Primordial Dark Energy from a Condensate of Spinors in a 5D Vacuum

Abstract

We explore the possibility that the expansion of the universe can be driven by a condensate of spinors which are free of interactions in a 5D relativistic vacuum defined in 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 studied here 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 subhorizon scales with the evolution of the inflationary expansion.

1. 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, 2] 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 in which we define a physical vacuum. The Campbell-Magaard theorem (CMT) [3] 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. (We shall consider that capital letters 𝐴 and 𝐵 run from 0 to 4 in a 5D extended de Sitter spacetime (where the 3D Euclidean space is in Cartesian coordinates), small letters 𝑎 and 𝑏 run from 0 to 5 in a 5D Minkowsky spacetime (in cartasian coordinates), Greek letters 𝛼 and 𝛽 run from 0 to 3, and latin letters 𝑖 and 𝑗 run from 1 to 3.). 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 [4–7]. An interesting problem in modern cosmology relies on explaining 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 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 the possibility that such expansion can be explained by a condensate of dark spinors has been explored [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.

2. The Effective Lagrangian in a 5D Riemann-Flat Spacetime

We are concerned with a 5D Riemann-flat spacetime with a line element given by 𝑑𝑆²=𝜓𝜓₀²𝑑𝑡²−𝑒^2𝑡/𝜓0𝑑𝑥²+𝑑𝑦²+𝑑𝑧²−𝑑𝜓², (1) where 𝑡,𝑥,𝑦 and 𝑧 are the usual local spacetime coordinate systems and 𝜓 is the noncompact space-like extra dimension.

We start from an effective Lagrangian density for non massive fermions in 5D: ℒ_eff1=−2∇_𝐴Ψ∇^𝐴Ψ. (2) 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 following functional derivatives: 𝛿ℒ_eff𝛿Ψ∇=0,_𝐴𝛿ℒ_eff𝛿∇_𝐴Ψ=12∇_𝐴∇^𝐴Ψ,𝛿ℒ_eff𝛿𝑔_𝑀𝑁=12∇^𝑀Ψ∇^𝑁Ψ+∇_𝑃𝐽^𝑀𝑁𝑃, (3)(4)(5) where the effective current 𝐽^𝑀𝑁𝑃 is symmetric with respect to permutations of 𝑀 and 𝑁, 𝐽^𝑀𝑁𝑃=18∇^𝑀Ψ𝑓^𝑁𝑃Ψ+Ψ∇^𝑀Ψ, (6) and 𝑓^𝑁𝑃=[𝛾^𝑁,𝛾^𝑃] is antisymmetric [14]. At this point we are in conditions of introducing the stress tensor 𝑇^𝑀𝑁=2(𝛿ℒ_eff/𝛿𝑔_𝑀𝑁)−𝑔^𝑀𝑁ℒ_eff: 𝑇^𝑀𝑁=∇^𝑀Ψ∇^𝑁Ψ+2∇_𝑃𝐽^𝑀𝑁𝑃+12𝑔^𝑀𝑁𝑔_𝐴𝐵∇^𝐴Ψ∇^𝐵Ψ. (7)

Applying the compatibility condition on the metric ∇_𝐶𝑔^𝐴𝐵=0, we obtain ∇_𝐴∇^𝐴Ψ=∇_𝐴𝑔^𝐴𝐵∇_𝐵Ψ=∇_𝐴𝑔^𝐴𝐵∇_𝐵Ψ+𝑔^𝐴𝐵∇_𝐴∇_𝐵Ψ=0, (8) and we obtain that the equation for the spinor Ψ takes the form 𝑔^𝐴𝐵∇_𝐴∇_𝐵Ψ=0. (9) The same procedure yields an identical equation for the field Ψ. On the other hand, the 5D Einstein equation for the Riemann-flat metric (1) is ⟨0|𝑇_𝐴𝐵|0⟩=0, (10) where ⟨0|𝑇_𝐴𝐵|0⟩ denotes the expectation value of 𝑇_𝐴𝐵 in the vacuum state |0⟩.

2.1. Tensorial Formulation of the Equation of Motion

Using the formalism previously introduced, the double Nabla can be expressed explicitly: ∇_𝐴∇_𝐵Ψ=𝜕_𝐴∇_𝐵Ψ+Γ_𝐴∇_𝐵Ψ−𝜔^𝐶_𝐴𝐵∇_𝐶Ψ=𝜕_𝐴𝜕_𝐵1Ψ−4𝜕_𝐴𝜔_𝐵^𝑎𝑏𝛾_𝑎𝛾_𝑏1Ψ−4𝜔_𝐵^𝑎𝑏𝛾_𝑎𝛾_𝑏𝜕_𝐴Ψ−14𝜔_𝐴^𝑎𝑏𝛾_𝑎𝛾_𝑏𝜕_𝐵Ψ+1𝜔16^𝐴𝑎𝑏𝜔_𝐵^𝑐𝑑𝛾_𝑎𝛾_𝑏𝛾_𝑐𝛾_𝑑Ψ−𝜔^𝐶_𝐴𝐵𝜕_𝐶Ψ+14𝜔^𝐶_𝐴𝐵𝜔_𝐶^𝑎𝑏𝛾_𝑎𝛾_𝑏Ψ, (11) where the spin connection is 𝜔_𝑀^𝑎𝑏=−𝑒^𝑎_𝑁[𝜕_𝑀𝑒^𝑏_𝐴𝑔^𝐴𝑁+𝑒^𝑏_𝐵𝑔^𝐴𝐵Γ^𝑁_𝐴𝑀] and Γ_𝑀=−(1/8)𝜔_𝑀^𝑎𝑏[𝛾_𝑎,𝛾_𝑏].

(The tensors can be written using the vielbein 𝑒^𝐴_𝑎 and its inverse 𝑒^𝑎_𝐴, such that, if 𝑒^𝐴_𝑎𝑒^𝑏_𝐴=𝛿^𝑏_𝑎, then 𝜂_𝑎𝑏=𝑒^𝐴_𝑎𝑒^𝐵_𝑏𝑔_𝐴𝐵, (12) where 𝜂_𝑎𝑏 is the 5D Minkowsky tensor metric with signature (+,−,−,−,−).)

Thus, after replacing the last expression in the equation of motion (9) we obtain 𝑔^𝐴𝐵𝜕_𝐴𝜕_𝐵1Ψ−2𝑔^𝐴𝐵𝜔^𝑎𝑏_(𝐴𝜎_𝑎𝑏𝜕_𝐵)Ψ−𝑔^𝐴𝐵𝜔^𝐶_𝐴𝐵𝜕_𝐶1Ψ+4𝑔^𝐴𝐵𝜕_𝐴𝜔_𝐵^𝑎𝑏𝜎_𝑎𝑏Ψ+1𝑔16^𝐴𝐵𝜔_𝐴^𝑎𝑏𝜔_𝐵^𝑐𝑑𝜎_𝑎𝑏𝜎_𝑐𝑑1Ψ−4𝑔^𝐴𝐵𝜔^𝐶_𝐴𝐵𝜔_𝐶^𝑎𝑏𝜎_𝑎𝑏Ψ=0. (13) Here, we have made use of the fact that 𝛾_𝑎𝛾_𝑏=(1/2){𝛾_𝑎,𝛾_𝑏}+(1/2)[𝛾_𝑎,𝛾_𝑏]=𝑔_𝑎𝑏𝕀+𝜎_𝑎𝑏, 𝜔_𝑀^𝑎𝑏=−𝜔_𝑀^𝑏𝑎 and 𝜔_𝑀^𝑎𝑏𝛾_𝑎𝛾_𝑏=𝜔_𝑀^𝑎𝑏𝜎_𝑎𝑏. Once we simplify some terms, we obtain 12𝑔^𝐴𝐵𝜔^𝑎𝑏_(𝐴𝜎_𝑎𝑏𝜕_𝐵)1Ψ=2𝑔^𝐴𝐵𝜔_𝐴^𝑎𝑏𝜎_𝑎𝑏𝜕_𝐵𝜔Ψ,_𝐶𝐴𝐵=𝜔_𝐴^𝐷𝐶𝑔_𝐷𝐵=𝜔_𝐴^𝑑𝑐𝑔_𝐷𝐵𝑒^𝐷_𝑑𝑒^𝐶_𝑐,𝑔^𝐴𝐵𝜔^𝐶_𝐴𝐵=𝑔^𝐴𝐵𝜔_𝐴^𝑑𝑐𝑔_𝐷𝐵𝑒^𝐷_𝑑𝑒^𝐶_𝑐=𝜔_𝐴^𝑑𝑐𝛿^𝐴_𝐷𝑒^𝐷_𝑑𝑒^𝐶_𝑐=𝜔_𝐴^𝑑𝑐𝑒^𝐴_𝑑𝑒^𝐶_𝑐. (14) Finally, the equation of motion for the spinors assumes its final form 𝑔^𝐴𝐵𝜕_𝐴𝜕_𝐵1Ψ−2𝑔^𝐴𝐵𝜔_𝐴^𝑎𝑏𝜎_𝑎𝑏𝜕_𝐵Ψ−𝜔_𝐴^𝑎𝑏𝑒^𝐴_𝑎𝑒^𝐶_𝑏𝜕_𝐶1Ψ+4𝑔^𝐴𝐵𝜕_𝐴𝜔_𝐵^𝑎𝑏𝜎_𝑎𝑏Ψ+1𝑔16^𝐴𝐵𝜔_𝐴^𝑎𝑏𝜔_𝐵^𝑐𝑑𝜎_𝑎𝑏𝜎_𝑐𝑑Ψ−14𝜔_𝐴^𝑎𝑏𝑒^𝐴_𝑎𝑒^𝐶_𝑏𝜔_𝐶^𝑐𝑑𝜎_𝑐𝑑Ψ=0, (15) which is very difficult to be resolved because the fields are coupled.

2.2. Conformal Mapping Based Solution

In order to simplify the structure of (15), we shall introduce the following transformation on the spinor components: 𝜑Ψ=¹𝜑₂, (16) where components are grouped as 𝜑₁=𝜓₁𝜓₂,𝜑₂=𝜓₃𝜓₄. (17) With this representation we obtain the equation of motion for 𝜑₁ and 𝜑₂: ∅𝜑₁+3𝜓₀𝜓²𝜕𝜑₁−4𝜕𝑡𝜓𝜕𝜑₁+1𝜕𝜓4𝜓²𝜑₁−𝑖𝜓₀𝜓²𝑒^{−𝑡/𝜓0}𝜎⋅∇𝜑₁=−𝑖𝜓₀𝜓²𝜕𝜑₂+𝜕𝑡3𝑖2𝜓²𝜑₂−𝜓₀𝜓²𝑒^{−𝑡/𝜓0}𝜎⋅∇𝜑₂,∅𝜑₂+3𝜓₀𝜓²𝜕𝜑₂−4𝜕𝑡𝜓𝜕𝜑₂+1𝜕𝜓4𝜓²𝜑₂+𝑖𝜓₀𝜓²𝑒^{−𝑡/𝜓0}𝜎⋅∇𝜑₂=𝑖𝜓₀𝜓²𝜕𝜑₁−𝜕𝑡3𝑖2𝜓²𝜑₁−𝜓₀𝜓²𝑒^{−𝑡/𝜓0}𝜎⋅∇𝜑₁. (18) Here, we have adopted the following conventions: 𝜓∅𝜑=⁰𝜓²𝜕²𝜑𝜕𝑡²−𝜓₀𝜓²𝑒^{−2𝑡/𝜓0}∇²𝜕𝜑−²𝜑𝜕𝜓²,𝜎=𝜎₁𝚤̂+𝜎₂𝚥̂+𝜎₃𝑘,𝜎⋅∇𝜑=𝜎₁𝜕𝜑𝜕𝑥+𝜎₂𝜕𝜑𝜕𝑦+𝜎₃𝜕𝜑.𝜕𝑧 (19) Now we can use the conformal mapping defining new complex fields Φ₊=𝜑₁+𝑖𝜑₂ and Φ₋=𝜑₁−𝑖𝜑₂. Rewriting (18) in terms of these new fields, it is possible to decouple the first equation, while the other coupling becomes a source for the second equation: ∅Φ₊+4𝜓₀𝜓²𝜕Φ₊−4𝜕𝑡𝜓𝜕Φ₊−5𝜕𝜓4𝜓²Φ₊=0,∅Φ₋+2𝜓₀𝜓²𝜕Φ₋−4𝜕𝑡𝜓𝜕Φ₋+7𝜕𝜓4𝜓²Φ₋=𝑖2𝜓₀𝜓²𝑒^{−𝑡/𝜓0}𝜎⋅∇Φ₊. (20)(21) Then, after few calculations, the Lagrangian density written in terms of the new fields takes the form ℒ_eff1=−2∇_𝐴𝜑₁∇^𝐴𝜑₁+∇_𝐴𝜑₂∇^𝐴𝜑₂ (22) or, alternatively, can be written as a function of the pair 𝜑₁=(1/2)(Φ₊+Φ₋),𝜑₂=(1/2𝑖)(Φ₊−Φ₋): ℒ_eff1=−4∇_𝐴Φ₊∇^𝐴Φ₊+∇_𝐴Φ₋∇^𝐴Φ₋. (23) On the other hand the 5D energy-momentum (EM) tensor is represented by ⁽⁵⁾𝑇_𝐴𝐵=2(𝛿ℒ_eff/𝛿𝑔^𝐴𝐵)−𝑔_𝐴𝐵ℒ_eff. This procedure takes 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.

2.3. Extra Dimensional Solution for Φ₊

We shall use the variable separation method in the homogeneous PDE (20); we obtain the following set of ODEs: ∇²𝑅+𝜅²𝜕𝑅=0,₂𝑇_𝜅⁽⁺⁾𝜕𝑡²+2𝜓₀𝜕𝑇_𝜅⁽⁺⁾+𝜅𝜕𝑡²𝑒^{−2𝑡/𝜓0}−𝑀²₁𝑇_𝜅⁽⁺⁾𝜓=0,₂𝜕²Λ𝜕𝜓²+4𝜓𝜕Λ+5𝜕𝜓4−𝑀²₁𝜓²₀Λ=0. (24)(25)(26) Equation (24) has a solution that can be written in terms of plane wavefront 𝑅(𝑟)⃗∼𝑒^{⃗𝑟±𝑖𝜅⋅}. (27) The second equation (25) has a general solution: Λ⁽⁺⁾𝜓=𝐶₁𝜓𝜓₀^{−((3/2)+1+𝑀21𝜓20)}+𝐶₂𝜓𝜓₀^{−((3/2)−1+𝑀21𝜓20)}. (28) Since we are interested in “localized” static solutions, that is, those that decay to zero when 𝜓 tends to infinity, we must choose 𝐶₂=0, so that 𝑛≡((3/2)+1+𝑀²₁𝜓²₀)>0. This choice makes 𝑀²₁=(((𝑛−(3/2))²−1)/𝜓²₀)≥0, with 𝑛≥3 and 𝑛∈ℝ, in order to get 1+𝑀²₁𝜓²₀≥0.

2.4. Extra Dimensional Solution for Φ₋

Now we are able to calculate the coupling term of the inhomogeneous equation (21) for each mode (see (37)):2𝑖𝜓₀𝜓²𝑒^{−𝑡/𝜓0}𝜎⋅∇Φ_+,𝜅2=−^𝜈Γ(𝜈)√𝜋𝜅^𝜈−1𝜓₀^(1/2)+𝜈×𝑒^{𝑖𝑥𝜅⋅}𝑒^{(𝜈−3)(𝑡/𝜓0)}𝜓𝜓₀^{√−((7/2)+𝜈2−3)}. (29) Using the last expression in (21), we obtain a degenerate two-component system; for the spinor Φ_−,𝜅. (Henceforth we are concerned with asymptotic solutions; that is, only the infrared limit has cosmological significance.) Again, a plane wavefront satisfies the spatial part. By inserting Φ_−,𝜅=𝐺_𝜅(𝑡,𝜓)𝑒^{𝑖⃗𝑟𝜅⋅} and multiplying by (𝜓/𝜓₀)², we obtain 𝜕²𝐺_𝜅⁽⁻⁾𝜕𝑡²+2𝜓₀𝜕𝐺_𝜅⁽⁻⁾+𝜅𝜕𝑡²𝑒^{−2𝑡/𝜓0}+74𝜓²₀𝐺_𝜅−𝜓𝜓₀²𝜕²𝐺_𝜅⁽⁻⁾𝜕𝜓²+4𝜓𝜓²₀𝜕𝐺_𝜅⁽⁻⁾2𝜕𝜓≃−^𝜈Γ(𝜈)𝜅^1−𝜈√𝜋𝜓₀^(1/2)+𝜈𝑒^{((𝜈−3)/𝜓0)𝑡}𝜓𝜓₀^{√−[(3/2)+𝜈2−3]}. (30) This inhomogeneous PDE can be converted to one with a constant coupling. In order to make the right side of (30) constant, we shall propose 𝐺_𝜅⁽⁻⁾𝑡,𝜓=𝜓₀⁻²𝑒^{((𝜈−3)/𝜓0)𝑡}𝜓𝜓₀^{√−[(3/2)+𝜈2−3]}𝐾_𝜅⁽⁻⁾𝑡,𝜓. (31) Finally, we must solve the follwing equation: 𝜕²𝐾_𝜅⁽⁻⁾𝜕𝑡²+2(𝜈−2)𝜓₀𝜕𝐾_𝜅⁽⁻⁾−𝜓𝜕𝑡𝜓₀²𝜕𝐾_𝜅⁽⁻⁾𝜕𝜓²−2𝜓²₀12−√𝜈²−3𝜓𝜕𝐾_𝜅⁽⁻⁾+𝜅𝜕𝜓₂𝑒^{−2𝑡/𝜓0}+10−4𝜈𝜓²₀𝐾_𝜅⁽⁻⁾2≃−^𝜈Γ(𝜈)√𝜋𝜅^𝜈−1𝜓₀^{−(3/2)+𝜈}. (32)

3. Effective Dynamics on the 4D Hypersurface 𝜓=1/𝐻₀

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=𝜓₀=1/𝐻₀ describes an effective 3D spatially flat, isotropic, and homogeneous de Sitter four-dimensional expanding universe with a constant Hubble parameter 𝐻₀, with a line element 𝑑𝑆²⟶𝑑𝑠²=𝑑𝑡²−𝑒^2𝐻0𝑡𝑑𝑟². (33) From the relativistic point of view, an observer moves in a comoving frame with the five-velocity 𝑈^𝜓=0 on a 4D hypersurface with a scalar curvature ⁽⁴⁾𝑅=12/𝜓²₀=12𝐻²₀, such that the Hubble parameter 𝐻₀ and thus also the cosmological constant Λ₀=3𝐻²₀/(8𝜋𝐺) are defined by the foliation 𝐻₀=𝜓₀⁻¹.

3.1. Time-Dependent Modes of Φ₊

The solution for the time-dependent equation (26) can be expanded in terms of first- and second-kind Hankel functions: 𝑇_𝜅⁽⁺⁾(𝑡)=𝑒^{−2𝐻0𝑡}𝐶₃ℋ_𝜈⁽¹⁾𝜅𝐻₀𝑒^−𝐻0𝑡+𝐶₄ℋ_𝜈⁽²⁾𝜅𝐻₀𝑒^−𝐻0𝑡, (34) where 𝜈=4+𝑀²₁𝜓²₀≥2. After making a Bunch-Davies normalization of the modes [15] we obtain the solution 𝑇_𝜅⁽⁺⁾(𝑡)=𝑖2𝜋𝐻₀𝑒^{−2𝐻0𝑡}ℋ_𝜈⁽²⁾𝜅𝐻₀𝑒^−𝐻0𝑡. (35) Since we are interested to describe the universe on super-Hubble cosmological scales, we must acquire 𝜅𝜓₀𝑒^{−𝑡/𝜓0}≪1; we reject solutions that go to zero at late times. The asymptotic behavior of 𝑇_𝜅⁽⁺⁾(𝑡) on cosmological scales will be 𝑇_𝜅⁽⁺⁾(𝑡)≃𝑖2𝜋𝐻₀Γ(𝜈)𝑒^{−2𝐻0𝑡}𝜅2𝐻₀𝑒^{−𝐻0𝑡−𝜈}. (36) Finally, the degenerate two-component spinor Φ₊ can be expanded as a function of the modes Φ_+,𝜅𝑡,𝑟,⃗𝜓₀=1𝐻₀≃𝑖𝐶₁2^𝜈−1Γ(𝜈)√𝜋𝐻₀^𝜈−(1/2)𝜅^−𝜈𝑒^{𝑖⃗𝑟𝜅⋅}𝑒^{(𝜈−2)𝐻0𝑡} (37) and their conjugated complex.

3.2. The Time-Dependent Modes for Φ₋

The homogeneous solution 𝐾_𝜅⁽⁻⁾(𝑡,𝜓)|_hom, of (32), is 𝐾_𝜅⁽⁻⁾𝑡,𝜓_hom=𝑒^{−(𝜈−2)𝑡/𝜓0}𝐶₃ℋ_𝜇⁽¹⁾𝜅𝜓₀𝑒^{−(𝑡/𝜓0)}+𝐶₄ℋ_𝜇⁽²⁾𝜅𝜓₀𝑒^{−(𝑡/𝜓0)}×𝐶₁𝜓𝜓₀^{(√𝜈2−3−𝜈2−4𝜈+7+𝑀22𝜓20)}+𝐶₂𝜓𝜓₀^{(√𝜈2−3+𝜈2−4𝜈+7+𝑀22𝜓20)}, (38) where 𝜇=𝜈²−4𝜈+4+𝑀²₂𝜓²₀≥0 and the squared mass of Φ₍₋₎ is [𝑀₂(𝑚,𝑛)]²=((𝑚−3/2)²−(𝑛−3/2)²+(4(𝑛−3/2)²+3−10))/𝜓²₀, which is definitely positive for 𝑚≥1 (with 𝑛≥3). In order get to the 𝜓-dependent solution of lim_𝜓→∞𝐾_𝜅⁽⁻⁾(𝑡,𝜓)|_hom→0, we shall require that 𝐶₂=0 and 𝑚>3/2+3+((𝑛−3/2)²+3−2)², for 𝑛≥3, such that (𝑚,𝑛)∈ℤ. After taking the asymptotic limit on cosmological scales we obtain that the modes Φ_−,𝜅(𝑡,𝑟,⃗𝜓₀=1/𝐻₀), for 𝜇=1, are Φ_−,𝜅𝑡,𝑟,⃗𝜓₀≃𝐴₂𝐻₀^1/2√𝑒𝜋𝜅^{𝑖⃗𝑟𝜅⋅}, (39) where 𝐴₂=𝐶₄[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_𝜅⁽⁻⁾(𝑡,𝜓₀)≃𝐾_𝜅⁽⁻⁾(𝑡,𝜓₀)|_hom on cosmological scales, once we consider 𝐻₀=1/𝜓₀=1×10⁻⁹𝑀_𝑝. Hence, the homogeneous solution 𝐾_𝜅⁽⁻⁾(𝑡,𝜓₀)|_hom is a very acceptable solution at the end of inflation for the time-dependent modes of Φ₋. In other words, at the end of inflation the effective 4D bosons Φ_± can be decoupled on cosmological scales.

3.3. 4D Einstein Equations

The effective 4D Lagrangian density (23) is expressed in terms of the fields Φ_±(𝑥^𝜇,𝜓₀), which can be thought of as two minimally coupled bosons; ℒ_eff1=−4∇_𝜇Φ₊∇^𝜇Φ₊+∇_𝜇Φ₋∇^𝜇Φ₋+𝑉Φ₊,Φ₋. (40) Since ∇₄Φ₊=𝜕Φ₊𝑛𝜕𝜓11=−𝜓Φ₊,∇11₄Φ₋=𝜕Φ₋𝑚𝜕𝜓11=−𝜓Φ₋,∇11₄Φ₊=𝜕Φ₊11𝑛𝜕𝜓=−𝜓Φ₊11,∇₄Φ₋=𝜕Φ₋11𝑚𝜕𝜓=−𝜓Φ₋11, (41) the effective 4D potential resulting is 𝑉Φ₊,Φ₋1=−4∇₄Φ₊∇⁴Φ₊+∇₄Φ₋∇⁴Φ_{−𝜓=1/𝐻0}=𝐻²₀4𝑛²Φ₊²+𝑚²Φ₋²_𝜓=1/𝐻0, (42) which is induced by the static foliation on the fifth coordinate 𝜓=𝜓₀=1/𝐻₀. This effective 4D potential is responsible to provide us with the dynamics of the fields Φ_±(𝑥^𝜇,𝜓₀) 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: 𝐸1𝜌=4∇₀Φ₊²+∇₀Φ₋²−𝑒^{−2𝐻0𝑡}4∇Φ₋⋅∇Φ₋+∇Φ₊⋅∇Φ₊+𝑉Φ₊,Φ₋+𝐹⁰₀𝐸_𝜓=1/𝐻0,𝐸1𝑃=4∇₀Φ₊²+∇₀Φ₋²−𝑒^{−2𝐻0𝑡}12∇Φ₋⋅∇Φ₋+∇Φ₊⋅∇Φ₊−𝑉Φ₊,Φ₋+𝐹^𝑖_𝑗𝛿^𝑖_𝑗𝐸_𝜓=1/𝐻0, (43) where |𝐸⟩ is some quantum state, 𝐹⁰₀=𝐶₃/𝜋[(𝐻⁷₀/8𝜅²)+(𝐻⁹₀/𝜅⁴)], 𝐹^𝑖_𝑗=𝐴₃/𝜋[(15𝐻⁷₀/32𝜅²)+(𝐻⁹₀/2𝜅⁴)]𝛿^𝑖_𝑗, and ∇₀Φ_±=𝜕₀∓14𝜓₀Φ_±11,∇_𝑗Φ₊=𝜕_𝑗Φ₊11,∇₁Φ₋=𝜕₁Φ₋𝐻+𝑖⁰𝑒^𝐻0𝑡2Φ₋11,∇₂Φ₋=𝜕₂Φ₋11𝐻+𝑖⁰𝑒^𝐻0𝑡2Φ₋𝑖,∇−𝑖₃Φ₋=𝜕₃Φ^∗₋11𝐻+𝑖⁰𝑒^𝐻0𝑡2Φ^∗₋1,∇−1₁Φ₋=𝜕₁Φ₋𝐻−𝑖⁰𝑒^𝐻0𝑡2Φ₋),∇(11₂Φ₋=𝜕₂Φ^∗₋()𝐻11−𝑖⁰𝑒^𝐻0𝑡2Φ^∗₋(),∇𝑖−𝑖₃Φ₋=𝜕₃Φ^∗₋()𝐻11−𝑖⁰𝑒^𝐻0𝑡2Φ^∗₋(1−1). (44) An interesting asymptotic solution can be obtained by considering the expectation values of, for instance, some quadratic scalar Σ²(𝑥,𝑡), as Σ²(𝑡)=𝐸Σ²(𝐸=1𝑥,𝑡)(2𝜋)^{3𝜖𝜅±0𝜅(𝑡)}_∗𝑑³𝜅Σ_𝜅(𝑥,𝑡)Σ^∗_𝜅(𝑡)(𝑥,𝑡), (45) where 𝜅_∗>0 is some minimum cut for the wavenumber to be determined and 𝜅⁺₀(𝑡)=𝐻₀𝑒^𝐻0𝑡 and 𝜅⁻₀(𝑡)=2𝐻₀𝑒^𝐻0𝑡 are the maximum wavenumbers to the modes of Φ₊ and Φ₋, respectively. The expectation values for the radiation energy density 𝜌 and the pressure 𝑃 are given by the follwing expressions: 𝜌=𝐴²₂173𝐻⁴₀𝜖128𝜋³−𝜖³𝐻⁴₀24𝜋³+𝐶₃𝐻⁸₀𝜖16𝜋³𝑒^𝐻0𝑡−𝐴²₂173𝑘_∗𝐻³₀128𝜋³+𝐶²₁101𝐻⁵₀32𝑘_∗𝜋³+𝐶₃−𝑘_∗𝐻⁷₀16𝜋³+𝐻⁹₀2𝑘_∗𝜋³−𝐶²₁101𝐻⁴₀64𝜋³𝜖+𝐻⁴₀𝜖𝜋³+𝐶₃𝐻⁸₀4𝜋³𝜖𝑒^−𝐻0𝑡+𝐶²₁𝑘_∗𝐻³₀2𝜋³+𝐴²₂𝑘³_∗𝐻₀24𝜋³𝑒^{−2𝐻0𝑡},𝑃=𝐴₃15𝐻⁸₀𝜖64𝜋³−𝐴²₂219𝐻⁴₀𝜖128𝜋³+𝐻⁴₀𝜖³24𝜋³𝑒^𝐻0𝑡−𝐶²₁99𝐻⁵₀32𝑘_∗𝜋³+𝐴²₂219𝑘_∗𝐻³₀128𝜋³+𝐴₃−15𝑘_∗𝐻⁷₀64𝜋³+𝐻⁹₀4𝑘_∗𝜋³+𝐶²₁99𝐻⁴₀64𝜋³𝜖−𝐻⁴₀𝜖𝜋³−𝐴₃𝐻⁸₀8𝜋³𝜖𝑒^−𝐻0𝑡+𝐴²₂𝑘³_∗𝐻₀24𝜋³+𝐶²₁𝑘_∗𝐻³₀2𝜋³𝑒^{−2𝐻0𝑡}. (46) Since we are interested to find solutions with 𝜇=1 and 𝜈=2 that correspond to 𝜕₀Φ_±=0, we must consider the values 𝑛=5/2 and 𝑚=7/2. In order to calculate the coefficients corresponding to the factors 𝑒^±𝐻0𝑡 and if we require that 𝜌=−𝑃=3𝐻²₀/(8𝜋𝐺), we obtain that 𝐶²₁=6𝑘_∗𝜋²−519+16𝜖²𝐻₀𝜖²32𝐻²₀−519+16𝜖²−𝑘²_∗101+64𝜖²=12𝑘_∗𝜋²657+16𝜖²𝐻₀𝜖²64𝐻²₀657+16𝜖²−15𝑘²_∗−99+64𝜖²,𝐴²₂=−3𝐶²₁101+64𝜖²2−519+16𝜖²=−45𝐶²₁−99+64𝜖²4657+16𝜖², (47)(48) such that from (47) we obtain that 𝜖=6.68586. Furthermore, due to the fact that the equation of state is 𝜌=−𝑃=3𝐻²₀/(8𝜋𝐺), we must require that 32𝐻²₀519−16𝜖²+𝑘²_∗101+64𝜖²𝑘_∗−519+16𝜖²=64𝐻²₀657+16𝜖²−15𝑘²_∗−99+64𝜖²2𝑘_∗657+16𝜖², (49) from which we obtain that 𝜅_∗=1.45598𝐻₀=1.45598×10⁻⁹𝑀_𝑝. Notice that we have neglected in 𝑃 and 𝜌 terms which are very small with respect to 3𝐻²₀/(8𝜋𝐺) and decrease as 𝑒^{−2𝐻0𝑡}. With the values earlier mentioned for 𝜅_∗, 𝐻₀, and 𝑀_𝑝, we arrive at the numerical values (𝐶₁)²=−2.63042×10³², (𝐴₂)²=5.95601×10³³, 𝐶₃=4.8693×10⁷⁰𝑀_𝑝⁻⁴, and 𝐴₃=9.081×10⁷⁰𝑀_𝑝⁻⁴, which correspond to 𝜌=−𝑃=1.19366×10⁻¹⁹𝑀⁴_𝑝. These values perfectly accord with that one expects during an 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 words the dark energy is concentrated in the range of scales (physical scales) 2𝜋/[𝜖𝜅^±₀(𝑡)]≃(𝜋/𝐻₀)𝑒^−𝐻0𝑡<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.

4. 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 from a condensate of two entangled spinors. The fields 𝜙_± are decoupled 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 is consistent 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/𝐻₀. 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) 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 subhorizon scales with the evolution of the inflationary expansion.

Acknowledgments

The authors acknowledge UNMdP and CONICET, Argentina, for financial support.