Decaying cosmological parameter in the early universe from NKK theory of gravity

Using a formalism recently introduced we study the decaying of the cosmological parameter during the early evolution of an universe, whose evolution is governed by a vacuum equation of state. We use a stochastic approach in a nonperturbative treatment of the inflaton field from a Noncompact Kaluza-Klein (NKK) theory, to study the evolution of energy density fluctuations in the early universe.


I. INTRODUCTION
Cosmological observations imply that there exists an extremely small upper limit on the vacuum energy density in the present state of our universe. This stands in sharp contradiction with theoretical predictions [1]. In fact, any mass scale in particle physics contributes to the vacuum energy density much larger than this upper bound [2]. According to modern quantum field theory, the structure of the vacuum is turned out to be interrelated with some spontaneous symmetry-breaking effects through the condensation of quantum scalar fields. This phenomenon gives rise to a non-vanishing vacuum energy density ρ vac ∼ M 4 p (M p = G −1/2 is the Planckian mass and G is the gravitational constant). The appearance of this characteristic mass scale may have an important effect on the cosmological constant because it receives potential contributions from this mass scale due to mass spectrum of corresponding physical fields in quantum field theory. By taking into account this contribution, an effective cosmological constant is defined as the sum of the bare cosmological constant λ and 8πGρ vac [3]. This type of contribution gives rise to an immediate difficulty called the cosmological constant problem. There are some possible solutions to this problem rendering Λ exactly or almost vanishing [4]. One of them consists to find some relaxation mechanism by which Λ could relax to its present day small value [5,6]. A credible mechanism for obtaining such a decay already exists, which is to assume the existence of a scalar field presently relaxing towards the minimum of its potential. Scalar fields are not only predicted to exist by some particle phsics theories that go beyond the Standard Model, but are also the most plausible engine behind a possible inflationary period in the very early universe [8][9][10][11]. In this work we shall study a possible mechanism for a decaying cosmological parameter from the KK formalism, but by considering the extra (spatial-like) dimension ψ as noncompact [12,13]. This theory, also called induced-matter theory is, in its simplest form, the basic 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. Thus the large extra dimension is 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. There has recently been an uprising interest in finding exact solutions of the KK field equations in 5D, where the fifth coordinate is considered as noncompact. 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, but also Riemann flat. This is because it is possible to have a flat 5D manifold which contains a curved 4D submanifold, as implied by the Campbell theorem. So, the universe may be "empty" and simple in 5D, but contain matter of complicated forms in 4D [14].
In this work we use a stochastic approach to study the dynamics of the inflaton field in the early universe, which is governed by a 4D vacuum equation of state p vac = −ρ vac = − Λ 8πG , being Λ the time dependent (decaying) cosmological parameter. To make it we shall use a 5D canonical metric which is Riemann flat (R A BCD = 0) and describes a 5D apparent vacuum (G AB = 0). To describe the system we shall propose a 5D action for a purely kinetic inflaton field which is minimally coupled to gravity.

II. 5D FORMALISM
We consider the 5D canonical metric where dr 2 = dx 2 + dy 2 + dz 2 , being x, y, z dimensionles spatial coordinates. Furthermore, t and ψ has spatial units (in this paper we shall consider c =h = 1. 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) is flat R A BCD = 0 and describes a 5D manifold in apparent vacuum (G AB = 0) and is a special case of the much-studied class of canonical metrics dS 2 = ψ 2 g µν dx µ dx ν − dψ 2 [15][16][17].
To describe neutral matter in a 5D geometrical vacuum (1) we can consider the Lagrangian where | (5) g| = ψ 8 (a/a 0 ) 6 , is the absolute value of the determinant for the 5D covariant metric tensor with components g AB (A, B take the values 0, 1, 2, 3, 4) and | (5) g 0 | = ψ 8 0 is a constant of dimensionalization determined by | (5) g| evaluated at ψ = ψ 0 and a 0 = a(t = t 0 ). Here, a(t) the scale factor of the universe such thatȧ/a = Λ/3. To describe the system we consider an action where ϕ is a scalar field minimally coupled to gravity. Furthermore, (5) R = 0 is the 5D Ricci scalar.
To describe the apparent vacuum, we shall consider the density Lagrangian L in (2) must to be for a free scalar field. The dynamics for ϕ(t, r, ψ) being given by the equation On the other hand, ϕ complies with the commutation expression where Π t = ∂L ∂ϕ,t = 3 Λψ 2φ and a 0 is the scale factor of the universe when inflation starts. As can be demonstrated ϕ(t, r, ψ) = e such that and ξ krk ψ (t, ψ) = e −i k ψ . ψξ krk ψ (t). The commutator (7) is satisfied for a krk ψ , a †

III. 4D DYNAMICS
To describe the 4D dynamics we can make a foliation on ψ = ψ 0 on the line element (1), such that the effective 4D metric holds: dS 2 | ef f = ds 2 , where In this section we shall study the dynamics described by the inflaton field ϕ, making emphasis on the long wavelength section, which describes this field on cosmological scales.

B. 4D stochastic dynamics of χ on cosmological scales
In order to describe separately the long and short wavelength sectors of the field χ we can define the fields χ L (t, r) and χ S (t, r) where c.c denotes the complex conjutate and k 0 (t) = e √ Λ/3dt 1 . The field that describes the dynamics of χ on the infrared sector (k 2 r ≪ k 2 0 ) is χ L . Its dynamics obeys the Kramers-like stochastic equation where and the stochastic operators η, κ and γ are This second order stochastic equation can be rewritten as two coupled stochastic equationṡ where we have introduced the auxiliar field u =χ L − ǫk 0 γ. The condition to can neglect the noise γ with respect to η, is˙ξ The Fokker-Planck equation for P (χ where D 11 (t) = ǫ 3k 0 k 2 0 4π 2 ξ ǫk 0 2 and P (χ L , u (0) ) to (χ L , u). Furthermore, ǫ ≃ 10 −3 is a dimensionless constant such that on cosmological scales holds k r /k 0 < ǫ. Hence, the equation of motion for χ 2 L = dχ L duχ 2 L P (χ L , u) will be d dt To describe the dynamics of the squared ϕ L -expectation value we return to the original field (26) can be rewritten as which has the following solution where ϕ 2 L 0 is a constant of integration. In order to understand better this result in the context of the inflaton field fluctuations φ( r, t), we can make the following semiclassical approach: where ϕ = φ c (t) and φ = 0. With this representation one obtains where φ c (t) is the solution of the zero mode equation in (12) The solutions of physical interest for φ c (t) should be decreasing with time, so that after inflation ends φ c (t → ∞) → 0. Hence, after inflation one obtains the following result: which means that for t t 0 ≫ 1 the following approximation is fulfilled: Furthermore, we can estimate the amplitude of density energy fluctuations on cosmological scales which is valid on cosmological scales. For late times one obtains φ 2 L IR ∼ t −1 , independently of the power p. The evolution of φ c (t) for this model is where φ 0 is φ c (t = t 0 ) and (A, B) are dimensionless constants such that A + B = 1. Note that φ c is monotonically decreasing. Finally, the energy density fluctuations for late times are which for very large p go as δρ

V. FINNAL COMMENTS
We have studied a cosmological model for the early universe from a NKK theory of gravity with a space-like extra dimension, where the cosmological parameter decreases with time. We have worked a stochastic treatment for the effective 4D inflaton field without the hypothesis of a slow -roll regime. Hence, the dynamics the field on large scales is described by a second order stochastic equation. In this framework the long -wavelength modes of the inflaton field reduces to a quantum system subject to a quantum noise which is originated by the short -wavelength sector. In this approach, the effective 4D potential is quadratic in ϕ and has a geometrical origin. As in STM theory [14] of gravity 4D source terms are induced from a 5D vacuum and the fifth dimension (here a space-like one) is noncompact. In our theory the 5D vacuum is represented by a 5D globally flat metric (which describes a 5D apparent vacuum G AB = 0) and a purely kinetic density Langrangian for a quantum scalar field minimally coupled to gravity.
In the example here studied for Λ = 3p 2 t −2 , we obtain that the energy density fluctuations decrease monotonically with time. In particular, we obtain that for very large p these fluctuations go as t −p .