravity in a 5D vacuum state, we consider the metric [11] with the action Here, Λ(t) is the decaying cosmological constant (Λ < 0), G = M −2 p is the gravitational constant (M p = 1.2 × 10 19 GeV is the Planckian mass), ξ is the coupling of ϕ with gravity and (5) R is the Ricci scalar, which is zero on the R A BCD = 0 (flat) metric (1). This metric is also 3D spatially isotropic, homogeneous and flat. In such metric dr 2 = dx 2 + dy 2 + dz 2 , ψ describes the fifth space-like coordinate and t is the cosmic time. The Lagrange equations give us the equation of motion for φ such that the last term in (3) is zero, because the metric (1) is flat. Furthermore, the commutation expression between ϕ and Π t = ∂L ∂ϕ,t = 3 Λψ 2φ , is where a 0 is the scale factor of the universe when inflation starts. The field ϕ(t, r, ψ) can be transformed as such that, due to the fact ∂ϕ ∂ψ = − 2 ψ ϕ, the equation (3) holds Using the transformation (5), we obtain the equation of motion for the field χ The field χ can be written as a Fourier expansion such that and ξ krk ψ (t, ψ) = e −ik ψ ψξ krk ψ (t). The commutator (9) is satisfied for = 0, if the following condition holds: whereξ krk ψ (t) satisfies the following equation of motion: Hence, since ξ krk ψ (t, ψ) = e −ik ψ ψξ krk ψ (t), the expansion (8) now can be written as being c.c the complex conjugate.

II. EFFECTIVE 4D DYNAMICS
We consider the metric (1). On the hypersurface ψ = 3 Λ(t) , the effective 4D metric that results is so that the effective 4D action for this metric is where 1/2 dt and ξ ef f is the effective coupling of ϕ with gravity on the metric (13). Moreover, (4) g is the determinant of g µν and (4) R is the Ricci scalar corresponding to the effective 4D metric (13). For this metric, we adopt the comoving frame given by the effective 4D velocities The condition for the metric (13) to be Lorentzian is g tt > 0, which is valid when Hence, in this letter we shall consider the case for which the condition (16) complies along all the expansion of the universe. The effective 4D Ricci scalar for the metric (13) is From eqs. (13) and (14), we obtain the equation of motion for ϕ(t, r) where Furthermore, we can make the following identification in eq. (18): From the action (14), the effective 4D scalar potential can be identified as Note that the effective 4D potential is due to the coupling of ϕ with gravity. On the other hand the additional kinetic term [ 1 2 g ψψ (ϕ ,ψ ) 2 ] in the 5D action (2) has a dissipative effect in the 4D equation of motion (18). The commutation relation between ϕ and Π t = ∂ (4) L ∂ϕ,t = g tt ϕ ,t is ϕ(t, r), Π t (t, r ′ ) = we obtainχ where tt R, being .
The field χ can be expanded in terms of their modes a kr e i kr. r η kr (t), such that the equation of motion of the time-dependent modes η kr (t) arë Hence, from eq. (23) and eq. (24), we obtain The effective 4D equation of state is p = ω ef f (t)ρ (p and ρ are respectively the pressure and the energy density), with ω ef f (t) = − 1 3 Note that whenΛ = 0, one obtains (4) R = 4Λ and the metric (13) describes exactly an effective FRW metric with a pressure p v = −ρ v = − Λ 8πG , in a de Sitter expansion.

III. AN EXAMPLE
We consider the case where Λ(t) = 3p 2 (t)/t 2 , such that p(t) is given by where a = 1 6 10 30n G n/2 , b = 8 7 10 15n G n/4 , C = 2 × 10 −61 G −1/2 and n = 0.352. There are at least four significative periods that we can identificate in this model.  In the following subsections we shall study with more detail these different epochs for the evolution of the universe.
A. Early (de Sitter) inflationary period: Λ ≃ Λ 0 The early inflationary period in which p(t) ≫ 1, can be approximated to a nearly de Sitter expansion whereΛ 2 /Λ 3 ≪ 1 and hence Λ ≃ Λ 0 ≃ 3p 2 /t 2 p . In this epoch, which describes the expansion of the universe for t ≪ 10 10 t p (in our model), the general solution is given by where ν 1 = 9 − 48ξ ef f /2 and y 1 (t) = k r e −( Λ 0 3 ) The normalized solution with A 1 = 0 and The power spectrum for the squared ϕ-fluctuations calculated on scales k r ≫ e q Λ 0 3 t (super Hubble scales), is It implies that this spectrum should be scale invariant only for ξ ef f = 0. In particular, we can calculate the range of validity for ξ ef f by comparing the spectrum (33) with observational data [12], for the spectral index n s n s = 0.97 ± 0.03.

D. Asymptotic de Sitter expansion
In our model the final asymptotic expansion of the universe can be approximated to a nearly de Sitter expansion whereΛ 2 /Λ 3 ≪ 1 such that Λ ≃ Λ f ≃ 3C 2 , where we are considering C = 2 × 10 −61 G −1/2 in the power (30). In this epoch, which describes in our model the expansion of the universe for t > 10 62 t p , the general solution is such that ν 4 = 9 − 48ξ ef f /2 and y 4 (t) = k r e −Ct .
The normalized solution with B 1 = 0 and The power of the spectrum for the squared ϕ-fluctuations on scales k r ≫ e Ct , is such that this spectrum become scale invariant for ξ ef f = 0. Finally, from (34) we obtain − 0.08 < ξ ef f < 0. (51)

IV. FINAL COMMENTS
In this letter we have studied a model which describes all the expansion of the universe governed by a decreasing cosmological constant from a 5D vacuum state. When we take a foliation on the fifth (space-like) coordinate ψ(t) = Λ(t) 3 , the effective 4D dynamics describes an universe which has a 4D equation of state p = ω ef f ρ, with ω ef f < 0. In this model, the expansion of the universe is due to the inflaton field, which is considered as nonminimally coupled to gravity. We have calculated the spectrum for the inflaton field fluctuations on cosmological scales in four different epochs of its evolution. • In the early inflationary expansion ω ef f ≃ −1 and we obtain that the spectrum of ϕ 2 | IR is nearly scale invariant for −0.08 < ξ ef f < 0.
• The third epoch describes the present day universe (which is considered as 1.5 × 10 10 years old), with ω ef f ≃ −0.68. For the spectrum of ϕ 2 | IR to be nearly scale invariant we obtain that the coupling must be −0.06 < ξ ef f < 0.08.
In view of these results, we conclude that ξ ef f cannot be constant along the evolution of the universe. However, ξ ef f should be very weak. In particular, in the present day quintessential epoch, the experimental data suggests that the coupling should be nearly zero (−0.06 < ξ ef f < 0.08). On the other hand, during the early and future inflationary expansions, observation suggests that ξ ef f should be negative, but during the (asymptotic) matter dominated epoch the coupling should be positive.