Primordial dark energy from a condensate of spinors in a 5D vacuum

We explore the possibility that the expansion of the universe can be driven by a condensate of spinors which are free of interactions on a 5D relativistic vacuum defined on an extended de Sitter spacetime which is Riemann-flat. The extra coordinate is considered as noncompact. After making a static foliation on the extra coordinate, we obtain an effective 4D (inflationary) de Sitter expansion which describes an inflationary universe. We found that the condensate of spinors here studied could be an interesting candidate to explain the presence of dark energy in the early universe. The dark energy density which we are talking about is poured into smaller sub-horizon scales with the evolution of the inflationary expansion.


I. INTRODUCTION
Modern versions of 5D General Relativity abandon the cylinder and compactification conditions used in original Kaluza-Klein (KK) theories, which caused problems with the cosmological constant and the masses of particles, and consider a large extra dimension. The main question that these approaches address is whether the four-dimensional properties of matter can be viewed as being purely geometrical in origin. In particular, the Induced Matter Theory (IMT)[1] is based on the assumption that ordinary matter and physical fields that we can observe in our 4D universe can be geometrically induced from a 5D Ricci-flat metric with a space-like noncompact extra dimension on which we define a physical vacuum. The Campbell-Magaard Theorem (CMT) [2] serves as a ladder to go between manifolds whose dimensionality differs by one. Due to this theorem one can say that every solution of the 4D Einstein equations with arbitrary energy momentum tensor can be embedded, at least locally, in a solution of the 5D Einstein field equations in a relativistic vacuum: G AB = 0 1 . Due to this fact the stress-energy may be a 4D manifestation of the embedding geometry and therefore, by making a static foliation on the space-like extra coordinate of an extended 5D de Sitter spacetime, it is possible to obtain an effective 4D universe that suffered an exponential accelerated expansion driven by an effective scalar field with an equation of state typically dominated by vacuum [4][5][6][7]. An interesting problem in modern cosmology relies to explain the physical origin of the cosmological constant, which is responsible for the exponential expansion of the early inflationary universe. The standard explanation for the early universe expansion is that it is driven by the inflaton field [8]. Many cosmologists mean that such acceleration (as well as the present day accelerated expansion of the universe) could be driven by some exotic energy called dark energy. Most versions of inflationary cosmology require of one scalar inflaton field which drives the accelerated expansion of the early universe with an equation of state governed by the vacuum [9]. The parameters of this scalar field must be rather finely tuned in order to allow adequate inflation and an acceptable magnitude for density perturbations. The need for this field is one of the less satisfactory features of inflationary models. Consequently, we believe that it is of interest to explore variations of inflation in which the role of the scalar field is played by some other field [10,11]. Recently has been explored the possibility that such expansion can be explained by a condensate of dark spinors [12]. This interesting idea was recently revived in the framework of the Induced Matter Theory (IMT) [13]. In this work we shall extend this idea.

II. THE EFFECTIVE LAGRANGIAN IN 5D RIEMANN-FLAT SPACETIME
We are concerned with a 5D Riemann-flat spacetime with a line element given by: where t, x, y, z are the usual local spacetime coordinate system and ψ is the noncompact space-like extra dimension.
We start from an effective Lagrangian density for non massive fermions in 5D: At this point it is easy to obtain the equations of motion from a variational principle.
The Euler-Lagrange equations for both Ψ and Ψ can be obtained making the functional derivatives: where the effective current J M N P is symmetric with respect to permutations of M and N and f N P = γ N , γ P is antisymmetric [14]. At this point we are in conditions of introduce the stress tensor T M N = 2 Applying the compatibility condition on the metric ∇ C g AB = 0, we obtain we obtain that the equation for the spinor Ψ takes the form The same procedure yields an identical equation for the field Ψ . On the other hand, the 5D Einstein equations for the Riemann-flat metric (1), is where 0 |T AB | 0 denotes the expectation value of T AB in the vacuum state |0 >.

A. Tensorial Formulation of the Equation of Motion
Using the formalism previously introduced, the double-Nabla can be expressed explicitly: where the spin connection is ω ab Thus, after replacing the last expression in the equation of motion (8) we obtain 2 The tensors can be written using the vielbein e A a and its inverse e a A , such that, if e A a e b A = δ b a and where η ab is the 5D Minkowsky tensor metric with signature (+, −, −, −, −).
Here, we have made use of the fact that and ω ab M γ a γ b = ω ab M σ ab . Once we simplify some terms, we obtain Finally, the equation of motion for the spinors assumes its final form which is very difficult to be resolved because the fields are coupled.

B. Conformal Mapping Based Solution
In order to simplify the structure of the equation (12), we shall introduce the following transformation on the spinor components: where components are grouped as With this representation we obtain the equation of motion for ϕ 1 and ϕ 2 Here, we have adopted the following conventions: Now we can use the conformal mapping defining new complex fields Φ + = ϕ 1 + iϕ 2 and Rewriting the equations (13) and (14) in terms of these new fields, it is possible to decouple the first equation, while that the other coupling becomes a source for the second equation Then, after few calculations, the Lagrangian density written in terms of the new fields takes the form or, alternatively, can be written as a function of the pair ϕ 1 = 1 On the other hand the 5D Energy-Momentum (EM) tensor is represented by (5) T AB = 2 δL ef f δg AB − g AB L ef f . This procedure take place in a 5D vacuum. Therefore, the effective Lagrangian and the EM tensor are involved directly with the cosmological observables we wish to evaluate. The observables to which we refer are energy density and pressure. Both come from the diagonal part of the EM tensor.

C. Extra dimensional solution for Φ +
We shall use the variable separation method to the homogeneous PDE (15), we obtain the following set of ODE's: The equation (18) has a solution that can be written in terms of plane wavefront The second equation (19) has a general solution Since we are interested in "localized" static solutions, i.e. those that decay to zero when ψ tends to infinity, we must choose C 2 = 0, so that n ≡ ( ≥ 0, with n ≥ 3 and n ∈ R, in order to 1 + M 2 1 ψ 2 0 ≥ 0. Using the last expression in eq. (16), we obtain a degenerate two-component system 3 for the spinor Φ −,κ . Again, a plane wavefront satisfies the spatial part. By inserting Φ −,κ = G κ (t, ψ)e i κ· r , and multiplying by ψ ψ 0 2 , we obtain This inhomogeneous PDE can be converted to one with a constant coupling. In order to make constant the right side of eq. (24), we shall propose Finally, we must solve the equation (25)

III. EFFECTIVE DYNAMICS ON THE 4D HYPERSURFACE ψ = 1/H 0
In order to describe the effective 4D dynamics of the physical system in the early inflationary universe with an effective 4D de Sitter expansion, we shall consider a static foliation on the 5D metric (1). The resulting 4D hypersurface after making the static foliation ψ = ψ 0 = 1/H 0 , describes an effective 3D spatially flat, isotropic and homogeneous de Sitter four-dimensional expanding universe with a constant Hubble parameter H 0 , with a line element From the relativistic point of view, an observer who moves in a co-moving frame with the fivevelocity U ψ = 0 on a 4D hypersurface with a scalar curvature (4) R = 12/ψ 2 0 = 12 H 2 0 , such that the Hubble parameter H 0 , and thus also the cosmological constant: Λ 0 = 3H 2 0 /(8πG), are defined by the foliation H 0 = ψ −1 0 .

A. Time dependent modes of Φ +
The solution for the time-dependent equation (20) can be expanded in terms of first and second kind Hankel functions where ν = 4 + M 2 1 ψ 2 0 ≥ 2. After make a Bunch-Davies normalization of the modes [15] we obtain the solution Since we are interested to describe the universe on super-Hubble cosmological scales we must require κ ψ 0 e − t ψ 0 ≪ 1, we reject solutions that goes to zero at late times. The asymptotic behavior of T (+) κ (t) on cosmological scales will be Finally, the degenerate two-component spinor Φ + can be expanded as a function of the and their complex conjugated.  , for n ≥ 3, such that (m, n) ∈ Z. After take the asymptotic limit on cosmological scales we obtain that the modes Φ −,κ (t, r, ψ 0 = 1/H 0 ), where A 2 =C 4[ C] 1 . Notice that we have neglected the inhomogenoues part of its solution because it is negligible on these large super-Hubble scales at the end of inflation.
As can be demonstrated the solution K

C. 4D Einstein equations
The effective 4D Lagrangian density (17) is expressed in terms of the fields Φ ± (x µ , ψ 0 ), which can be thought of as two minimally coupled bosons Since hence the effective 4D potential results to be which is induced by the static foliation on the fifth coordinate ψ = ψ 0 = 1/H 0 . This effective 4D potential is the responsible to provide us the dynamics of the fields Φ ± (x µ , ψ 0 ) on the effective 4D hypersurface on which the equation of state is ω = P/ρ = −1. The energy density and pressure related to these fields are obtained from the diagonal part of the energy-momentum tensor written in a mixed manner where |E is some quantum state, are given by the expressions Since we are interested to find solutions with µ = 1 and ν = 2 that correspond to ∂ 0 Φ ± = 0, we must consider the values n = 5/2, m = 7/2. In order to cancelate the coefficients corresponding to the factors e ±H 0 t and if we require that ρ = −P = 3H 2 0 /(8πG), we obtain that such that from eq. (53) we obtain that ǫ = 6.68586. Furthermore, due to the fact the equation of state is ρ = −P = 3H 2 0 /(8πG), we must require that from which we obtain that κ * = 1.45598 H 0 = 1.45598 × 10 −9 M p . Notice that we have neglected in P and ρ terms which are very small with respect 3H 2 0 /(8πG) and decrease as e −2H 0 t . With the values earlier mentioned for κ * , H 0 and M p , we arrive at the numerical values (C 1 ) 2 = −2.63042 × 10 32 , (A 2 ) 2 = 5.95601 × 10 33 , C 3 = 4.8693 × 10 70 M −4 p , A 3 = 9.081 × 10 70 M −4 p , that correspond to ρ = −P = 1.19366 × 10 −19 M 4 p . These values are perfectly according to which one expects during a inflationary vacuum dominated expansion of the early universe. A very important fact is that the dark energy is outside the horizon at the beginning of inflation, but during the inflationary epoch enters to causally connected regions. In other word the dark energy is concentrated on the range of scales (physical scales) 2π/[ǫκ ± 0 (t)] ≃ (π/H 0 )e −H 0 t < λ phys < 2π/κ * . Hence, the effective 4D scalar (massive) field Φ − should be an interesting candidate to explain dark energy in the early inflationary universe.

IV. FINAL REMARKS
We have explored the possibility that the expansion of the universe during the primordial inflationary phase of the universe can be driven by a condensate of spinor fields. In our picture φ ± are effective fields which became from a condensate of two entangled spinors.
The fields φ ± decouple at the end of inflation. In all our analysis we have neglected the role of the inflaton field, which (in a de Sitter expansion) is freezed in amplitude and nearly scale invariant, but decays at the end of inflation into other fields. The point here is how we explain the existence of dark energy once the inflaton field energy density goes to zero.
Our proposal consist to prove that the dark energy could be physically explained though the entanglement of spinor fields that behave as effective 1-spin and 0-spin bosons on a 4D hypersurface on which the universe suffers a vacuum dominated expansion. The equation of state of the universe is determined by the static foliation ψ = 1/H 0 . Our calculations show that the vector boson φ + is massless and with spin 1, and therefore compatible with the properties of a massless vector boson. On the other hand the field φ − is a scalar boson which could be (jointly with the inflaton) the responsible for the expansion of the universe and would be a good candidate to explain the existence of the dark energy. [Other fields such as the curvaton field [16], have been proposed in the literature to explain it.] A very interesting fact is that the (dark) energy density which we are talking about is poured into smaller sub-horizon scales with the evolution of the inflationary expansion.