Mauricio Bellini

We develop a nonperturbative quantum scalar field formalism from a noncompact Kaluza-Klein (KK) theory using the induced-matter theory of gravity during inflation. We study the particular case of a de Sitter expansion for the universe.


I. INTRODUCTION
The two current versions of 5D gravity theory are membrane theory [1][2][3] and inducedmatter theory [4]. In the former, gravity propagates freely into the bulk, while the interactions of particle physics are confined to a hypersurface (the brane). The induced-matter theory in its simplest form is the basic Kaluza-Klein (KK) theory in which the fifth dimension is not compactified and the field equations of general relativity in 4D follow from the fact that the 5D manifold is Ricci-flat; the large extra dimension is thus responsible for the appearance of sources in 4D general relativity. Hence, the 4D world of general relativity is embedded in a 5D Ricci-flat manifold. An interesting result of the induced-matter theory is that if ds 2 = g µν (x)dx µ dx ν is the 4D metric of any matter-free spacetime in general relativity, the dS 2 = (ψ/ψ 0 ) 2 ds 2 − dψ 2 is the metric of a 5D manifold that is Ricci-flat [5,6].
During the last two decades the inflationary paradigm has become an almost universally accepted scenario to explain the observed large scale flatness and homogeneity of the universe [7]. In particular, stochastic inflation [8][9][10][11] (or, in general, for a semiclassical treatment for the inflaton field during inflation) has been subject of great interest in the last years. However, one of the problems with this approach is that one must to make a perturbative expansion of the scalar field potential in terms of the quantum fluctuations of the inflaton field to finally give a solution for a first order expansion in the equation of motion for these fluctuations [12]. This is a good approximation because such that fluctuations are small during inflation on cosmological scales. However, the predictions of the inflationary theory could be significatively improved by using a nonperturbative calculation for the inflaton field ϕ. Of course, it is impossible to make from a 4D quantum field formalism, but could be developed from a scalar quantum (inflaton) field in a 5D vacuum state with a purely kinetic density Lagrangian. The aim of this work consists to develop a nonperturbative scalar quantum field theory from a 5D apparent vacuum state defined as a purely kinetic 5D density Lagrangian of a scalar field minimally coupled to gravity in a 5D Ricci-flat canonical metric [13]. To make it, we consider the 5D canonical metric [14] where dr 2 = dx 2 + dy 2 + dz 2 . Here, the coordinates (N, r) are dimensionless and the fifth coordinate ψ has spatial unities. We shall assume in what follows that the extra dimension is spacelike and that the universe is 3D spatially flat, isotropic and homogeneous. The metric (1) describes a flat 5D manifold in apparent vacuum (G AB = 0). We consider a diagonal metric because we are dealing only with gravitational effects, which are the important ones during inflation. To describe neutral matter in a 5D geometrical vacuum (1) we can consider the Lagrangian where | (5) g| = ψ 8 e 6N is the absolute value of the determinant for the 5D metric tensor with components g AB (A, B take the values 0, 1, 2, 3, 4) and | (5) g 0 | = ψ 8 0 e 6N 0 is a constant of dimensionalization determined by | (5) g| evaluated at ψ = ψ 0 and N = N 0 . In this work we shall consider N 0 = 0, so that (5) Here, the index "0" denotes the values at the end of inflation. Furthermore, we shall consider an action where ϕ is a scalar field minimally coupled to gravity and G is the gravitational constant. Furthermore, (5) R is the 5D Ricci scalar, which of course, is cero for the 5D flat metric (1). Since the 5D metric (1) describes a manifold in apparent vacuum, the density Lagrangian L in (2) must to be which represents a free scalar field. In other words, we define the vacuum as a purely kinetic 5D-lagrangian on a globally 5D-flat metric [in our case, the metric (1)]. To describe the metric in physical coordinates we can make the following transformations: such that we obtain the 5D metric where t is the cosmic time and R 2 = X 2 + Y 2 + Z 2 . This metric is the Ponce de Leon one [15], and describes a 3D spatially flat, isotropic and homogeneous extended (to 5D) FRW metric in a de Sitter expansion [4].

II. QUANTUM FIELD THEORY IN A 5D APPARENT VACUUM
Taking into account the metric (1) and the Lagrangian (2), we obtain the equation of motion for ϕ where ∂N ∂ψ is zero because the coordinates (N, r, ψ) are independents. Hence, we obtain where the overstar denotes the derivative with respect to N and ϕ ≡ ϕ(N, R, ψ). To simplify its structure we can make the transformation ϕ = χe −3N/2 ψ 0 ψ 2 , so that we obtain the 5D generalized Klein-Gordon like equation for the redefined field χ(N, r, ψ): The field χ can be written in terms of a Fourier expansion where the asterisk denotes the complex conjugate and (a † krk ψ ,a krk ψ ) are the creation and annihilation operators such that Furthermore, the commutation relation between χ and ⋆ χ is In order to the commutation equation (12) holds the following renormalization condition must to be fulfilled: Hence, the equation for the modes ξ krk ψ (N, ψ) that complies with the condition (13) in a 4D de Sitter expansion will be The general solution for this equation is where . Furthermore, the functions G 1 (ψ) and G 2 (ψ) are arbitrary functions constrained by the renormalization condition (13) In this paper we shall choose the generalized Bunch-Davis vacuum: The squared ϕ fluctuations are given by A k r -scale invariant power spectrum results from ν = 3/2, for which

III. 4D DE SITTER EXPANSION
We can take a foliation ψ = ψ 0 in the metric (5), such that the effective 4D metric results which describes 4D globally isotropic and homogeneous expansion of a 3D spatially flat, isotropic and homogeneous universe that expands with a Hubble parameter H = 1/ψ 0 (in our case a constant) and a 4D scalar curvature (4) R = 6(Ḣ + 2H 2 ). Note that in this particular case the Hubble parameter is constant so thatḢ = 0. The 4D energy density ρ and the pressure p are [14] 8πG where G = M −2 p is the gravitational constant and M p = 1.2 10 19 GeV is the Planckian mass. Furthermore, the universe describes a vacuum equation of state: p = −ρ, such that where the brackets denote the 4D expectation vacuum and the cosmological constant Λ gives the vacuum energy density ρ = Λ 8πG . Thus, Λ is related with the fifth coordinate by means of Λ = 3/ψ 2 0 [15]. Furthermore, the 4D Lagrangian is given by where the effective potential for the 4D FRW metric [13], is In our case this potential takes the form where k ψ 0 is the wavenumber for ψ = ψ 0 . Furthermore the effective 4D motion equation for ϕ isφ which means that the effective derivative (with respect to ϕ) for the potential, is Now we can make the following transformation: Note that now ϕ ≡ ϕ( R = ψ 0 r, t = ψ 0 N, ψ = ψ 0 ) = e −3t/(2ψ 0 ) χ( R, t), where [see eq. (9)] χ( R, t) = χ( R = ψ 0 r, t = ψ 0 N, ψ = ψ 0 ): Hence, we obtain the following 4D Klein-Gordon equation for χ The equation of motion for the time dependent modes ξ k R k ψ 0 (t) will bë where the effective squared mass µ 2 of the modes ξ k R k ψ 0 is given by Here, α describes the self-interaction squared mass of the redefined inflaton field χ due to the expansion of the universe, and the term 9 4ψ 2 0 represents its bare squared mass due to the expansion. Note that both terms in (33) has a geometrical origin because they are induced by the fifth coordinate.
The general solution of eq. (32) is Here, (3−ns) 2 = 9 4 − (2 + k 2 ψ 0 ψ 2 0 ) and n s is the spectral index of the ϕ 2 -spectrum on super Hubble (SH) scales when it is considered the Bunch-Davis vacuum: where A(t) is a time dependent function, k 0 (t) = e t/ψ 0 ψ 0 9 4 − αψ 2 0 , ǫ ≪ 1, P(t, k) is the power of the spectrum and k * is the absolute value for the wave vector related to the physical wavelength at the moment the horizon entry. This maximal physical scale is supported by causal arguments. Note that this result corresponds to an effective self-interaction squared mass α for the inflaton field when the standard semiclassical approach in a de Sitter expansion is considered [12]. For n s ≪ 1, it can be approximated to n s ≃ 2(2+k 2 ψ 0 It is well known that the universe has a power spectrum which is very close to a scale invariant one on cosmological scales [16], so that the equation (37) is a good approximation.
In such that case we obtain the following valued expressions for V (ϕ) and V ′ (ϕ): where the equation (39) corresponds to α = 0 in (31). Therefore, α becomes zero for a super Hubble scale invariant power spectrum of ϕ 2 . Note that this result disagrees with the result obtained using a semiclassical 4D treatment for the inflaton field in a de Sitter expansion [12]. Furthermore, we can see from eq. (39) that the effective 4D parameter of mass for ϕ is nonzero. It describes the inflaton ϕ-squared mass geometrically induced by the fifth coordinate ψ on the hypersurface ψ = ψ 0 , when the self-interaction is ausent in a de Sitter expansion: α = 0. It is easy to see that m 2 ef f > µ 2 | α=0 . However, both masses has a different origin, because µ 2 is the squared mass related to each ξ k R k ψ 0 mode of χ and m 2 ef f is the effective squared mass of the nonperturbative field ϕ( R, t, ψ 0 ), with back-reaction effects included.

A. Energy density fluctuations
Once the time dependent modes ξ k R k ψ 0 are known, we can to obtain the effective 4D expectation value for the energy density where k p = G −1/2 is the absolute value of the Planckian wave vector and k * is the inverse of the Hubble's radius at the moment (t = t * ) the horizon entry.
On the other hand, if we define ρ (0) ψ=ψ 0 the k R -zero mode 4D expectation value for the energy density on the effective 4D spatially isotropic and homogeneous (de Sitter) FRW metric (19), hence the energy density fluctuations (on arbitrary scales) will be

B. Super Hubble energy density fluctuations
It is well known that the universe is nearly scale invariant on cosmological scales [16]: |n s | ≪ 1. We shall consider this range for the spectrum on the range k * < k R < ǫk 0 (t). In order to compute the SH density energy fluctuations we can remember the small-argument limit for the second kind Hankel function: H (2) ν [x] ≃ − i π Γ(ν)(x/2) −ν , for ν > 0, which in our case is valid to describe the super Hubble (cosmological) ϕ-field time dependent modes ξ krk ψ 0 in eq. (34). With this approach, we obtain

IV. FINAL COMMENTS
We have developed a 5D nonperturbative quantum scalar field formalism and studied the particular case of a 4D de Sitter inflationary expansion in the framework of the inducedmatter theory developed by Wesson and co-workers. The results are very interesting, because we note some differences with respect to whose obtained by means of the semiclassical expansion in a de Sitter expansion for the universe. 1) First all, the effective scalar 4D potential appears to be quadratic in ϕ (see eq. (25), meanwhile in the standard semiclassical treatment is a constant V 0 (see for example [12]). 2) The spectral index n s < 1/60 depends on the self-interaction squared mass α (which should be very small if the slow-roll conditions hold), but not on the mass of the inflaton field m ef f (as in the standard semiclassical approach to inflation). Notice this result suggests that k ψ 0 ≃ i √ 2/ψ 0 = i √ 2H. 3) Furthermore, the Hubble parameter in a de Sitter expansion (as in the semiclassical approach to inflation) is a constant, but in our case, its value being given by the inverse of the fifth coordinate H = 1/ψ 0 for a foliation ψ = ψ 0 of the 5D Ricci-flat metric (5).
Finally, the nonperturbative treatment for ϕ here developed can be extended to other (more realistic) inflationary models with time dependent Hubble parameters and, more generally, to cosmological models dinamically governed by scalar fields.