Scalar Metric ﬂuctuations in Space Time Matter inﬂation

Using the Ponce de Leon background metric, which describes a 5D universe in an apparent vacuum: ¯ G AB = 0, we study the eﬀective 4D evolution of both, the inﬂaton and gauge-invariant scalar metric ﬂuctuations, in the recently introduced model of space time matter inﬂation.

amplitude for the scalar metric perturbations on the evolution of the background Friedmann-Robertson-Walker (FRW) universe in a coordinate-independent manner at every moment in time.
On the other hand, cosmological theories with extra dimensions are already known to be of great importance in cosmology [9,10]. During the last years there were many attempts to construct a consistent brane world (BW) cosmology [11]. The induced-matter, or space-time-matter (STM) theory stands out for their closeness to the Einstein's project of considering matter and radiation as manifestations of pure geometry [12]. The basic extension of 4D Einstein theory and the low-energy limit of higher-dimensional theories is the modern incarnation of noncompact 5D Kaluza-Klein theory. In the STM theory, the conjecture is that ordinary matter and fields that we observe in 4D are induced geometrically by the extra dimension [13]. In this letter we develop a consistent first-order formalism to study the inflaton and scalar metric fluctuations [14] in STM inflation, which was recently introduced [15].

II. 5D FORMALISM
We consider a 5D background metric, which is 3D spatially isotropic, homogeneous and flat. In this paper we shall consider the background metric: dS 2 (b) =ḡ AB dx A dx B 1 , introduced by Ponce de Leon [9] where dr 2 = dx 2 +dy 2 +dz 2 and l is the fifth coordinate (which is space-like). The background metric (1) represents a 5D apparent vacuumḠ AB = 0, but however it is no flat. The absolute value for the determinant of the background metric tensorḡ AB is To describe the system, we consider the action where (5)R = 0 is the 5D background Ricci scalar for the background metric (1) and G = M −2 p is the Newton's constant (M p = 1.2 × 10 19 GeV is the Planckian mass). We shall use the unities c =h = 1, being c and h the speed of light and the Planck's constant. The Lagrangian density in (2) is given by 1 In our conventions, capital Latin indices run from 0 to 4.
which is only kinetic because we are dealing with a 5D free scalar field ϕ in an apparent vacuum state. Furthermore, g AB =ḡ AB + δg AB is the perturbed contravariant metric tensor andḡ AB is the background contravariant metric tensor. In this letter we shall consider δg AB as the scalar perturbations of the metric at first order, because we are dealing with weak gravitational fields. The diagonal perturbed metric [with respect to the background metric (1)], is where Φ(t, r, l), Ψ(t, r, l) and Q(t, r, l) are the scalar metric fluctuations. In particular for Φ = Ψ and Q = 2Ψ, we obtain the following line element: The contravariant metric tensor, after make a Ψ-first order approximation, is given by which can be written as g AB =ḡ AB + δg AB , beingḡ AB the contravariant background metric tensor. Furthermore, the Lagrangian (5) L = 1 2 g AB ϕ ,A ϕ ,B , with the metric fluctuations Ψ included in g AB , now can be written as where the fields ϕ and Ψ play the role of coordinates and g AB is given by (6). The relevant components of the linearized Einstein tensor are where the overstar denotes the derivative with respect to the fifth coordinate l. On the other hand, the energy momentum tensor can also be written as a linear one, with background componentsT AB , plus the first order perturbations δT AB : Notice that all the terms δT AB (with A = B) are zero, because δg AB is diagonal. The relevant perturbed components are which are nonlinear. In order to solve the perturbed and background Einstein equations, we consider the following semiclassical expansion for the inflaton field: ϕ(t, r, l) = ϕ (b) (t, l) + δϕ(t, r, l), being ϕ (b) the background inflaton field and δϕ the quantum fluctuations such that δϕ = 0. If we take into account this semiclassical expansion in in eqs. (13), (14) and (15), the linearized Einstein equations Combinding the eqs. (16), (17) and (18), we obtain an equation of motion for Ψ The Lagrange equation for Ψ: where (5) L. In absence of the fluctuations δϕ and Ψ, the eq. (20) give us Furthermore, the Lagrange equation for ϕ: Notice that the equations (20) and (22) are very difficult to be solved because ϕ and Ψ are quantum operators. However, the eq. (20) can be treated on the background [see eq. (21)] and the eq. (22) can be linearized using the semiclassical expansion ϕ(t, r, l) = ϕ (b) (t, l) + δϕ(t, r, l). The linearized Lagrangian equations take the form The only solution of eq. (23) on the manifold (1) that is solution of eq. (21) is ϕ (b) (t, l) = C, where C is a constant. Hence, the right side of eq. (24) becomes zero. If we take into account the expression (21), the eq. (19) holds In order to simplify the structure of this equation we propose the transformation Ψ(t, r, l) = We propose the following Fourier's expansion for χ χ(t, r, l) = 1 (2π) 3/2 d 3 k r dk l a krk l e i( kr. r+k l l) ξ krk l (t, l) + a † krk l e −i( kr. r+k l l) ξ * krk l (t, l) , where the creation and annihilation operators a krk l and a † krk l describe the algebra The dynamics of the modes ξ krk l (t, l) is given bÿ Using the transformation ξ krk l = e −i k l . lξ krk l , we obtain the following equation of motion for ξ krk lξ The problem with this equation is that it is not separable. However, as we shall see later, this equation can be worked in the limit p ≫ 1, which is relevant for inflation.
A. Inflationary case: p ≫ 1 In the limit p ≫ 1 the equation (29) can be simplified tö which, once normalized, has the following solution: where ν = √ and s 2 is a separation constant. Furthermore, the only solution of (23) that complies with the expression (21), is ϕ (b) = C, where C is a constant. The equation of motion for the inflaton field (24), for p ≫ 1, can be approximated tö We can make the transformation δϕ(t, r, l) = t 2(p−1) Π(t, r, l), so that the equation of motion for Π in the limit p ≫ 1, is The field Π can be expressed as a Fourier expansion in terms of its modes Π krk l (t, r, l) = e i( kr. r+k l l) θ krk l (t, l), such that the dynamics for θ krk l is described by the equation which has the following normalized solution Here, µ = √ , and (F 1 , E 1 , E 2 , q 2 = (p − 1) 2 s 2 ) are constants.

III. EFFECTIVE 4D DYNAMICS
We consider the background metric (1). If we take a foliation such that l = l 0 , the effective 4D background metric that results is and the effective 4D Lagrangian is (4) L(ϕ, ϕ ,µ ) = 1 2 g µν ϕ ,µ ϕ ,ν − V (ϕ), such that the effective 4D background potential V (ϕ (b) ) induced on the metric (36) is In the limit p ≫ 1, which is relevant for inflation, the metric (36) can be approximated to The effective 4D potential (37) can be founded by solving the effective equation of motion on the effective background metric (38). If we make ϕ (b) (t, l) = ϕ 1 (t)ϕ 2 (l), we obtain The general solutions for these equations are where A 1 , A 2 , B 1 , B 2 are constants of integration and m is a separation constant. If we choose A 1 = B 2 = m = 0, we obtain that ϕ 1 (t) = B 1 and ϕ 2 (l) = A 2 l 3p/(p−1) , such that . Therefore, the induced 4D background potential will be The metric (38), with the change of variables τ = l 0 t and R = l 0 r, becomes Due to the fact ∂ϕ (b) ∂τ = 0, the effective 4D background energy density is Here, the Hubble parameter H(τ ) = 1 a da dτ = p/τ , which is related to the scale factor a(τ ) = a 0 (τ /τ 0 ) 2p and ϕ (b) (τ, l = l 0 ) is a constant:

A. Effective 4D dynamics of metric fluctuations
Using the fact that the solution of eq. (25), in the limital case p ≫ 1, can be written as and that on the hypersurface l = l 0 one obtains Hence, the effective equation of motion for ψ(τ = l 0 t, R = l 0 r) on the hypersurface l = l 0 (which only is valid for p ≫ 1), will be when we have used B 1 = B 2 . The equation (51) can be simplified using the transformation χ(τ, R), such that the equation of motion for χ is The field χ(τ, R) can be expanded in terms of their modes χ k R (τ, R) = e i k R . Rξ k R (τ ), and the equation of motion for the τ -dependent modesξ k R (τ ), is The normalized solution for this equation is .

B. Energy density fluctuations
The energy density fluctuations on the effective 4D FRW metric is [14] δρ where the brackets < ... > denote the expectation value on the 3D hypersurface R(X, Y, Z). This approximation is valid during inflation on super Hubble scales. The amplitude for the 4D gauge-invariant metric fluctuations on cosmological scales is so that being ǫ ≃ 10 −3 a dimensionless constant and P Ψ (k R ) ∼ k 3−2ν R the power spectrum of Ψ 2 . It is known from experimental data [16], that the spectral n s index for this spectrum is where, in our case, n s = 4 − 8(p − 1) 2 s 2 − 16p 2 . Hence, from the condition (58), we obtain which provide us a cut for the separation constant s in terms of the value of the fifth coordinate l 0 . Note that for each possible foliation l (i) 0 there is a range of possible separation constants s (i) .

IV. FINAL COMMENTS
In this letter we have studied 4D gauge-invariant (scalar) metric fluctuations in space time matter inflation, using the 5D Ponce de Leon background metric. This metric describes a 5D universe in an apparent vacuumḠ AB = 0, but however it is not Riemann flat. In general (i.e., for an arbitrary power of expansion p), the equations of motion for δϕ and Ψ are not factorizable, so that its treatment is very difficult. However, in the limital case p ≫ 1 the treatment is possible because both, δϕ(t, r, l) and Ψ(t, r, l) can be written as product of functions of t, r and l (i.e., both functions are factorizable). In such a case we have found that Ψ and δϕ become independents (as in an effective 4D de Sitter expansion studied in [14]), because the effective 4D background field ϕ (b) becomes a constant of τ = l 0 t. This is not surprasing because the case p ≫ 1 describes an effective 4D asymptotic de Sitter expansion. In particular, using experimental values of n s (for the δρ/ρ-spectrum), we have found that the fifth coordinate used for the foliation l = l 0 could take values close to l 0 ≃ 1.55 in a scale invariant P δϕ -power spectrum.