Integrability and Cosmological Solutions in Einstein-aether-Weyl theory
aa r X i v : . [ g r- q c ] J a n Integrability and Cosmological Solutions in Einstein-æther-Weyl theory
Andronikos Paliathanasis ∗ and Genly Leon † Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa Departamento de Matem´aticas, Universidad Cat´olica del Norte,Avda. Angamos 0610, Casilla 1280 Antofagasta, Chile (Dated: January 6, 2021)We consider a Lorentz violating scalar field cosmological model given by the modified Einstein-æther theory defined in Weyl integrable geometry. The existence of exact and analytic solutionsis investigated for the case of a spatially flat Friedmann–Lemaˆıtre–Robertson–Walker backgroundspace. We show that the theory admits cosmological solutions of special interests. In addition, weprove that the cosmological field equations admit the Lewis invariant as a second conservation law,which indicates the integrability of the field equations.
PACS numbers: 98.80.-k, 95.35.+d, 95.36.+xKeywords: Einstein-æther; Weyl theory; Cosmology; Scalar field; Exact solutions
1. INTRODUCTION
A plethora of modified or alternative theories to Einstein’s gravity [1, 2] have been proposed during the last yearsin order to explain the cosmological observations. There is a family of theories which violate the Lorentz symmetry.The main representatives of the Lorentz violating gravitational theories are the Hoˇrava-Lifshitz theory [3] and theEinstein-æther theory [4, 5].Hoˇrava-Lifshitz gravity is a power-counting renormalizable theory with consistent ultra-violet behaviour exhibitingan anisotropic Lifshitz scaling between time and space at the ultra-violet limit [3]. On the other hand, in Einstein-æther theory, the quadratic invariants of the kinematic quantities of a unitary time-like vector field, which is calledæther field, are introduced in the gravitational Action integral; modifying the Einstein-Hilbert Action [4]. TheEinstein-æther action is the most general second-order theory which is defined by the spacetime metric and the ætherfield involving no more than two derivatives [6] (not including total derivatives). There are several gravitational andcosmological applications for both of these theories in the literature, for instance see [7–24] and references therein.Scalar fields play a significant role in the explanation of the early acceleration phase of the universe known asinflation. Lorentz violating scalar field theories have been studied in Hoˇrava-Lifshitz gravity [25–28] and in theEinstein-æther theory [5, 29–34]. In [29] it has been proposed an Einstein-æther scalar field model in which thecoupling coefficients of the æther field with gravity are functions of the scalar fields. From the latter an interactionbetween the scalar field and the æther field it follows. For that model it was found that the inflationary era is dividedinto two parts, a Lorentz-violating stage and the standard slow-roll stage. In the Lorentz-violating stage the universeexpands as an exact de Sitter spacetime, although the inflaton field is rolling down the potential.In this work we are interest on the existence of exact and analytic solutions for a Lorentz-violating scalar fieldcosmological model. We consider the Einstein-æther theory defined in Weyl integrable geometry [35]. In this specifictheory the Action Integral of the Einstein-æther is modified such that a scalar field coupled to the æther fieldis introduced in a geometric way. The global dynamics of the background space were studied in [35] for variouscosmological models in the absence or in the presence of matter. It was found that the model provides severalcosmological eras in agreement with the cosmological history. In addition, a Weyl manifold is a conformal manifoldequipped with a connection which preserves the conformal structure and is torsion-free. In Weyl integrable theory thegeometry is supported by the metric tensor and a connection structure which differs from the conformal equivalentmetric by a scalar field [36, 37]. Cosmological and gravitational applications of the Weyl geometry can be found forinstance in [46–54]. The novelty of the Weyl geometry is that the scalar field in the gravitational Action integral isintroduced by the geometry.The context of integrability is essential in all areas of physics. A set of differential equations describing a physicalsystem is said to be integrable if there exist a sufficient number of invariant functions such that the dynamical systemcan be written in algebraic form. When the latter algebraic system is explicitly solved the solution of the dynamical ∗ Electronic address: [email protected] † Electronic address: [email protected] system can be expressed in closed-form [56, 57]. The study of integrability properties of nonlinear dynamical systemsis important, because analytical techniques can be applied for the better understanding of the physical phenomena.Although, nowadays we have powerful computers and numerical techniques to solve nonlinear differential equations,as Arscott discussed on the preface of his book [58] “[...] fall back on numerical techniques savours somewhat ofbreaking a door with a hammer when one could, with a little trouble, find the key”.The plan of the paper is as follows. In Section 2 we present the cosmological model under consideration which isthat of Einstein-æther defined in Weyl integrable geometry assuming a spatially flat Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) background space without any matter source terms. In Section 3 we present for the first timeanalytic and exact solutions for this cosmological model, we focus on the existence of exact solutions where thescale factor describes inflationary models of special interests. We obtain those solutions which are determined as thegeneral analytic solutions for the corresponding scalar field potentials. In Section 4 we show that this is possiblebecause the cosmological field equations form an integrable dynamical system, where the conservation laws are theconstraint cosmological equation, i.e. the modified first Friedmann’s equation and the Lewis invariant. The laterinvariant is essential for the study of integrability of time-dependent classical or quantum systems. We show thatthe field equations form an integrable dynamical system for an arbitrary potential function. Finally, in Section 5 wesummarize our results and we draw our conclusions.
2. EINSTEIN-ÆTHER-WEYL THEORY
The Einstein-æther-Weyl gravitational model is an extension of the Lorentz violating Einstein-æther theory inWeyl integrable geometry. It is a scalar field Lorentz violating theory where there is a coupling between the scalarfield and the æther field. The corresponding gravitational Action integral has the form of Einstein-æther gravity,thus it is generalized in Weyl integrable geometry. The latter generalization provides a geometric mechanism for theintroduction of the scalar field into the gravitational theory.Weyl geometry is a generalization of Riemannian geometry where the metric tensor and the covariant derivative { g µν , ∇ µ } , are generalized to n ˜ g µν , ˜ ∇ µ o , where ˜ ∇ µ is not defined by the Levi-Civita connection of g µν , but by theaffine connection ˜Γ κµν (˜ g ) with the property ˜ ∇ κ g µν = ω κ g µν , (1)and ˜ g µν is the metric compatible with ˜ ∇ µ . We study Weyl integrable geometry, where the gauge vector field ω µ whichdefines the geometry is a gradient vector field, i.e., it satisfies ω µ = φ ,µ for an scalar field φ . Then, it is defined thenew metric tensor ˜ g µν = e − φ g µν as the conformal related metric compatible with the covariant derivative ˜ ∇ µ , i.e,˜ ∇ κ ˜ g µν = 0. Connections ˜Γ κµν can be constructed from the Christoffel symbols Γ κµν ( g ) of the metric tensor g µν asfollows [55]: ˜Γ κµν = Γ κµν − φ , ( µ δ κν ) + 12 φ ,κ g µν . (2)The gravitational integral of the Einstein-æther-Weyl theory is [35]: S AEW (cid:16) g µν , ˜Γ κµν ; u µ (cid:17) = S W (cid:16) g µν , ˜Γ κµν (cid:17) + S AE (cid:16) g µν , ˜Γ κµν ; u µ (cid:17) , (3)where S W (cid:16) g µν , ˜Γ κµν (cid:17) is the extension of the Einstein-Hilbert action in Weyl geometry [55]: S W (cid:16) g µν , ˜Γ κµν (cid:17) = Z dx √− g (cid:16) ˜ R + ξ (cid:16) ˜ ∇ ν (cid:16) ˜ ∇ µ φ (cid:17)(cid:17) g µν (cid:17) , (4)with ˜ R denoting the Weylian scalar curvature˜ R = R − √− g (cid:0) g µν √− gφ (cid:1) ,µν + 32 (cid:16) ˜ ∇ µ φ (cid:17) (cid:16) ˜ ∇ ν φ (cid:17) , (5)and ξ is an arbitrary coupling constant. S AE (cid:16) ˜ g µν , ˜Γ κµν ; u µ (cid:17) is the Action Integral for the æther field u µ defined inWeyl geometry, that is: S AE (cid:16) g µν , ˜Γ κµν ; u µ (cid:17) = Z d x p − ˜ g (cid:16) ˜ K αβµν ˜ ∇ α u µ ˜ ∇ β u ν + λ (˜ g µν u µ u ν + 1) (cid:17) , (6)where ˜ g µν = e − φ g µν is the conformal related metric associated with the covariant derivative ˜ ∇ µ .Parameters c , c , c and c are dimensionless constants and define the coupling between the æther field andthe conformal metric through a kinetic term. Lagrange multiplier λ ensures the unitarity of the æther field, i.e.˜ g µν u µ u ν = − , while the fourth-rank tensor ˜ K αβµν is defined as˜ K αβµν ≡ c ˜ g αβ ˜ g µν + c ˜ g αµ ˜ g βν + c ˜ g αν ˜ g βµ + c ˜ g µν u α u β . (7)Equivalently, the Action Integral (6) can expressed in terms of the kinematic quantities n ˜ θ, ˜ σ µν , ˜ ω µν , ˜ α µ o as follows S AE (cid:16) g µν , ˜Γ κµν ; u µ (cid:17) = Z p − ˜ gdx (cid:16) c θ θ + c σ ˜ σ + c ω ˜ ω + c α ˜ α (cid:17) , (8)where the new parameters c θ , c σ , c ω , c a are functions of c , c , c and c , that is, c θ = ( c + 3 c + c ) , c σ = c + c , c ω = c − c , c a = c − c .From (3) we find the gravitational field equations to be˜ G µν + ˜ ∇ ν (cid:16) ˜ ∇ µ φ (cid:17) − (2 ξ − (cid:16) ˜ ∇ µ φ (cid:17) (cid:16) ˜ ∇ ν φ (cid:17) + ξg µν g κλ (cid:16) ˜ ∇ κ φ (cid:17) (cid:16) ˜ ∇ λ φ (cid:17) − g µν U ( φ ) = T æ µν , (9)where T æ ab is the energy momentum tensor which corresponds to the æther field and ˜ G µν is the Einstein tensor inWeyl theory, that is, ˜ G µν = ˜ R µν − ˜ Rg µν . The rhs of equation (9) corresponds to the energy-momentum tensor ofthe æther field: T æ µν = 2 c ( ˜ ∇ µ u α ˜ ∇ ν u α − ˜ ∇ α u µ ˜ ∇ β u ν ˜ g αβ ) + 2 λu µ u ν + ˜ K µβαµ ˜ ∇ µ u α ˜ ∇ β u µ g µν −
2[ ˜ ∇ α ( u ( µ J αν ) ) + ˜ ∇ α ( u α J ( aν ) ) − ˜ ∇ α ( u ( µ J ν ) α )] − c (cid:16) ˜ ∇ α u µ u α (cid:17) (cid:16) ˜ ∇ β u ν u β (cid:17) , (10)where J µν = − ˜ K µβνα ˜ ∇ β u α . In the case of a spatially flat FLRW geometry, with line element ds = − dt + a ( t ) (cid:0) dx + dy + dz (cid:1) , (11)for the comoving æther field we calculate˜ θ = θ − ˙ φ , ˜ σ = 0 , ˜ ω = 0 and ˜ α = 0 (12)where θ is the Riemannian expansion rate defined as θ = 3 ˙ aa . The gravitational field equations are expressed as follows θ − ρ φ − ρ æ = 0 (13)˙ θ + θ ρ φ + 3 p φ ) + 12 ( ρ æ + 3 p æ ) = 0 (14)where ρ φ , p φ are the energy density and pressure of the scalar field, that is, ρ φ (cid:16) φ, ˙ φ (cid:17) = ζ φ − U ( φ ) , p φ (cid:16) φ, ˙ φ (cid:17) = ζ φ + U ( φ ) , (15)where parameter ζ := 2 ξ − is a coupling parameter between the scalar field and the gravity. Furthermore, ρ æ , p æ are the density and pressure of the æther field, defined as ρ æ = − c θ θ , p æ = c θ (cid:16) θ ,t + ˜ θ (cid:17) (16)in which h µν = g µν + u µ u ν , that is ρ æ = − c θ (cid:16) θ − ˙ φ (cid:17) , p æ = c θ (cid:18) (cid:16) ˙ θ − ¨ φ (cid:17) + (cid:16) θ − ˙ φ (cid:17) (cid:19) . (17)By replacing in (13), (14) we derive the gravitational field equations(1 + c θ ) θ − c θ θ ˙ φ − (cid:18) ζ − c θ (cid:19) ˙ φ − U ( φ ) = 0 , (18)(1 + c θ ) ˙ θ + (1 + c θ )3 θ − c θ θ ˙ φ + (cid:16) ζ + c θ (cid:17) ˙ φ − c θ ¨ φ − U ( φ ) = 0 , (19)while the equation of motion for the scalar field is(2 c θ − c θ ) ζ ) ¨ φ + 3 ζc θ ˙ φ − c θ ) ζθ ˙ φ − c θ ) U ,φ = 0 . (20)In the following we investigate the existence of analytic solutions for the dynamical system (18)-(20). It is importantto mention that the three equations are not independent, indeed equation (18) is a conservation law for the higher-orderequations (19), (20).It is important to mention that in the latter dynamical system for ζ = 0 the system is degenerated and it has onlyone dependent variable, hence, in the following we consider that ζ = 0.
3. EXACT SOLUTIONS
Before to proceed with the derivation of the exact solution we perform the following change of variable φ ( t ) = − c θ ζ ln ψ ( t ), and define V ( ψ ) = U ( − c θ ζ ln ψ ( t )), where now the gravitational field equations becomes(1 + c θ ) θ c θ ) ζ ˙ ψψ θ + 2 ( c θ ) (2 c θ − ζ )27 ζ ˙ ψψ ! − V ( ψ ) = 0 , (21)(1 + c θ ) (cid:18) ˙ θ + 13 θ (cid:19) + 49 ( c θ ) ζ ˙ ψψ θ + 2 ( c θ ) (2 c θ − ζ )27 ζ ˙ ψψ ! + 23 ( c θ ) ζ ¨ ψψ − V ( ψ ) = 0 , (22)4 c θ (3( c θ + 1) ζ − c θ )27( c θ + 1) ζ ¨ ψψ + 4 c θ (2 c θ − ζ )27 ζ ˙ ψψ ! + 4 c θ ζ ˙ ψψ θ + ψV ′ ( ψ ) = 0 . (23)Finally, the gravitational equations reduce to V ( ψ ) = 2 c θ (2 c θ − ζ ) ˙ ψ ζ ψ + 4 c θ θ ˙ ψ ζψ + 13 ( c θ + 1) θ , (24) V ′ ( ψ ) = 4 c θ (3 ζ − c θ ) ˙ ψ ζ ψ − c θ θ ˙ ψ ζψ + 2(3( c θ + 1) ζ − c θ ) ˙ θ ζψ , (25)¨ ψ = − c θ + 1) ζψ ˙ θ c θ . (26)We proceed by study the existence of exact solution for the scalar field for specific forms of the scalar factor a ( t )which describe exact solutions of special interests. Consider the power-law solution θ ( t ) = c θ c θ ) ζ θ t , which describe a universe dominated by an ideal gas withconstant equation of state parameter w and scale factor a ( t ) = a t w ) , with w = − ζ (1+ c θ ) θ c θ , the solutiondescribes an inflationary universe when w < − , while in the special cases where θ = ζ (1+ c θ ) c θ , θ = ζ (1+ c θ )4 c θ or θ = ζ (1+ c θ )2 c θ , the ideal gas is that of dust fluid, radiation or stiff fluid respectively. ψ V ( ψ ) ψ - V ( ψ ) FIG. 1: Qualitative behaviour of the scalar field potential V ( ψ ) for various values of the free parameters ψ , ψ . Solid lineis for ( ψ , ψ ) = (cid:0) , (cid:1) , dotted line is for ( ψ , ψ ) = (cid:0) , (cid:1) , dashed line is for ( ψ , ψ ) = (cid:0) , − (cid:1) . From the plots we observethat the potential behaves like a power-law function. The plots are for θ = 1 , ζ = 1 and c θ = 6 where the power-law solutiondescribes an accelerated universe. The potentials are for the power-law scale factor By replacing in (21) and (22) we calculate for the scalar field ψ ( t ) = ψ t p + + ψ t p − , p ± = 12 (cid:16) ± p θ (cid:17) , (27)27 (1 + c θ ) ζ c θ t ψV ( ψ ( t )) = 2 ( c θ ) ( ψ ( p + + θ ) t p + + ψ ( p − + θ ) t p − )+ (2 c θ − c θ ) ζ ) ( p ψ t p + + p ψ t p − ) . (28)In Fig. 1, we present the parametric plot for the scalar field potential V ( ψ ) as it is expressed by (28). In the speciallimiting case where ψ ψ = 0, lets say that ψ = 0, the exact solution for the scalar field potential becomes V ( ψ ) = V A t with V A = c θ ζ ( c θ p + (2 − ζ ) − p ζ +2 c θ ( p + θ ) ) c θ ) ; thus, we end up with the power-law potential V ( ψ ) = V A ( ψ ) p + ψ − p + . (29) We assume now that the expansion rate θ ( t ) is a constant, i.e. θ ( t ) = θ . That solution describes the de Sitteruniverse with scale factor a ( t ) = a e θ t . Hence from (21) and (22) we find the closed-form solution for the scalar field ψ ( t ) = ψ ( t − t ) , (30)where ψ , t are integration constants; for the scalar field potential it follows V ( ψ ( t )) = (1 + c θ )3 θ + 4 c θ ζ θ ( t − t ) + 2 ( c θ ) (2 c θ − ζ )27 ζ ( t − t ) . (31)Finally we end with the functional form of the potential V ( ψ ( t )) = V B + V B ψ − + V C ψ − . (32)where V B = (1+ c θ )3 θ , V B = c θ ζ ψ θ and V C = c θ ) (2 c θ − ζ )27 ζ ( ψ ) . Let us consider the scale factor a ( t ) = a exp (cid:0) − a t (cid:1) which describes an exact solution of Einstein’s GeneralRelativity with quadratic corrections terms [59]. This solution can also be recovered by a modified Chaplygin gas inGeneral Relativity [60]. For this scale factor we calculate θ ( t ) = − θ c θ ) ζ t , where we have set a = − θ c θ c θ ) ζ .Therefore, from the field equations (21) and (22) it follows ψ ( t ) = ψ e √ θ t + ψ e −√ θ t , (33)27 (1 + c θ ) ζ θ c θ (cid:16) e √ θ t ψ + ψ (cid:17) V ( ψ ( t )) = 2 c θ (cid:16) e √ θ t (cid:16)p θ t − (cid:17) ψ + ψ (cid:16)p θ t + 1 (cid:17)(cid:17) + (3 (1 + c θ ) ζ − c θ ) (cid:16) e √ θ t ψ − ψ (cid:17) . (34)For ψ = 0, the scalar field potential is written as V ( ψ ) = V C + V C ln (cid:18) ψψ (cid:19) + V C (cid:18) ln (cid:18) ψψ (cid:19)(cid:19) , (35)where V C = θ (2 c θ (1+ c θ ) − ζ )27(1+ c θ ) ζ , V C = − θ c θ c θ ) ζ and V C = θ c θ c θ ) ζ . a ( t ) = a t α e α t Scale factor of the form a ( t ) = a t α e α t has been studied before in [60]. For this solution we find θ ( t ) = 3 (cid:0) a t + a (cid:1) ,and ˙ θ ( t ) = 3 a t . For simplicity we replace a = θ c θ ζ (1+ c θ ) , while for the scalar field it follows ψ ( t ) = ψ t q + + ψ t q − , q ± = 12 (cid:16) p θ (cid:17) , (36)27 t ζ V ( ψ ( t )) = 4 c θ q − − c θ q − ( q − − a t ) ζ + 81 a (1 + c θ ) t ζ + 2 c θ (cid:0) c θ q − + 9 a (1 + c θ ) tζ (cid:1) c θ θ + c θ c θ ++ 4 c θ ( q + − q − ) t q + (cid:0) (1 + c θ ) (2 c θ q − − q − ζ + 9 a tζ ) + θ c θ (cid:1) (1 + c θ ) ψ ( t ) ψ ++ 2 c θ ( q + − q − ) t q (2 c θ − ζ ) ψ ( t ) ψ . (37)For ψ = 0 , it follows U ( φ ) = V D + V D (cid:18) ψψ (cid:19) − q + + V D (cid:18) ψψ (cid:19) − q + , (38)where V D = 3 a (1 + c θ ), V D = a c θ (2 q + + θ )3 ζ and V D = c θ (1+ c θ ) q (2 c θ − ζ )+4 c θ q + θ + c θ θ c θ ) ζ . The parametric plot of thescalar field potential V ( ψ ) is presented in Fig. 2 for various values of the free parameters. ψ V ( ψ ) ψ - V ( ψ ) FIG. 2: Qualitative behaviour of the scalar field potential V ( ψ ) for various values of the free parameters ψ , ψ . Solid line isfor ( ψ , ψ ) = (cid:0) , (cid:1) , dotted line is for ( ψ , ψ ) = (cid:0) , (cid:1) , dashed line is for ( ψ , ψ ) = (cid:0) , − (cid:1) . From the plots we observe thatthe potential behaves like a power-law function. The plots are for θ = 1 , ζ = 1 and c θ = 6 and a = 1 . The potential is forthe scale factor a ( t ) = a t α e α t . Consider the power-law solution θ ( t ) = Af t − (1 − f ) , A > , < f < q = − (1 − f ) Af t − f and scale factor a ( t ) = a e At f . The solution describes an inflationaryuniverse [38–40]. The expansion of the universe with this scale factor is slower than the de Sitter inflation ( a ( t ) = a e θ t ), but faster than the power law inflation ( a ( t ) = a t q where q > V ( ψ ( t )) = 127 A ( c θ + 1) f t f − + 4 Ac θ f t f − ˙ ψ ζψ ( t ) + 2 c θ (2 c θ − ζ ) ˙ ψ ζ ψ , (39) V ′ ( ψ ) = − Ac θ f t f − ˙ ψ ζψ + 2 A ( f − f (3( c θ + 1) ζ − c θ ) t f − ζψ + 4 c θ (3 ζ − c θ ) ˙ ψ ζ ψ , (40)¨ ψ = − A ( c θ + 1)( f − f ζt f − ψ c θ . (41)Choosing 0 < f < , A >
0, we obtain the exact solution ψ = 2 − f √ t ( c θ f ) − /f (cid:0) − A ( f − f ζ (cid:1) f | c θ + 1 | f × " ψ Γ (cid:18) f − f (cid:19) I − f √ At f/ p − ( f − f ζ p | c θ + 1 | c θ f ! + ψ ( − f Γ (cid:18) f (cid:19) I f √ At f/ p − ( f − f ζ p | c θ + 1 | c θ f ! , (42) ψ V ( ψ ) ψ - V ( ψ ) FIG. 3: Qualitative behaviour of the scalar field potential V ( ψ ) for the intermediated inflation model, plot is for( ψ , ψ , c θ , ζ, f, A ) = (0 , , , − , / , where ψ , ψ are integration constants and I /f ( t ) denotes the Bessel function. Considering the condition 0 < f < ψ = 0 to obtain real solutions. In Fig. 3 the qualitative behaviour of the latter scalar field potential ispresented. Consider the solution θ ( t ) = Aλ ln λ − ( t )3 t , where λ and A are dimensionless constant parameters such that λ > A >
0, with deceleration parameter q = − ln − λ ( t ) Aλ − ( λ −
1) ln − λ ( t ) Aλ and scale factor a ( t ) = exp[ A (ln t ) λ ]. Thisgeneralized model of the expansion of the universe is called log-mediate inflation [40, 44]. Note that for the specialcase in which λ = 1 , A = p , the log-mediate inflation model becomes a power-law inflation model [80].With these assumptions the field equations (21), (22), (23) becomes: V ( ψ ( t )) = A ( c θ + 1) λ ln λ − ( t )27 t + 4 Ac θ λ ln λ − ( t ) ˙ ψ ζtψ + 2 c θ (2 c θ − ζ ) ˙ ψ ζ ψ , (43) V ′ ( ψ ( t )) = − Ac θ λ ln λ − ( t ) ˙ ψ ζtψ + 2 Aλ (3( c θ + 1) ζ − c θ )( λ − ln( t ) −
1) ln λ − ( t )27 ζt ψ + 4 c θ (3 ζ − c θ ) ˙ ψ ζ ψ , (44)¨ ψ = A ( c θ + 1) ζλψ ln λ − ( t )( − λ + ln( t ) + 1)2 c θ t . (45)However, equation (45) is not integrable in closed form. We propose the asymptotic expansion ψ ( t ) ∼ c t α ln( t ) + ǫ ( ψ t + ψ ) t α − + O (cid:0) ǫ (cid:1) , for tǫ < B for some B > , and 0 < ǫ ≪ . (46)The equation (45) becomes0 = R ( t, α ; ǫ, λ, ζ ) := c t α − (cid:16) Aζλ ( λ − ln( t ) −
1) ln λ ( t ) + ln( t )(2 α + ( α − α ln( t ) − (cid:17) ln( t )+ ǫt α − (cid:18) Aζλ ( ψ t + ψ ) ( λ − ln( t ) −
1) ln λ − ( t ) + 14 (2 α −
1) ((2 α − ψ + (2 α + 1) ψ t ) (cid:19) + O (cid:0) ǫ (cid:1) , (47)For α ∈ (cid:8) , , , − , − , (cid:0) − √ λ (cid:1)(cid:9) , it is verifiedlim t →∞ R ( t, α ; ǫ, λ, ζ ) = 0 . Moreover, setting α = − k, k >
0, it follows R ( t, α ; ǫ, λ, ζ ) := c t − k − (cid:16) Aζλ ( λ − ln( t ) −
1) ln λ ( t ) + ln( t )((4( k − k + 3) ln( t ) − k + 8) (cid:17) t )+ ǫt − k − (cid:16) Aζλ ( ψ t + ψ ) ( λ − ln( t ) −
1) ln λ ( t ) + ( k −
1) ln ( t ) ( ψ ( k − t + ψ k ) (cid:17) ln ( t ) + O (cid:0) ǫ (cid:1) , (48)lim t →∞ R ( t, α ; ǫ, λ, ζ ) = 0 . (49)Hence, the gravitational field equations in Einstein-æther-Weyl theory in a spatially flat FLRW background spacedescribed by the set of differential equations (18)-(19) with scale factor a ( t ) = exp[ A (ln t ) λ ] admits an asymptoticsolution ψ ( t ) ∼ c t α ln( t ) , (50) V ( ψ ( t )) ∼ A ( c θ + 1) ζ λ ln λ ( t ) + 4 Ac θ ζλ ( α ln( t ) + 1) ln λ ( t ) + 2(2 c θ − ζ )( αc θ ln( t ) + c θ ) ζ t ln ( t ) , (51)as t → ∞ , for any α < .That, is V ∼ O (cid:18) t (cid:19) ! (cid:16) ln λ ( t ) + ln λ ( t ) + 1 (cid:17) , (52) ψ ∼ t α − c ln (cid:18) t (cid:19) + O (cid:18) t (cid:19) !! . (53) Λ CDM universe
As a final application we consider the scale factor which describes the Λ-cosmology, i.e. a ( t ) = a sinh ( ωt ). Thus,for this exact solution the scalar field is found to be expressed in terms of the hypergeometric function ψ ( t ) = ψ (tanh ( ωt )) − µ F (cid:18) − µ , − µ , − µ, tanh ( ωt ) (cid:19) + (54)+ ψ (tanh ( ωt )) + µ F (cid:18)
14 + µ ,
34 + µ , µ, tanh ( ωt ) (cid:19) , (55)where µ = √ c θ +12 ζ (1+ c θ )2 c θ , and ψ , ψ are two integration constants. For simplicity we omit the presentation of theexact form of the scalar field potential V ( ψ ). Thus for specific values of the free parameters we present the parametricevolution of V ( ψ ) in Fig. 4.
4. INTEGRABILITY OF THE GRAVITATIONAL FIELD EQUATIONS
In Section 3 we solved the gravitational field equations for different scale factors, which are of interests as cosmologi-cal solutions. The exact solutions of our analysis have the sufficient number of initial constants of integration, they arethe constants ψ , ψ and the non-essential constant of the time translation t → t + t which we have omitted. Hence,the solutions that we have found are the general analytic solutions of the nonlinear dynamical system which providethese specific scale factors. Note, that we have not considered any functional form for the scalar field potential butfor all the cases that we have studied, a scalar field potential can be found. Our analysis is motivated by the originalwork on cosmological solutions in scalar field theory by Ellis and Madsen [61]. There, the solutions that have been0 (cid:27)(cid:28)(cid:29) (cid:30)(cid:31) !" ψ / ψ V ( ψ ) / V ψ - V ( ψ ) FIG. 4: Qualitative behaviour of the scalar field potential V ( ψ ) for various values of the free parameters ψ , ψ . Solid line isfor ( ψ , ψ ) = (cid:0) , (cid:1) , dotted line is for ( ψ , ψ ) = (cid:0) , (cid:1) , dashed line is for ( ψ , ψ ) = (cid:0) , − (cid:1) . From the plots we observe thatthe potential behaves like a power-law function. The plots are for ω = 1 , ζ = 1 and c θ = 20. The axes has been normalized.The potential is for the ΛCDM scale factor. found are exact solutions and particularly, they are special solutions and not the complete solution of the dynamicalsystem. Some analytic solutions in scalar field cosmology can be found by using techniques of analytic mechanicssuch is the theory of invariant transformations [62–64]. However in this study we have not applied any symmetry inorder to find the solutions, that indicates that except from the constraint equation another conservation law shouldalways exists for any functional form of the scalar field potential.The new scalar field ψ ( t ) = exp (cid:16) − ζ c θ φ ( t ) (cid:17) that we defined it was not an ad hoc selection. Indeed, in thesecoordinates by replacing V ( ψ ) from (21) in (22) we end with the second-order differential equation of the form¨ ψ + ω ( t ) ψ = 0 . (56)where ω ( t ) = (cid:16)
32 (1+ c θ ) c θ ζ ˙ θ (cid:17) .The second-order differential equation is a linear equation also known as the time-dependent oscillator [65]. Thedifferential equation (56) admits the conservation law [66] I = 12 (cid:16) y ˙ ψ − ˙ yψ (cid:17) + (cid:18) ψy (cid:19) ! , (57)where y = y ( t ) is any solution of the Ermakov-Pinney equation¨ y + ω ( t ) y − y − = 0 . (58)Conservation law (57) it is known as Lewis invariant and it was derived for the first time as an adiabatic invariant[67]. Alternatively, the conservation law (57) can be constructed through a set of canonical transformations [68] orwith the use of Noether’s theorem [65]. The set of equations (56)-(58) it is also known as the Ermakov system whichcan be found in many applications in physical science [69–72].Hence, for the gravitational field equations (19)-(20) the following theorem holds. Theorem:
The gravitational field equations in Einstein-æther-Weyl theory in a spatially flat FLRW backgroundspace described by the set of differential equations (18)-(19) form an integrable dynamical system for arbitrary potential. The two conservation laws are the constraint equation (18) and the Lewis invariant I (cid:16) φ, ˙ φ, y (cid:17) = 12 e − ζcθ φ (cid:18) ζ c θ y ˙ φ + ˙ y (cid:19) + y − ! , (59) where y ( t ) satisfies the Ermakov-Pinney equation (58). It is important to mention at this point that in another lapse function dt = N ( τ ) dτ in the metric tensor (11) ourresults are valid. In such a case, the equivalent equation (56) it is of the form d ψdτ + α ( τ ) dψdτ + β ( τ ) ψ = 0 , (60)which also admits an invariant function [68] similar to the Lewis invariant.Except from the Lewis invariant, the linear differential equation (60) is maximally symmetric and admits eight Liepoint symmetries which form the SL (3 , R ) Lie algebra [73], for arbitrary functions α ( τ ) and β ( t ). Hence, accordingto S. Lie theorem the differential equation (60) is equivalent to the free particle Y ′′ = 0 and there exists a pointtransformation { τ, ψ ( τ ) } → { χ, Y ( χ ) } which transform equation (60) into that of the free particle, for more detailswe refer the reader to the review article [74]. That is an alternative way to prove the integrability of the gravitationalfield equations for the cosmological model of our consideration.
5. CONCLUSIONS
In this work we considered a spatially flat FLRW background space in Einstein-æther theory defined in Weylintegrable geometry. The novelty of this approach is that a scalar field coupled to the æther field is introduced in ageometric way. For this model we investigated the existence of exact solutions of special interests, in particular wefocused on exact solutions which can describe the inflationary epoch of our universe.Indeed, we proved that the cosmological model of our consideration can provide exact solutions such that thepower-law inflation, de Sitter expansion, quadratic Lagrangian inflation and others. For these specific scale factorswe were able to calculate the closed-form expression of the scalar field solution and of the scalar field potential.Moreover, we investigate also the possibility of Einstein-æther-Weyl cosmological model to admit a cosmologicalsolution where the scalar field unify the dark matter and the dark energy of the universe, and for that investigation weproved that there exists a scalar field potential which can describe explicitly the ΛCDM universe. Scalar field modelswhich unify the dark components of the universe have been drawn the attention of the academic society because theyprovide a simple mechanism for the observable universe, see [75–79] and references therein.However, the main result of this work is that we were able to prove the integrability of the field equations of ourcosmological model for arbitrary potential function. In particular we found a point transformation which reduce oneof the two equations to the linear equation of the time-dependent oscillator, and to prove that the Lewis invariant is aconservation law for the field equations for arbitrary scalar field potential. This is an interesting result which we didnot expect it, assuming the nonlinearity form of the field equations and mainly that according to our knowledge thereis not any effective Lagrangian description for the cosmological field equations in order to apply techniques for theinvestigation of conservation laws similar with that applied before for the quintessence or the scalar tensor theories.From this work it is clear that in the background space the Einstein-æther-Weyl cosmological model is cosmologicalviable. Thus in a future work we plan to investigate further the physical properties of this theory as an inflationarymodel and as a unified model for the dark components of the universe.
Acknowledgments
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