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The Solution of the Cosmological Constant Problem: the Cosmological Constant Exponential Decrease in the Super-Early Universe

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The Solution of the Cosmological Constant Problem: the Cosmological Constant Exponential Decrease in the Super-Early Universe universe Communication The Solution of the Cosmological Constant Problem: The Cosmological Constant Exponential Decrease in the Super-Early Universe Ol’ga Babourova 1 and Boris Frolov 2,* 1 Department ‘Physics’, Moscow Automobile and Road Construction State Technical University (MADI), Leningradsky pr., 64, 125319 Moscow, Russia; [email protected] 2 Institute of Physics, Technology and Information Systems, Moscow Pedagogical State University (MPGU), M. Pirogovskaya ul., 29, 119992 Moscow, Russia * Correspondence: [email protected] or [email protected]; Tel.: +7-903-267-7107 Received: 30 September 2020; Accepted: 29 November 2020; Published: 4 December 2020 Abstract: The stage of a super-early (primordial) scale-invariant Universe is considered on the basis of the Poincaré–Weyl gauge theory of gravity in a Cartan–Weyl space-time. An approximate solution has been found that demonstrates an inflationary behavior of the scale factor and, at the same time, a sharp exponential decrease in the effective cosmological constant from a huge value at the beginning of the Big Bang to an extremely small (but not zero) value in the modern era, which solves the well-known “cosmological constant problem.” Keywords: Poincaré–Weylgaugetheory; super-early(primordial)Universe; effectivecosmologicalconstant 1. Introduction In [1], the gravitational field as the field of curvature and torsion was introduced as the gauge field of the localized Poincaré group. This led to the construction of the Einstein–Cartan theory of gravity [2], based on the Lagrangian function in the form of a curvature scalar of the Riemann–Cartan space. The next step towards the construction of a Poincaré-gauge theory of gravity was made in [3–5], where the Lagrangian quadratic in the curvature tensor was introduced, and the theorem on the sources of a gauge field was formulated and proven in a general form, as well as in [6,7], where in addition, the Lagrangian quadratic in the torsion tensor was also introduced (for the Poincaré-gauge theory of gravity, see [8–12]). Then, in [13–17], the Poincaré–gauge local symmetry was supplemented by transformations of the Weyl symmetry subgroup—stretching and contraction (dilatations) of space-time. In [15–17], the gauge theory of the Poincaré–Weyl group was developed. In this theory, tetrad coefficients (which cannot be gauge fields, since they are tensor components), as well as connection, are some functions of real translational, rotational, and dilatational gauge fields. It has been shown that with the Poincaré–Weyl gauge theory of gravitation, the space-time acquires the properties of a Cartan–Weyl space with a curvature 2-form a and a torsion 2-form a, as well as R b T a nonmetricity 1-form with the Weyl condition, = (1/4)g . In this case, in addition to the Qab Qab ab Q metric tensor, a scalar field β appears, which by its properties coincides with the scalar field introduced earlier by Dirac [18]. In [19–22], it was hypothesized that the group of symmetry of space-time at the primordial stage of the evolution of the Universe—the epoch of the Big Bang and subsequent inflation—was not the Poincaré group, but the Poincaré–Weyl group. At this stage, the rest masses of elementary particles had not yet appeared, and all interactions were carried out by massless quanta; therefore, these interactions had the property of scale invariance. Universe 2020, 6, 230; doi:10.3390/universe6120230 www.mdpi.com/journal/universe Universe 2020, 6, 230 2 of 9 The proposed hypothesis is based on the assumption made by Harrison and Zel’dovich about the approximate scale invariance of the early stage of the evolution of the Universe, which underlies the calculation of the initial part of the spectrum of primary fluctuations of matter density in the early Universe (Harrison–Zel’dovich spectrum, see [23]) and has been confirmed by results of the COBE experiment for the study of the anisotropy of the brightness of the background radiation (see [24]). This paper presents preliminary results of a possible solution to the problem of the cosmological constant, which is one of the most important problems of modern fundamental physics. The solution of this problem will allow to reconcile the theory of the evolution of the Universe with modern physical concepts. 2. Lagrangian Density If we omit the terms with squares of curvature, the Lagrangian density 4-form of the Poincaré–Weyl gauge theory of gravity takes the form [20–22]: 2h a b 2 a a b = 2 f β (1/2) ηa β L η+ ρ a + ρ ( θ ) ( θa) L 0 R b ^ − 0 1T ^ ∗T 2 T ^ b ^ ∗ T ^ a b a 2 + ρ ( θa) ( θ ) + ξ + ζ θ a + l β dβ dβ (1) 3 T ^ ^ ∗ T ^ b Q ^ ∗Q Q ^ ^ ∗T 1 − ^ ∗ 1 α 1 i 4 ab + l β dβ θ α + l β dβ + β L ( (1/4)g ), 2 − ^ ^ ∗T 3 − ^ ∗Q ^ Qab − abQ where θa (a = 0, 1, 2, 3) are basis 1-forms, is the Hodge dualization operation, Lab are the Lagrange ∗ multipliers, η is a volume 4-form with components η , and η = ( 1/2!)θc θdη . abcd ab ^ abcd According to Gliner [25,26], the cosmological constant L in the Einstein equations is interpreted as the energy of the hypothetical vacuum-like medium, which is currently called dark energy. The term 2 β L0 in (1) describes the effective cosmological constant [20–22] (interpreted here as dark energy), depending on the Weyl–Dirac scalar field β (L0 is the theory parameter providing the correct rate of 2 = 120 inflation). The equality β0L0 10 L (see [27]) should be fulfilled, where L is the modern value of the cosmological constant and β0 is the value of the Weyl–Dirac scalar field at t = 0. 3. The Derivation of the Field Equations and Its Consequences The derivation of the variational field equations is based on the use of the following lemma on commutation of variation and dualization operations of external forms in the Cartan–Weyl space, proved in [20,28]: Lemma 1. Let F and Y be arbitrary p-forms on n-dimensional manifold, then the following relation holds for the commutator of variation and the operation of dual Hodge conjugation: F δ Y = δ Y F ^ ∗ ^ ∗ 1 ρ p(n 1)+s+1 σ ρ + δgρ g F Y + ( 1) − θ ( Y θ ) F + (2) 2 ^ ∗ − ^ ∗ ∗ ^ ^ ∗ σ p p(n 1)+s+1 + δθ ( 1) F (Y θσ) + ( 1) − ( Y θσ) F . ^ − ^ ∗ ^ − ∗ ∗ ^ ^ ∗ here, n = 4 is the dimension of space-time and s = 3 is the index of the metric form of the Minkowski space. Then, this Lemma was modified, taking into account the presence of the scalar Weyl–Dirac field, a ab and the following important relationships were obtained (F b, Fa, F are arbitrary 2-forms) [20]: a a a a b δ ( f (β)Fa) = d(δθ f (β)Fa) + δθ D( f (β)Fa) + δG b θ f (β)Fa , T ^ ^ ^ ^ ^ (3) δ ( f (β)Fab) = d( δg f (β)Fab) + δg D( f (β)Fab) + δGa 2 f (β)F b). Qab ^ − ab ab b ^ (a Universe 2020, 6, 230 3 of 9 We present here the result of using Lemma to vary some terms of the Lagrangian density (1): 1 2 a b a 1 2 b a b δ( β ηa ) = d(δG β ηa ) + βdβ ηa + 2 R b ^ b ^ 2 R b ^ a 2 1 b 1 bc 1 bc b +δG β ( Q ηa + Tc ηa + ηac Q + d ln β ηa )+ (4) b ^ − 4 ^ 2 ^ 2 ^ ^ 1 ab 2 c d 1 2 (a c b) a 1 2 b c +δg ( g β ηc + β θ θc ) + δθ ( β c η a) , ab 4 R d ^ 2 ^ ^ ∗Rj j 2 R ^ b 4 3 a 4 1 b 1 b c d δ(β L η) = 4δβ β L η + δθ β L ηa , ηa = θ η = θ θ θ η . (5) 0 0 ^ 0 3 ^ ab 3! ^ ^ abcd δ(dβ dβ) = δβ 2(d d ln β + d ln β d ln β) + d(2δβ dβ)+ ^ ∗ − · ∗ ^ ∗ +δg (( 1 gabdβ θ(a ( dβ θb))) dβ)+ (6) ab 2 − ^ ∗ ∗ ^ ^ ∗ a +δθ ( dβ (dβ θa) ( dβ θa) dβ) , ^ − ^ ∗ ^ − ∗ ∗ ^ ^ ∗ δ(βdβ ) = δβ βd + d(β δβ δg βgab dβ) + δGa 2β δb dβ ^ ∗Q − · ∗ Q ∗ Q − ab ∗ b ^ a ∗ +δg (β(gabD dβ + 1 gabdβ θ(a ( θb)) dβ) (7) ab ∗ 2 ^ ∗Q − ∗ ∗Q ^ ∗ a +δθ β( dβ ( θa) ( θa) dβ), ^ − ^ ∗ Q ^ − ∗ ∗Q ^ ∗ As a result of the variational procedure using the Lemma, we have obtained three field equations: Г-equation, θ-equation, and β-equation: Г-equation: h b bc bc b f (1/4) ηa + (1/2) c ηa + (1/2)ηac + d ln β ηa 0 − Q ^ T ^ ^ Q ^ b b c b c + 2ρ1θ a + 2ρ2θ θc ( θa) + 2ρ3θ θa ( θc) ^ ∗T ^ ^ ∗ T ^ ^ ^ ∗ T ^ (8) b b c b + 4ξδ [2, 3, 4] + ζ(2δ θ c + θ ( θa)) a Q a ^ ∗T ^ ∗ Q ^ b b i 2 b + l θ (d ln β θa) + l 2δ d ln β β La = 0 . 2 ^ ∗ ^ 3 a ∗ − θ-equation: 1 b c 2 c η a β L ηa 2 R ^ b − 0 h b b i + ρ 2D a + ( θa) + ( θa) + 4d ln β a 1 ∗ T Tb ^ ∗ T ^ ∗ ∗T ^ ^ ∗Tb ^ ∗T h b b b c + ρ 2D(θ ( θa)) + 2 (θ a) ( θc θa)( θ ) 2 b ^ ∗ T ^ T ^ ∗ b ^ T − ∗ T ^ ^ T ^ b c d b i ( ( θ ) θa) ( θc) + 4d ln β θ ( θa) − ∗ ∗ T ^ d ^ ^ ∗ T ^ ^ b ^ ∗ T ^ h b b b c + ρ 2D(θa ( θ )) + 2 a ( θ ) ( θ θa)( θc) 3 ^ ∗ T ^ b T ^ ∗ T ^ b − ∗ T ^ b ^ T ^ b c b i ( ( θ ) θa) ( θc) + 4d ln β θa ( θ ) (9) − ∗ ∗ T ^ b ^ ^ ∗ T ^ ^ ^ ∗ T ^ b b + ξ[ ( θa) ( θa) ] + ζ[D θa) a + θ ( θa) −Q ^ ∗ Q ^ − ∗ ∗Q ^ ∗ Q ∗ Q ^ − Q ^ ∗T Q ^ ^ ∗ Tb ^ b + ( θa) ( θ ) + 2d ln β ( θa)] ∗ ∗Tb ^ ^ ∗ Q ^ ^ ∗ Q ^ +l [ d ln β (d ln β θa) ( d ln β θa) d ln β)] 1 − ^ ∗ ^ − ∗ ∗ ^ ^ ∗ b b +l [D (d ln β θa) + d ln β θ ( θa) d ln β a + ( θa) (d ln β θ ) 2 ∗ ^ ^ ^ ∗ Tb ^ − ^ ∗T ∗ ∗Tb ^ ^ ∗ ^ + 2d ln β (d ln β θa)] + l [ d ln β ( θa) ( θa) d ln β] = 0 .
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