Dark energy as a cosmological consequence of existence of the Dirac scalar field
aa r X i v : . [ g r- q c ] O c t Dark energy as a cosmological consequence of existenceof the Dirac scalar field
O.V. Babourova ∗ , B.N. Frolov † Moscow Pedagogical State University,Faculty of Physics and Informational Technologies,M. Pirogovskaya ul. 29, Moscow 119992, Russian Federation
Abstract
The solution of the field equations of the conformal theory of gravitation withDirac scalar field in Cartan–Weyl spacetime at the very early Universe is obtained.In this theory dark energy (describing by an effective cosmological constant) is afunction of the Dirac scalar field β . This solution describes the exponential decreas-ing of β at the inflation stage and has a limit to a constant value of the dark energyat large time. This can give a way to solving the fundamental cosmological constantproblem as a consequence of the fields dynamics in the early Universe. Pacs 04.50.Kd, 04.20.Fy, 98.80.Jk
1. Introduction
The Poincar´e–Weyl gauge theory of gravitation (PWTG) has been developed in [1]. Thistheory is invariant both concerning the Poincare subgroup and the Weyl subgroup –extensions and compressions (dilatations) of spacetime. Dilatations are equivalent inthe mathematical sence to the transformations of the group of length calibres changes,which is the gauge group of the H. Weyl theory developed in 1918 [2]. The gauge fieldintroduced by the subgroup of dilatations is named as dilatation field. Its vector-potentialis the Weyl vector, and its strength is the Weyl’s segmental curvature tensor arising in thegeometrical interpretation of the theory together with the curvature and torsion tensors.The dilatation gauge field does not coinside with electromagnetic field (that has beenasserted by Weyl in his basic work [2]), but represents a field of another type [3]. Inparticular, quanta of this field can have nonzero rest masses. ∗ E-mail: [email protected] † E-mail: [email protected] β ( x ) is introduced in PWTGas an essential geometrical addendum to the metric tensor. The properties of this fieldcoincide with those of the scalar field introduced by Dirac in his well-known article [4]and also by S. Deser in [5]. We shall name this field as Dirac scalar field. The Diracscalar field plays an important role in construction of the gravitation Lagrangian, somemembers of which have structure of the Higgs Lagrangian and after spontaneous brakingof 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 thedark energy (described by the cosmological constant) is of dominant importance in dynam-ics of the universe. In this connection the major unsolved problem of modern fundamentalphysics is very large difference of around 120 orders of magnitude between a very smallvalue of Einstein cosmological constant Λ, which can be estimated on the basis of mod-ern observations in cosmology, and the value of the cosmological constant in the earlyUniverse, which has been estimated by theoretical calculations in quantum field theory ofquantum fluctuation contributions to the vacuum energy [7]–[10]. In the present work wetry to understand the cosmological constant problem as the effect of the gravitational fieldand 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 acurvature 2-form R ab , a torsion 2-form T a and a nonmetricity 1-form Q ab of the Weyltype, Q ab = g ab Q , where Q is a Weyl 1-form.On the basis of PWTG, a conformal theory of gravitation in Cartan–Weyl spacetimewith the Dirac scalar field has been developed in the external form formalism with theLagrangian density 4-form L [11]–[13], L = L G + L m + β Λ ab ∧ (cid:18) Q ab − g ab Q (cid:19) , (1)where L m is the matter Lagrangian density 4-form, Λ ab are the Lagrange multipliers andthe gravitational field Lagrangian density 4-form L G reads, L G = 2 f (cid:20) β R ab ∧ η ab + β Λ η + 14 λ R aa ∧ ∗R bb ++ τ R [ ab ] ∧ ∗R ba + τ ( R [ ab ] ∧ θ a ) ∧ ∗ ( R [ cb ] ∧ θ c )++ τ ( R [ ab ] ∧ θ c ) ∧ ∗ ( R [ cb ] ∧ θ a ) + τ ( R [ ab ] ∧ θ a ∧ θ b ) ∧ ∗ ( R [ cd ] ∧ θ c ∧ θ d )++ τ ( R [ ab ] ∧ θ a ∧ θ d ) ∧ ∗ ( R [ cd ] ∧ θ c ∧ θ b )++ τ ( R ab ∧ θ c ∧ θ d ) ∧ ∗ ( R cd ∧ θ a ∧ θ b )++ ρ β T a ∧ ∗T a + ρ β ( T a ∧ θ b ) ∧ ∗ ( T b ∧ θ a )++ ρ β ( T a ∧ θ a ) ∧ ∗ ( T b ∧ θ b ) + ξβ Q ∧ ∗Q + ζ β Q ∧ θ a ∧ ∗T a ++ l dβ ∧ ∗ dβ + l βdβ ∧ θ a ∧ ∗T a + l βdβ ∧ ∗Q i ++ β Λ ab ∧ ( Q ab − g ab Q ) . (2)Here ∧ is the exterior product sign, d is the exterior derivative operator, ∗ is the Hodgedual conjugation. The second term in (2) is the effective cosmological constant which isinterpreted as the dark energy density (Λ is the Einstein cosmological constant).Variational field equations in the Cartan–Weyl spacetime have been derived from L by exterior form variational formalism [14]. Independent variables are basis 1-forms θ a ,a nonholonomic connection 1-form Γ ab , the scalar field β and Lagrange multipliers Λ ab .As a result we have Γ-, θ - and β -equations, which have the following forms in vacuum( L m ≈
0) [11]–[13],Γ–equation:2 f (cid:20) β (cid:18) − Q ∧ η ab + 12 T c ∧ η abc + 12 η ae ∧ Q be + d ln β ∧ η ab (cid:19) ++ λ D ( δ ba ∗ R cc ) + τ D (cid:0) ∗R [ ba ] (cid:1) + τ D (cid:0) δ [ da δ bc ] θ d ∧ ∗ ( R [ fc ] ∧ θ f ) (cid:1) ++ τ D (cid:0) δ [ da δ bc ] θ f ∧ ∗ ( R [ fc ] ∧ θ d ) (cid:1) + τ D (cid:0) ∗ ( R cd ∧ θ c ∧ θ d ) θ a ∧ θ b (cid:1) ++ τ D (cid:0) ∗ ( R [ cd ] ∧ θ c ∧ θ [ b ) θ a ] ∧ θ d (cid:1) + τ D (cid:0) ∗ ( R cd ∧ θ a ∧ θ b ) θ c ∧ θ d (cid:1) ++ ρ β θ b ∧ ∗T a + ρ β θ b ∧ θ c ∧ ∗ ( T c ∧ θ a ) + ρ β θ b ∧ θ a ∧ ∗ ( T c ∧ θ c )++ ξ β δ ba ∗ Q + ζ β (cid:0) δ ba θ c ∧ ∗T c + θ b ∧ ∗ ( Q ∧ θ a ) (cid:1) + l βθ b ∧ ∗ ( dβ ∧ θ a )++ l βδ ba ∗ dβ i − β Λ ab = 0 . (3) θ –equation:2 f (cid:20) β (cid:18) R bc ∧ η bca (cid:19) + β Λ η a ++ λ (cid:18) R cc ∧ ∗ ( R bb ∧ θ a ) + 14 ∗ ( ∗R bb ∧ θ a ) ∧ ∗R cc (cid:19) ++ τ (cid:0) R [ ab ] ∧ ∗ (cid:0) R ba ∧ θ c (cid:1) + ∗ (cid:0) ∗R ba ∧ θ c (cid:1) ∧ ∗R [ ab ] (cid:1) ++ τ (cid:0) R [ ab ] ∧ ∗ ( R [ cb ] ∧ θ c ) − ∗ ( R [ bc ] ∧ θ b ∧ θ a ) R [ dc ] ∧ θ d −− ∗ ( ∗ ( R [ bc ] ∧ θ b ) ∧ θ a ) ∧ ∗ ( R [ dc ] ∧ θ d ) (cid:1) ++ τ (cid:0) R [ bc ] ∧ ∗ ( R [ ac ] ∧ θ b ) − ∗ ( R [ bc ] ∧ θ d ∧ θ a ) ∧ R [ dc ] ∧ θ b −− ∗ ( ∗ ( R [ bc ] ∧ θ d ) ∧ θ a ) ∧ ∗ ( R [ dc ] ∧ θ b ) (cid:1) ++ τ (cid:0) ∗ ( R [ bc ] ∧ θ b ∧ θ c )(4 R [ af ] ∧ θ f + ∗ ( R [ ef ] ∧ θ e ∧ θ f ) η a ) (cid:1) ++ τ (cid:0) ∗ ( R [ bc ] ∧ θ b ∧ θ d )(2 R [ ad ] ∧ θ c − δ ca R [ fd ] ∧ θ f ++ ∗ ( R [ f d ] ∧ θ f ∧ θ c ) η a ) (cid:1) ++ τ (cid:0) ∗ ( R [ bc ] ∧ θ e ∧ θ f )( ∗ ( R ef ∧ θ b ∧ θ c ) η a + 2 g ab R ef ∧ θ c −− δ ca R ef ∧ θ b ) (cid:17) ++ ρ β (cid:16) D ( ∗T a ) + T c ∧ ∗ ( T c ∧ θ a ) + ∗ ( ∗T c ∧ θ a ) ∧ ∗T c ++4 d ln β ∧ ∗T a (cid:17) ++ ρ β (cid:16) T d ∧ ∗ ( T a ∧ θ d ) + 2 D ( θ b ∧ ∗ ( T b ∧ θ a ))++ 4 d ln β ∧ θ b ∧ ∗ ( T b ∧ θ a ) −− ∗ ( ∗ ( T c ∧ θ d ) ∧ θ a ) ∧ ∗ ( T d ∧ θ c ) − ∗ ( T b ∧ θ c ∧ θ a )( T c ∧ θ b ) (cid:1) ++ ρ β (cid:16) D ( θ a ∧ ∗ ( T b ∧ θ b )) + 2 T a ∧ ∗ ( T b ∧ θ b ) −− ∗ ( T b ∧ θ b ∧ θ a )( T c ∧ θ c ) − ∗ ( ∗ ( T b ∧ θ b ) ∧ θ a ) ∧ ∗ ( T c ∧ θ c )++ 4 d ln β ∧ θ a ∧ ∗ ( T b ∧ θ b ) (cid:1) ++ ξβ (cid:16) − Q ∧ ∗ ( Q ∧ θ a ) − ∗ ( ∗Q ∧ θ a ) ∗ Q (cid:17) ++ ζ β (cid:16) D ∗ ( Q ∧ θ a ) − Q ∧ ∗T a + Q ∧ θ b ∧ ∗ ( T b ∧ θ a )++ 2 d ln β ∧ ∗ ( Q ∧ θ a ) + ∗ ( ∗T b ∧ θ a ) ∧ ∗ ( Q ∧ θ b ) (cid:17) ++ l (cid:16) − dβ ∧ ∗ ( dβ ∧ θ a ) − ∗ ( ∗ dβ ∧ θ a ) ∧ ∗ dβ (cid:17) ++ l (cid:16) β ( D ∗ ( dβ ∧ θ a ) + + dβ ∧ θ b ∧ ∗ ( T b ∧ θ a ) − dβ ∧ ∗T a ++ ∗ ( ∗T b ∧ θ a ) ∧ ∗ ( dβ ∧ θ b ))++ dβ ∧ ∗ ( dβ ∧ θ a ) (cid:17) + l β (cid:16) − dβ ∧ ∗ ( Q ∧ θ a ) − ∗ ( ∗Q ∧ θ a ) ∗ dβ (cid:17)i = 0 . (4) β –equation:2 f h β R ab ∧ η ab − β Λ η + 2 ρ β T a ∧ ∗T a ++ 2 ρ β ( T a ∧ θ b ) ∧ ∗ ( T b ∧ θ a ) + 2 ρ β ( T a ∧ θ a ) ∧ ∗ ( T b ∧ θ b )++ 2 ξβ Q ∧ ∗Q + 2 ζ β
Q ∧ θ a ∧ ∗T a + l ( − d ∗ dβ ) + l ( − βd ( θ a ∧ ∗T a ))++ l ( − βd ∗ Q ) i + 4 β Λ ab ∧ ( Q ab − g ab Q ) = 0 . (5)The variation with respect to the Lagrange multipliers Λ ab gives the Weyl conditionfor the nonmetricity 1-form Q ab , Q ab − g ab Q = 0 . (6)
3. Solutions of the field equations at ultra-early Universe
We shall solve the field equations for the scale factor a ( t ) and the scalar Dirac field β atthe very early stage of evolution of universe, when a matter density has been very small, L m ≈
0. We shall omit the terms quadratic in curvature for symplicity.In homogeneous and isotropic spacetime the conditions, T a = T ∧ θ a are valid, andwe shall find, as the consequence of the field equations, the torsion and nonmetricity inthe forms, T µ = χ T d ln β , Q µ = χ Q d ln β , whete the coefficients χ T , χ Q are expressed bycouple constants of the Lagrangian density (2).We consider the spatially flat Friedman–Robertson–Walker (FRW) metric ds = dt − a ( t )( dx + dy + dz ) . (7)Taking into account that L m ≈
0, we obtain from the θ -equation together with the β − equation the following system of equations [11]–[13],(0 ,
0) : 3 ˙ a a + 6 ˙ aa ˙ ββ + 3 B (cid:16) ˙ ββ (cid:17) = Λ β , (8)(1 ,
1) : 2 ¨ aa + 2 ¨ ββ + 4 ˙ aa ˙ ββ + (cid:0) ˙ aa (cid:1) +( B − (cid:16) ˙ ββ (cid:17) = Λ β , (9) β : A (cid:16) ¨ ββ + 3 ˙ aa ˙ ββ (cid:17) + ( B − A ) (cid:16) ˙ ββ (cid:17) = 0 , (10)where the constants A , B , B = (2 B + B ), B , B are expressed through the param-eters of the Lagrangian density (2), the components (2, 2) and (3, 3) being equal to thecomponent (1, 1).The system of equations (8)–(10) is inconsistent, because we have three equations fortwo unknown functions a ( t ) and β ( t ). Let us put in this system, B = B = B = 1, andalso u = ln a , v = ln β . Then substract Eq. (8) from Eq. (10). As a result we obtain thefollowing system of equations, ( ˙ u ) + 2 ˙ u ˙ v + ( ˙ v ) = Λ3 e v (11)¨ u + ¨ v − ˙ u ˙ v − ( ˙ v ) = 0 , (12)¨ v + 3 ˙ u ˙ v + BA ( ˙ v ) = 0 . (13)Eq. (11) is equivalent to the equation,˙ u + ˙ v = ± λ e v , λ = √ . (14)It is easy to check that Eq. (12) is fulfilled identically as a consequence of Eq. (14).Therefore we have only 2 equations (13) and (14) for 2 unknown functions a ( t ), β ( t ), andthis system of equations is consistent. In what follows we choose the sign ”+” in Eq. (14).Let us find ˙ u from Eq. (14) and put it in Eq.(13). We obtain the equation,¨ v − λe v ˙ v + ω ( ˙ v ) = 0 , ω = BA − . (15)The first integral of this equation is the following,˙ v = λ e − ωv − λ ω e v , (16)where λ is a constant of integration.The system of equation (14), (16) have a large variety of integrable solutions parametrizedby ω and λ . Let us obtain the solution for the case ω = 0. If we put in Eq. (16) λ = λ ,then this equation reads, ˙ v = λ (1 − e v ), and we have a solution [13], β ( t ) = e v ( t ) = 11 − e − λ ( t + t ) , a ( t ) = a e λ ( t + t ) (1 − e − λ ( t + t ) ) / . (17)We assume that the value of β is very large, when t = 0. Therefore the constant ofintegration t should be very small (0 < t ≪ λ − ). Then from Eq. (17) under t ≫ t onehas approximately, β ( t ) = ( β ) exp ( − λt ) , a ( t ) = ( a ) e λ t . (18)These solutions realize exponential diminution of a field β (see Figure 1) for the func-tion (17)), and thus sharp exponential decrease of physical vacuum energy (dark energy)by many orders. We have Λ eff = β Λ → Λ in a limit at t → ∞ . Thus, the effectivecosmological constant can slightly differ already by the end of inflation from the limit-ing value equal to its modern size Λ that provides the subsequent transition from theFriedman epoch to the epoch of the accelerated expansion in accordance with the modernobservant cosmological data. f(t) t Figure 1: The solution (17) for the Dirac scalar field in the early UniverseWe have for the solutions (17), (18), β → , Λ eff = β Λ → Λ , when t → ∞ . (19)Therefore the limit of the effective cosmological constant for large time is not zero (seeFigure 1) and is equal to the value of the Einstein cosmological constant that ensures anaccelerating expansion of the modern Universe.Our solutions are realized, if the following conditions are valid, B = 3 A , B = 1, B = 1. These conditions are determined in rather complicated manner by the 16 couplingconstants of the gravitational Lagrangian density (2), and can be easily fulfilled.
4. Discussion and final remarks