Integrability and Cosmological Solutions in Einstein-aether-Weyl theory

We consider a Lorentz violating scalar field cosmological model given by the modified Einstein-aether theory defined in Weyl integrable geometry. The existence of exact and analytic solutions is investigated for the case of a spatially flat Friedmann--Lema\^{\i}tre--Robertson--Walker background space. We show that the theory admits cosmological solutions of special interests. In addition, we prove that the cosmological field equations admit the Lewis invariant as a second conservation law, which indicates the integrability of the field equations.


INTRODUCTION
A plethora of modified or alternative theories to Einstein's gravity [1,2] have been proposed during the last years in 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 Hořava-Lifshitz theory [3] and the Einstein-aether theory [4,5].
Hořava-Lifshitz gravity is a power-counting renormalizable theory with consistent ultra-violet behaviour exhibiting an anisotropic Lifshitz scaling between time and space at the ultra-violet limit [3]. On the other hand, in Einsteinaether theory, the quadratic invariants of the kinematic quantities of a unitary time-like vector field, which is called aether field, are introduced in the gravitational Action integral; modifying the Einstein-Hilbert Action [4]. The Einstein-aether action is the most general second-order theory which is defined by the spacetime metric and the aether field involving no more than two derivatives [6] (not including total derivatives). There are several gravitational and cosmological applications for both of these theories in the literature, for instance see [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and references therein. Scalar fields play a significant role in the explanation of the early acceleration phase of the universe known as inflation. Lorentz violating scalar field theories have been studied in Hořava-Lifshitz gravity [25][26][27][28] and in the Einstein-aether theory [5,[29][30][31][32][33][34]. In [29] it has been proposed an Einstein-aether scalar field model in which the coupling coefficients of the aether field with gravity are functions of the scalar fields. From the latter an interaction between the scalar field and the aether field it follows. For that model it was found that the inflationary era is divided into two parts, a Lorentz-violating stage and the standard slow-roll stage. In the Lorentz-violating stage the universe expands 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 field cosmological model. We consider the Einstein-aether theory defined in Weyl integrable geometry [35]. In this specific theory the Action Integral of the Einstein-aether is modified such that a scalar field coupled to the aether field is introduced in a geometric way. The global dynamics of the background space were studied in [35] for various cosmological models in the absence or in the presence of matter. It was found that the model provides several cosmological eras in agreement with the cosmological history. In addition, a Weyl manifold is a conformal manifold equipped with a connection which preserves the conformal structure and is torsion-free. In Weyl integrable theory the geometry is supported by the metric tensor and a connection structure which differs from the conformal equivalent metric by a scalar field [36,37]. Cosmological and gravitational applications of the Weyl geometry can be found for instance in [46][47][48][49][50][51][52][53][54]. The novelty of the Weyl geometry is that the scalar field in the gravitational Action integral is introduced by the geometry.
The context of integrability is essential in all areas of physics. A set of differential equations describing a physical system is said to be integrable if there exist a sufficient number of invariant functions such that the dynamical system can be written in algebraic form. When the latter algebraic system is explicitly solved the solution of the dynamical system can be expressed in closed-form [56,57]. The study of integrability properties of nonlinear dynamical systems is 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 of breaking 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 is that of Einstein-aether defined in Weyl integrable geometry assuming a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) background space without any matter source terms. In Section 3 we present for the first time analytic and exact solutions for this cosmological model, we focus on the existence of exact solutions where the scale factor describes inflationary models of special interests. We obtain those solutions which are determined as the general analytic solutions for the corresponding scalar field potentials. In Section 4 we show that this is possible because the cosmological field equations form an integrable dynamical system, where the conservation laws are the constraint cosmological equation, i.e. the modified first Friedmann's equation and the Lewis invariant. The later invariant is essential for the study of integrability of time-dependent classical or quantum systems. We show that the field equations form an integrable dynamical system for an arbitrary potential function. Finally, in Section 5 we summarize our results and we draw our conclusions.

EINSTEIN-AETHER-WEYL THEORY
The Einstein-aether-Weyl gravitational model is an extension of the Lorentz violating Einstein-aether theory in Weyl integrable geometry. It is a scalar field Lorentz violating theory where there is a coupling between the scalar field and the aether field. The corresponding gravitational Action integral has the form of Einstein-aether gravity, thus it is generalized in Weyl integrable geometry. The latter generalization provides a geometric mechanism for the introduction 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 g µν ,∇ µ , where∇ µ is not defined by the Levi-Civita connection of g µν , but by the affine connectionΓ κ µν (g) with the property∇ andg µν is the metric compatible with∇ µ . We study Weyl integrable geometry, where the gauge vector field ω µ which defines the geometry is a gradient vector field, i.e., it satisfies ω µ = φ ,µ for an scalar field φ. Then, it is defined the new metric tensorg µν = e −φ g µν as the conformal related metric compatible with the covariant derivative∇ µ , i.e, ∇ κgµν = 0. ConnectionsΓ κ µν can be constructed from the Christoffel symbols Γ κ µν (g) of the metric tensor g µν as follows [55]: The gravitational integral of the Einstein-aether-Weyl theory is [35]: where S W g µν ,Γ κ µν is the extension of the Einstein-Hilbert action in Weyl geometry [55]: withR denoting the Weylian scalar curvaturẽ and ξ is an arbitrary coupling constant. S AE g µν ,Γ κ µν ; u µ is the Action Integral for the aether field u µ defined in Weyl geometry, that is: whereg µν = e −φ g µν is the conformal related metric associated with the covariant derivative∇ µ . Parameters c 1 , c 2 , c 3 and c 4 are dimensionless constants and define the coupling between the aether field and the conformal metric through a kinetic term. Lagrange multiplier λ ensures the unitarity of the aether field, i.e. g µν u µ u ν = −1, while the fourth-rank tensorK αβµν is defined as Equivalently, the Action Integral (6) can expressed in terms of the kinematic quantities θ ,σ µν ,ω µν ,α µ as follows where the new parameters c θ , c σ , c ω , c a are functions of c 1 , c 2 , c 3 and c 4 , that is, From (3) we find the gravitational field equations to bẽ where T ae ab is the energy momentum tensor which corresponds to the aether field andG µν is the Einstein tensor in Weyl theory, that is,G µν =R µν − 1 2R g µν . The rhs of equation (9) corresponds to the energy-momentum tensor of the aether field: where J µ ν = −K µβ να∇β u α .

FLRW spacetime
In the case of a spatially flat FLRW geometry, with line element for the comoving aether field we calculatẽ where θ is the Riemannian expansion rate defined as θ = 3ȧ a . The gravitational field equations are expressed as follows where ρ φ , p φ are the energy density and pressure of the scalar field, that is, where parameter ζ := 2ξ − 3 2 is a coupling parameter between the scalar field and the gravity. Furthermore, ρ ae , p ae are the density and pressure of the aether field, defined as By replacing in (13), (14) we derive the gravitational field equations while the equation of motion for the scalar field is In the following we investigate the existence of analytic solutions for the dynamical system (18)- (20). It is important to mention that the three equations are not independent, indeed equation (18) is a conservation law for the higher-order equations (19), (20).
It is important to mention that in the latter dynamical system for ζ = 0 the system is degenerated and it has only one dependent variable, hence, in the following we consider that ζ = 0.

EXACT SOLUTIONS
Before to proceed with the derivation of the exact solution we perform the following change of variable , where now the gravitational field equations becomes Finally, the gravitational equations reduce to 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.

Power-law solution
Consider the power-law solution θ (t) = , the ideal gas is that of dust fluid, radiation or stiff fluid respectively.
From the plots we observe that the potential behaves like a power-law function. The plots are for θ0 = 1, ζ = 1 and c θ = 6 where the power-law solution describes an accelerated universe. The potentials are for the power-law scale factor By replacing in (21) and (22) we calculate for the scalar field In Fig. 1, we present the parametric plot for the scalar field potential V (ψ) as it is expressed by (28). In the special limiting case where ψ 1 ψ 2 = 0, lets say that ψ 2 = 0, the exact solution for the scalar field potential becomes V (ψ) = ; thus, we end up with the power-law potential

de Sitter spacetime
We assume now that the expansion rate θ (t) is a constant, i.e. θ (t) = θ 0 . That solution describes the de Sitter universe with scale factor a (t) = a 0 e θ 0 3 t . Hence from (21) and (22) we find the closed-form solution for the scalar field where ψ 1 , t 0 are integration constants; for the scalar field potential it follows Finally we end with the functional form of the potential where

Quadratic Lagrangian inflation
Let us consider the scale factor a (t) = a 0 exp −a 1 t 2 which describes an exact solution of Einstein's General Relativity with quadratic corrections terms [59]. This solution can also be recovered by a modified Chaplygin gas in General Relativity [60]. For this scale factor we calculate θ (t) = − 2θ0 3(1+c θ )ζ t, where we have set a 1 = − θ0c 2 θ 9(1+c θ )ζ . Therefore, from the field equations (21) and (22) it follows For ψ 2 = 0, the scalar field potential is written as where 3.4. Scale factor a (t) = a0t α 1 e α 2 t Scale factor of the form a (t) = a 0 t α1 e α2t has been studied before in [60]. For this solution we find θ (t) = 3 a1 t + a 2 , andθ (t) = 3 a1 t . For simplicity we replace a 1 = θ0c 2 θ 9ζ(1+c θ ) , while for the scalar field it follows For ψ 2 = 0, it follows where . The parametric plot of the scalar field potential V (ψ) is presented in Fig. 2 for various values of the free parameters.
From the plots we observe that the potential behaves like a power-law function. The plots are for θ0 = 1, ζ = 1 and c θ = 6 and a2 = 1. The potential is for the scale factor a (t) = a0t α 1 e α 2 t .

Intermediate inflation
Consider the power-law solution θ (t) = 1 3 Af t −(1−f ) , A > 0, 0 < f < 1 which describes intermediated inflation with deceleration parameter q = −1 + (1−f ) Af t −f and scale factor a (t) = a 0 e At f . The solution describes an inflationary universe [38][39][40]. The expansion of the universe with this scale factor is slower than the de Sitter inflation (a (t) = a 0 e θ 0 3 t ), but faster than the power law inflation (a(t) = a 0 t q where q > 1). It was shown that the intermediate inflation arises as the slow-roll solution to potentials which fall off asymptotically as an inverse power law inflation in the standard canonical framework and can be modelled by an exact cosmological solution [39,40]. The intermediate inflation has also been studied in some warm inflationary scenarios in order to examine its predictions for inflationary observables [41][42][43][44]. With these assumptions the field equations (21), (22), (23) becomes: Choosing 0 < f < 1, A > 0, we obtain the exact solution where ψ 1 , ψ 2 are integration constants and I 1/f (t) denotes the Bessel function. Considering the condition 0 < f < 1 we set ψ 1 = 0 to obtain real solutions. In Fig. 3 the qualitative behaviour of the latter scalar field potential is presented.

3t
, where λ and A are dimensionless constant parameters such that λ > 1 and A > 0, with deceleration parameter q = −1 + ln 1−λ (t) Aλ and scale factor a(t) = exp[A(ln t) λ ]. This generalized model of the expansion of the universe is called log-mediate inflation [40,44]. Note that for the special case in which λ = 1, A = p, the log-mediate inflation model becomes a power-law inflation model [80].
Hence, the gravitational field equations in Einstein-aether-Weyl theory in a spatially flat FLRW background space described by the set of differential equations (18)- (19) with scale factor a(t) = exp[A(ln t) λ ] admits an asymptotic solution as t → ∞, for any α < 3 2 . That, is

ΛCDM universe
As a final application we consider the scale factor which describes the Λ-cosmology, i.e. a (t) = a 0 sinh 2 3 (ωt). Thus, for this exact solution the scalar field is found to be expressed in terms of the hypergeometric function where µ = √ c 2 θ +12ζ(1+c θ ) 2c θ , and ψ 1 , ψ 2 are two integration constants. For simplicity we omit the presentation of the exact form of the scalar field potential V (ψ). Thus for specific values of the free parameters we present the parametric evolution of V (ψ) in Fig. 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 cosmological solutions. The exact solutions of our analysis have the sufficient number of initial constants of integration, they are the constants ψ 1 , ψ 2 and the non-essential constant of the time translation t → t + t 1 which we have omitted. Hence, the solutions that we have found are the general analytic solutions of the nonlinear dynamical system which provide these specific scale factors. Note, that we have not considered any functional form for the scalar field potential but for all the cases that we have studied, a scalar field potential can be found. Our analysis is motivated by the original work on cosmological solutions in scalar field theory by Ellis and Madsen [61]. There, the solutions that have been found are exact solutions and particularly, they are special solutions and not the complete solution of the dynamical system. Some analytic solutions in scalar field cosmology can be found by using techniques of analytic mechanics such is the theory of invariant transformations [62][63][64]. However in this study we have not applied any symmetry in order to find the solutions, that indicates that except from the constraint equation another conservation law should always exists for any functional form of the scalar field potential.
The new scalar field ψ (t) = exp − 3ζ 2c θ φ (t) that we defined it was not an ad hoc selection. Indeed, in these coordinates by replacing V (ψ) from (21) in (22) we end with the second-order differential equation of the form where ω (t) = 3 2 (1+c θ ) c 2 θ ζθ . The second-order differential equation is a linear equation also known as the time-dependent oscillator [65]. The differential equation (56) admits the conservation law [66] where y = y (t) is any solution of the Ermakov-Pinney equation 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] or with the use of Noether's theorem [65]. The set of equations (56)- (58) it is also known as the Ermakov system which can be found in many applications in physical science [69][70][71][72].
Hence, for the gravitational field equations (19)-(20) the following theorem holds. Theorem: The gravitational field equations in Einstein-aether-Weyl theory in a spatially flat FLRW background space 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 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) our results are valid. In such a case, the equivalent equation (56) it is of the form 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 Lie point symmetries which form the SL (3, R) Lie algebra [73], for arbitrary functions α (τ ) and β (t). Hence, according to S. Lie theorem the differential equation (60) is equivalent to the free particle Y ′′ = 0 and there exists a point transformation {τ, ψ (τ )} → {χ, Y (χ)} which transform equation (60) into that of the free particle, for more details we refer the reader to the review article [74]. That is an alternative way to prove the integrability of the gravitational field equations for the cosmological model of our consideration.

CONCLUSIONS
In this work we considered a spatially flat FLRW background space in Einstein-aether theory defined in Weyl integrable geometry. The novelty of this approach is that a scalar field coupled to the aether field is introduced in a geometric way. For this model we investigated the existence of exact solutions of special interests, in particular we focused 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 the power-law inflation, de Sitter expansion, quadratic Lagrangian inflation and others. For these specific scale factors we 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-aether-Weyl cosmological model to admit a cosmological solution where the scalar field unify the dark matter and the dark energy of the universe, and for that investigation we proved that there exists a scalar field potential which can describe explicitly the ΛCDM universe. Scalar field models which unify the dark components of the universe have been drawn the attention of the academic society because they provide a simple mechanism for the observable universe, see [75][76][77][78][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 our cosmological model for arbitrary potential function. In particular we found a point transformation which reduce one of the two equations to the linear equation of the time-dependent oscillator, and to prove that the Lewis invariant is a conservation law for the field equations for arbitrary scalar field potential. This is an interesting result which we did not expect it, assuming the nonlinearity form of the field equations and mainly that according to our knowledge there is not any effective Lagrangian description for the cosmological field equations in order to apply techniques for the investigation 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-aether-Weyl cosmological model is cosmological viable. Thus in a future work we plan to investigate further the physical properties of this theory as an inflationary model and as a unified model for the dark components of the universe.