# Dark Energy as a Cosmological Consequence of Existence of the Dirac Scalar Field in Nature

Authors: O. V. Babourova; B. N. Frolov
Journal: Physics Research International
 (2015)

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## Abstract

The solution of the field equations of the conformal theory of gravitation with Dirac scalar field in Cartan-Weyl spacetime at the very early Universe is obtained. In this theory dark energy (described by an effective cosmological constant) is a function of the Dirac scalar fieldβ. This solution describes the exponential decreasing of β at the inflation stage and has a limit to a constant value of the dark energy at large time. This can give a way to solving the fundamental cosmological constant problem as a consequence of the fields dynamics in the early Universe.


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## Body

## 1. Introduction

The Poincaré-Weyl gauge theory of gravitation (PWTG) has been developed in [1]. This theory is invariant concerning both the Poincare subgroup and the Weyl subgroup—extensions and compressions (dilatations) of spacetime. Dilatations are equivalent in the mathematical sense to the transformations of the group of length calibres changes, which is the gauge group of the Weyl theory developed in 1918 [2]. The gauge field introduced by the subgroup of dilatations is named as dilatation field. Its vector potential is the Weyl vector, and its strength is Weyl’s segmental curvature tensor arising in the geometrical interpretation of the theory together with the curvature and torsion tensors. The dilatation gauge field does not coincide with electromagnetic field (that has been asserted by Weyl in his basic work [2]) but represents a field of another type [3]. In particular, quanta of this field can have nonzero rest masses.As it has been shown in [1], an additional scalar field β(x) is introduced in PWTG as an essential geometrical addendum to the metric tensor. The properties of this field coincide with those of the scalar field introduced by Dirac in his well-known article [4] and also by Deser in [5]. We will name this field as Dirac scalar field. The Dirac scalar field plays an important role in construction of the gravitation Lagrangian, some members of which have structure of the Higgs Lagrangian and after spontaneous braking of dilatational invariance can cause an appearance of nonzero rest masses of particles [6].On the basis of the observational data, it is accepted in modern cosmology that the dark energy (described by the cosmological constant) is of dominant importance in dynamics of the Universe. In this connection the major unsolved problem of modern fundamental physics is very large difference of around 120 orders of magnitude between a very small value of Einstein cosmological constantΛ, which can be estimated on the basis of modern observations in cosmology, and the value of the cosmological constant in the early Universe, which has been estimated by theoretical calculations in quantum field theory of quantum fluctuation contributions to the vacuum energy [7–10]. In the present work we try to understand the cosmological constant problem as the effect of the gravitational field and the Dirac scalar field dynamics in the Cartan-Weyl spacetime in the early Universe.


## 2.  Gravitational Lagrangian

Spacetime in PWTG has the geometrical structure of the Cartan-Weyl space with a curvature 2-formRab, a torsion 2-form Ta, and a nonmetricity 1-form Qab of the Weyl type, Qab=1/4gabQ, where Q is a Weyl 1-form.On the basis of PWTG, a conformal theory of gravitation in Cartan-Weyl spacetime with the Dirac scalar field has been developed in the external form formalism with the Lagrangian density 4-formL [11–13]:(1)L=LG+Lm+β4Λab∧Qab-14gabQ,where Lm is the matter Lagrangian density 4-form and Λab are the Lagrange multipliers and the gravitational field Lagrangian density 4-form LG reads:(2)LG=2f012β2Rab∧ηab+β4Ληhhhhhhh+14λRaa∧*Rbbhhhhhhh+τ1R[ab]∧*Rbahhhhhhh+τ2(R[ab]∧θa)∧*(R[cb]∧θc)hhhhhhh+τ3(R[ab]∧θc)∧*(R[cb]∧θa)hhhhhhh+τ4(R[ab]∧θa∧θb)∧*(R[cd]∧θc∧θd)hhhhhhh+τ5(R[ab]∧θa∧θd)∧*R[cd]∧θc∧θbhhhhhhh+τ6(Rab∧θc∧θd)∧*(Rcd∧θa∧θb)hhhhhhh+ρ1β2Ta∧*Tahhhhhhh+ρ2β2Ta∧θb∧*(Tb∧θa)hhhhhhh+ρ3β2Ta∧θa∧*(Tb∧θb)hhhhhhh+ξβ2Q∧*Q+ζβ2Q∧θa∧*Tahhhhhhh+l1dβ∧*dβ+l2βdβ∧θa∧*Tahhhhhhh12+l3βdβ∧*Q.Here ∧ is the exterior product sign, d is the exterior derivative operator, and * is the Hodge dual conjugation. The second term in (2) is the effective cosmological constant which is interpreted as the dark energy density (Λ is the Einstein cosmological constant).Variational field equations in the Cartan-Weyl spacetime have been derived fromL by exterior form variational formalism [14]. Independent variables are basis 1-form θa, a nonholonomic connection 1-form Γab, the scalar field β, and Lagrange multipliers Λab. As a result we have Γ-, θ-, and β-equations, which have the following forms in vacuum (Lm≈0) [11–13]:Γ-equation:(3)2f0β2-14Q∧ηab+12Tc∧ηabchhhhhhh+12ηae∧Qbe+dln⁡β∧ηabhhh+λ12Dδab*Rcc+τ12D*R[ba]hhh+τ2D2δa[dδc]bθd∧*(R[fc]∧θf)hhh+τ3D2δa[dδc]bθf∧*(R[fc]∧θd)hhh+τ42D*(Rcd∧θc∧θd)θa∧θbhhh+τ52D*(R[cd]∧θc∧θ[b)θa]∧θdhhh+τ62D*(Rcd∧θa∧θb)θc∧θdhhh+ρ12β2θb∧*Tahhh+ρ22β2θb∧θc∧*Tc∧θahhh+ρ32β2θb∧θa∧*Tc∧θchhh+ξ4β2δab*Qhhh+ζβ22δabθc∧*Tc+θb∧*(Q∧θa)hhh+l2βθb∧*dβ∧θahhh14+l32βδab*dβ-2β4Λab=0.θ-equation:(4)2f0β212Rbc∧ηbca+β4Ληahhh+λ14Rcc∧*Rbb∧θahhhhhhh+14**Rbb∧θa∧*Rcchhh+τ1R[ab]∧*Rba∧θchhhhhhhh+**Rba∧θc∧*R[ab]hhh+τ22Rab∧*Rcb∧θchhhhhhh-*R[bc]∧θb∧θaRdc∧θdhhhhhhh-**R[bc]∧θb∧θa∧*Rdc∧θdhhh+τ32R[bc]∧*R[ac]∧θbRdc∧θbhhhhhhhh-*R[bc]∧θd∧θa∧R[dc]∧θbhhhhhhhh-**R[bc]∧θd∧θahhhhhhhh∧*Rdc∧θbhhh+τ4*R[bc]∧θb∧θc4Raf∧θf+*R[ef]∧θe∧θfηahhhhhhhh·4Raf∧θf+*R[ef]∧θe∧θfηahhh+τ5*R[bc]∧θb∧θdhhhhhhhhh·2R[ad]∧θc-2δacR[fd]∧θfhhhhhhhhhhh+*R[fd]∧θf∧θcηahhh+τ6*R[bc]∧θe∧θf+2gabRef∧θc-2δacRef∧θbhhhhhhhh·*Ref∧θb∧θcηahhhhhhhhhh+2gabRef∧θc-2δacRef∧θbhhh+ρ1β22D*Ta+Tc∧*Tc∧θahhhhhhhhhh+**Tc∧θa∧*Tchhhhhhhhhh+4dln⁡β∧*Tahhh+ρ2β22Td∧*Ta∧θdTb∧θc∧θahhhhhhhhh+2Dθb∧*Tb∧θahhhhhhhhh+4dln⁡β∧θb∧*Tb∧θahhhhhhhhh-**Tc∧θd∧θahhhhhhhhh∧*Td∧θchhhhhhhhh-*Tb∧θc∧θaTc∧θbhhh+ρ3β22Dθa∧*Tb∧θbhhhhhhhhhh+2Ta∧*Tb∧θbhhhhhhhhhh-*Tb∧θb∧θaTc∧θchhhhhhhhhh-**Tb∧θb∧θa∧*Tc∧θchhhhhhhhhh+4dln⁡β∧θa∧*Tb∧θbhhh+ξβ2-Q∧*Q∧θa-**Q∧θa*Qhhh+ζβ2D*Q∧θa-Q∧*TaQ∧θbhhhhhhhhh+Q∧θb∧*Tb∧θahhhhhhhhh+2dln⁡β∧*Q∧θahhhhhhhhh+**Tb∧θa∧*Q∧θbhhh+l1-dβ∧*dβ∧θahhhhhhhhh-**dβ∧θa∧*dβhhh+l2βD*dβ∧θa+dβ∧θb∧*Tb∧θadβ∧θbhhhhhhhhh-dβ∧*Tahhhhhhhhh+**Tb∧θa∧*dβ∧θbhhhhhhhdβ∧θb+dβ∧*dβ∧θahhh12+l3β-dβ∧*Q∧θa-**Q∧θa*dβ=0.β-equation:(5)2f0βRab∧ηab-4β3Ληhhh+2ρ1βTa∧*Tahhh+2ρ2βTa∧θb∧*Tb∧θahhh+2ρ3βTa∧θa∧*Tb∧θbhhh+2ξβQ∧*Q+2ζβQ∧θa∧*Tahhh+l1-2d*dβ+l2-βdθa∧*Tahhhηab+l3-βd*Q+4β3Λab∧Qab-14gabQ=0.The variation with respect to the Lagrange multipliersΛab gives the Weyl condition for the nonmetricity 1-form Qab:(6)Qab-14gabQ=0.


## 3. Solutions of the Field Equations at Ultra-Early Universe

We will solve the field equations for the scale factora(t) and the scalar Dirac field β at the very early stage of evolution of Universe, when a matter density has been very small, Lm≈0. We will omit the terms quadratic in curvature for simplicity.In homogeneous and isotropic spacetime the conditionsTa=1/3T∧θa are valid, and we will find, as the consequence of the field equations, the torsion and nonmetricity in the forms, Tμ=χTdln⁡β, Qμ=χQdln⁡β, where the coefficients χT, χQ are expressed by couple constants of Lagrangian density (2) (see the Appendix).We consider the spatially flat Friedmann-Robertson-Walker (FRW) metric:(7)ds2=dt2-a2tdx2+dy2+dz2.Taking into account that Lm≈0, we obtain from the θ-equation together with the β-equation the following system of equations [11–13]:(8)(0,0):3a˙2a2+6a˙aβ˙β+3B3β˙β2=Λβ2,(9)(1,1):2a¨a+2β¨β+4a˙aβ˙β+a˙a2+(B2-2)β˙β2=Λβ2,(10)β:Aβ¨β+3a˙aβ˙β+(B-A)β˙β2=0,where the constants A, B, B3=1/3(2B1+B2), B1, and B2 are expressed through the parameters of Lagrangian density (2), the components (2, 2) and (3, 3) being equal to the component (1, 1).The system of (8)–(10) is inconsistent, because we have three equations for two unknown functions a(t) and β(t). Let us put in this system B1=B2=B3=1 and also u=ln⁡a, v=ln⁡β. Then subtract (8) from (10). As a result we obtain the following system of equations:(11)u˙2+2u˙v˙+v˙2=Λ3e2v,(12)u¨+v¨-u˙v˙-v˙2=0,(13)v¨+3u˙v˙+BAv˙2=0.Equation (11) is equivalent to the equation(14)u˙+v˙=±λ3ev,λ=3Λ.It is easy to check that (12) is fulfilled identically as a consequence of (14). Therefore we have only 2 equations (13) and (14) for 2 unknown functions a(t), β(t), and this system of equations is consistent. In what follows we choose the sign “+” in (14).Let us findu˙ from (14) and put it in (13). We obtain the equation(15)v¨+λevv˙+ωv˙2=0,ω=BA-3.The first integral of this equation is the following:(16)v˙=λ0e-ωv-λ1+ωev,where λ0 is a constant of integration.The system of (14), (16) has a large variety of integrable solutions parameterized by ω and λ0. Let us obtain the solution for the case ω=0. If we put in (16) λ0=λ, then this equation reads, v˙=λ(1-ev), and we have a solution [13](17)β(t)=ev(t)=11-e-λ(t+t0),a(t)=a0e(λ/3)(t+t0)1-e-λ(t+t0)4/3.We assume that the value ofβ is very large, when t=0. Therefore the constant of integration t0 should be very small (0<t0≪λ-1). Then from (17) under t≫t0 one has approximately(18)β(t)=β01exp⁡⁡-λt,a(t)=a01eλ/3t.These solutions realize exponential diminution of a fieldβ (see Figure 1 for function (17)), and thus sharp exponential decrease of physical vacuum energy (dark energy) by many orders. We have β→1, Λeff=β2Λ→Λ in a limit at t→∞. Thus, the effective cosmological constant can slightly differ already by the end of inflation from the limiting value equal to its modern size Λ that provides the subsequent transition from the Friedmann epoch to the epoch of the accelerated expansion in accordance with the modern observed cosmological data.Figure 1
Solution (17) for the Dirac scalar field in the early Universe.Our solutions are realized, if the conditions are valid,B=3A, B1=1, and B2=1. These conditions are determined in rather complicated manner by the 16 coupling constants of gravitational Lagrangian density (2) and can be easily fulfilled (see the Appendix).


## 4. Discussion and Final Remarks

Here the field equations for the spatially flat homogeneous and isotropic ultra-early Universe are investigated and the solution for the Dirac scalar field (and therefore for the effective cosmological constant) is found. This solution for the scalar field has the behavior of the very extensive diminishing exponent, whose limits are the modern value of the Einstein cosmological constant. Solution (17) could be realized at the very beginning of the Universe evolution, when the cosmological constant Λ0 estimated by quantum field theory was equal to Λ0/Λ=β02~10120, and the number β0~1060 was very large.Thus our result can explain the exponential decrease in time at very early Universe of the dark energy (the energy of physical vacuum), described by the effective cosmological constant. This can give a way to solving the problem of cosmological constant as a consequence of fields dynamics at the early Universe. It is well known that this problem is one of the fundamental problems of the modern theoretical physics [7–10].We point out that the ultra-rapid decrease of the energy of physical vacuum according to law (17) occurs only prior to the Friedmann era evolution of the Universe. Further evolution of the Universe is determined not only by a scalar field only but also by the born ultra-relativistic matter and the radiation interacting with it.The Dirac scalar field condenses near massive objects. As a consequence of this, the authors formulate the hypothesis thatthe Dirac scalar field is realized itself not only as the “dark energy” but also as one of the components of the “dark matter” [13].



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