Quantization of Einstein-aether Scalar field Cosmology

We present, for the first time, the quantization process for the Einstein-aether scalar field cosmology. We consider a cosmological theory proposed as a Lorentz violating inflationary model, where the aether and scalar fields interact through the assumption that the aether action constants are ultra-local functions of the scalar field. For this specific theory there is a valid minisuperspace description which we use to quantize. For a particular relation between the two free functions entering the reduced Lagrangian the solution to the Wheeler-DeWitt equation as also the generic classical solution are presented for any given arbitrary potential function.


INTRODUCTION
The interest in Lorentz violating cosmological theories comes together with the idea of a variable speed of light. The Horava-Lifshitz gravity is certainly violating Lorentz invariance by construction, since arbitrary time re-definitions are not among its covariances [1]. In the Einstein-aether theory, a unit time-like rotationally invariant vector field, called the "aether" is also responsible for the same effect [2]. Specifically, in Einstein-aether theory quadratic kinematic quantities of the unitary time-like vector fields are introduced in the gravitational Action Integral [3,4]. These new terms break the Lorentz symmetry [5], by selecting a preferred frame at each space time point, while keeping the field equations of second-order as in the case of General Relativity.
On the other hand, scalar fields play a prominent role in modern cosmology. The main mechanism for the description of the inflation is based on the domination of a scalar field potential, known as inflaton [6]. Moreover, scalar fields have also been proposed as dark energy models, while they can attribute the geometrodynamical degrees of freedom provided by higher-order theories which belong to the class of modified theories of gravity, for more details we refer the reader to [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].
One can contemplate a non-trivial coupling of the scalar to the aether field by allowing the coefficients of its kinetic action to be functions of the scalar field [23]. Such a theory has been proposed before as an alternative inflationary model which provides two periods of inflation [24]. A classical slow-roll era and a Lorentz violating epoch. In this work we realize this idea for the case of a spatially flat FLRW universe and a scalar field with arbitrary potential and we present, for the first time, the quantization of the Einstein-aether scalar field cosmology; we also derive the generic algebraic classical solution to the field equations. Einstein-aether scalar field theory are of special interest in the scientific society and there are various studies in the literature on the subject, some of these studies for homogeneous and inhomogeneous spacetimes can be found in [25][26][27][28][29][30][31][32]. The structure of the paper is as follows.
In Section 2, we define the cosmological model that we focus on in this work; it is the Einstein-aether scalar field cosmology in a homogeneous and isotropic geometric background space where only quadratic terms of the derivatives exist in the Action Integral. The latter property is essential for enabling a minisuperspace description of the gravitational field equations. In Section 3 we present the point-like Lagrangian of our model. This specific model has been proposed before as an alternative model for inflation. The quantization process is presented in Section 4. In Section 5 we study the classical limit while we discuss our results and we draw our conclusions in Section 6.

EINSTEIN-AETHER SCALAR FIELD COSMOLOGY
Kanno and Soda in [24] proposed a Lorentz violating Einstein-aether Action Integral while assuming the Einsteinaether coupling parameters to be functions of this scalar field, thus generating a coupling between the scalar and the aether field. The main characteristic of this model is that the inflationary epoch can be described by two stages; the usual slow-roll stage and the new Lorentz violating stage.
A more general Einstein-aether scalar field model was later proposed in [33], where a scalar field is introduced in the Einstein-aether Action Integral with the scalar field potential being a function of the field and the kinematic invariants of the aether field. The model of Kanno and Soda is included as a special case in that of [33]. While the model proposed in [33] describes a second-order theory, only in the limit of [24] the Action Integral depends only on quadratic terms of the derivatives. As we shall see in the following this is an essential property in order for the dynamical field equations to admit a minisuperspace description.
The Einstein-aether scalar field model proposed in [24] is described by the Action Integral where S Aether describes the terms of the aether field u µ as follows Function λ is the Lagrange multiplier which is introduced to ensure the unitarity of the aether field, i.e. u µ u µ +1 = 0. Coefficients β 1 , β 2 , β 3 and β 4 define the coupling between the aether and the gravitational field. While in the Einsteinaether theory the coefficients are constants in this specific theory they are functions of the scalar field φ.
We select the case of a homogeneous and isotropic spacetime described by the spatially flat FLRW line element where a (t) is the scale factor, N (t) is the lapse function. The corresponding Hubble function is defined as H (t) = 1 Nȧ a (where a dot denotes total derivative with respect to the variable t) and for the aether field we have u µ = 1 N δ µ t . These assumptions imply that φ = φ(t) and then the gravitational field equations follow from the variation of the action [24] where the new function B (φ) is expressed as B (φ) = β 1 (φ) + 3β 2 (φ) + β 3 (φ) + 1. We observe that in the limit where B (φ) =const., the field equations take the form of the quintessence scalar field model in General Relativity, which means that the Lorentz violating inflationary stage does not exist. Thus in the following we consider the case where B (φ) ,φ = 0. The cosmological field equations (4)-(5) can be written by using the Hubble function H (t) as follows where the new functions ρ ef f and p ef f are the energy density and pressure for the effective fluid defined as ρ φ = 1 2φ is an effect that is observed in the Jordan frame and in scalar tensor theories, however here this effect follows from the time-dependent coupling function for the aether field.

MINISUPERSPACE DESCRIPTION
The superspace is an infinite-dimensional space serving as the basic configuration space of canonical quantum gravity [34]. As defined in the canonical formulation of General Relativity, it consists of all Riemannian 3-dimensional metrics and the matter fields. In cosmology, due to the spacetime symmetries of the geometry, the infinite degrees of freedom of the corresponding superspace are truncated to a finite number and thus a particular minisuperspace model is achieved: In the second integral the spatial dependence has been integrated out of the action (due to the space-time symmetries), leaving only a multiplicative constant V 0 symbolizing the volume of a finite three-space cell. One has always to ensure though that the variation of the new action of finite degrees of freedom gives rise to Euler-Lagrange equations that are equivalent to those of the original field theory, under the assumed ansatz for the metric and the matter fields. In cosmology the L is a singular Lagrangian given, for matter actions quadratic in the field derivatives, by the following expression Functions q α (t), N (t) are the unknown functions which describe the spacetime and the kinematic quantities of the matter source (N (t) correspond to the lapse-function). The G αβ (q) transforms as a second-rank tensor under arbitrary redefinitions of the q µ 's. It is the so-called minisuperspace metric, while U (q) is the effective potential which describes the dynamical interactions of the gravitational field and of the matter source. The Lagrangian function L is a singular Lagrangian since det Not all the cosmological models in General Relativity have a minisuperspace description. For a full scale factor matrix and non vanishing shift, only the Bianchi models which belong to the Class A and the Bianchi V admit a minisuperspace description. Moreover, there are some inhomogeneous models where the field equations follow from a point-like Lagrangian of the form (10). In the context of alternative theories of gravity not all the proposed theories have a minisuperspace description.
The existence of a Lagrangian function for a given dynamical system, known also as the inverse problem, is essential in physics. In addition, the existence of a point-like Lagrangian for the given dynamical system can be used for the quantization process, which is the main approach applied in quantum cosmology. From there various approaches can be followed, e.g. canonical theory, loop quantum cosmology, path integrals etc. [35][36][37][38][39][40][41]. The importance of the existence of a Lagrangian description of a given set of equations lies in the reach methods of analytical Mechanics that can be applied in order to study the evolution of the field equations and their integrability. In the minisuperspace approach, the quantum analogs of the classical integrals of motion can be used as supplementary conditions in conjunction to the Wheeler-DeWitt equation, so that the wave function describing quantum states is defined up to constants.
As far as the Einstein-aether theory is concerned in general, the gravitational field equations do not admit a pointlike Lagrangian. The determination of a Lagrangian description for the field equations in Einstein-aether theory was the subject of study in [42,43].
For the cosmological model of our consideration, the unknown functions of the spacetime (3) are the scale factor a and the lapse function N, while from the matter source the dynamical variable is function φ. We observe that the field equations (4)-(6) follow from the variation of the Action Integral where now the Lagrangian function L N, a,ȧ, φ,φ is the point-like Lagrangian [44] L N, a,ȧ, φ,φ where q α = (a, φ) , the minisuperspace metric is and the effective potential is U (q) = a 3 V (φ). The metric defined by the kinetic part of the point-like Lagrangian has dimension two, i.e. dim G αβ = 2, which means that it admits an infinite number of conformal symmetries, independently of the functional form of B (φ). Recall that we assume that B ,φ (φ) = 0.

QUANTIZATION
We can exploit the parametrization invariance of Lagrangian (10) to bring it into an equivalent form which resembles the motion of a free relativistic particle in a (generally) curved space. To this end, we reparametrize the lapse function in (12) in order to obtain Note that L n and L are equivalent, i.e. they reproduce the same set of Euler-Lagrange equations. Having obtained L n in this form allows us to interpret V 0 as the "mass" of the supposed relativistic particle and as the scaled mini-superspace metric corresponding to a "constant (effective) potential" in the Lagrangian, i.e. the metric of the space in which the motion of the free particle takes place, where G αβ is given by (13). In the particular problem we are studying,Ḡ αβ will generally designate a two dimensional curved manifold of hyperbolic signature. However, there exists a large class of models for which this space becomes flat, thus leading to a straightforward quantum description. Specifically, it is easy to see that, if the potential V (φ) and the coupling function B(φ) are related through or equivalently where the V i , B i , i = 1, 2 are constants, then the corresponding metricḠ αβ is that of a flat space. Relations (16) and (17) guarantee that the Riemann curvature tensor of the mini-superspace is zero. As a result, whenever (16) (or equivalently (17)) holds, the system is equivalent to a motion of a free relativistic particle in a two dimensional flat space. Consequently there exist three classical integrals of motion, whose quantum counterparts can be used as observables in a canonical quantum description together with the Wheeler-DeWitt equation. Before proceeding with the quantum description, let us briefly give the connection with the Cartesian coordinates, say (u, v), with respect to which the solution of all this class of models can be obtained straightforwardly. The mini-superspace line element corresponding to the metricḠ αβ is By using (17) and introducing a new variable φ → ψ = V (φ)dφ, the expression (18) for which of course we need to assume B 2 = 0. In the special case where B 2 = 0 the corresponding transformation is easily derived to be We may now proceed with the quantization of the system. The classical Hamiltonian constraint that emerges from Lagrangian (14) is where p α = ∂Ln ∂q α are the momenta and the symbol "≈" denotes a weak equality in the Dirac sense [45]. In the canonical description we assign to the momenta the differential operators p α →p α = −i ∂ ∂q α , while for the factor ordering in the kinetic term of the Hamiltonian constraint we choose to make use of the Laplacian 1 and thus have the quantum constraint operator which -following Dirac's prescription of quantizing constrained systems [46] -we demand to annihilate Ψ, i.e. HΨ = 0 must hold for all the states of the system. The latter defines the Wheeler-DeWitt equation of the mini-superspace model. In this flat two-dimensional configuration space we are studying, there are two well known quantization algebras: one involving the constant translations generators and another the boost in the u − v plane. To utilize the first, we start from the Cartesian coordinates where ds 2 2D = −du 2 + dv 2 and use the two classical integrals of motion, which in these coordinates are just p u and p v . Their quantum counterparts are the commuting operatorsp u = −i ∂ ∂u and p v = −i ∂ ∂v which can be used to define the eigenvalue equationŝ admitting the plane wave solution Ψ(u, v) = Ψ µν (u, v) = 1 2π e i (µu+νv) , which normalizes to a product of Dirac delta functions since The spectrum is continuous and the quantum numbers µ, ν can take values in the entire R domain. However, the quantum Hamiltonian constraint sets the additional condition which forces us to assume that |ν| < |µ|.
The second way to proceed with the canonical quantization is to use the quantum equivalent of the third classical integral of motion, which in these variable is Q = vp u + up v . In this case, it is far more convenient to utilize coordinates in which the corresponding symmetry generator assumes a normal form. In particular we may adopt the transformation u = r cosh θ, v = r sinh θ, which makes the flat space line element ds 2 2D = −dr 2 + r 2 dθ 2 and the aforementioned integral of motion Q = p θ . At the quantum level we can thus write the eigenvalue equation which leads to the solution Ψ(r, θ) = 1 √ 2π e i κθ ψ(r). Note that here we have no reason to consider θ as a periodic variable. As a result we take κ to have a continuous spectrum and be normalized to a Dirac delta function, like the eigenvalues µ, ν previously. The ψ(r) part is to be obtained by the Wheeler-DeWitt equation which results to The latter is the Bessel equation with general solution where the J n (z), Y n (z) are the Bessel equations of the first and second kind respectively. An orthogonal set of states normalized in a distributional sense to a delta function can be constructed with the help of the function which was firstly defined in [48]. Using the typical procedure of deriving the orthogonality condition in a Sturm-Liouville problem, it has been shown that (see the appendix of [49]) under the condition Quantization approaches involving similar Wheeler-DeWitt equations can also be found in [50,51]. Note that the weight r in the integral in (31) is exactly what emerges from using the natural measure for the inner product between states, i.e. the square root of the determinant of the mini-superspace metric (in the (r, θ) coordinates | det(Ḡ αβ )| 1/2 = r). The necessary for the orthogonality of states condition (32), ensures at the same time the Hermiticity of the operatorĤ under the assumption that the wave function vanishes at the boundary of the half line (0, +∞). As a result we may write the full wave function of this case as subject to the condition As it is evident from (33)- (35), we choose to interpret V 0 as some short of "eigenvalue". In this manner we observe that for a fixed κ, a discretization is introduced in the fiducial volume of the three space through the orthogonality condition (35). It is interesting that the study of the reduced mini-superspace system can yield such an information about the three space, which is usually discarded through the process of the reduction.

CLASSICAL SOLUTION
We lastly proceed with the presentation of the classical solution for the gravitational field equations for arbitrary potential . For the sake of simplicity of the resulting expressions we here present the case where B (φ) = B1 V (φ) . For the lapse parameterization N (t) =N (t) a 3 V (φ) −1 (in which the potential of the relevant Lagrangian (12) is free of a, φ ) the aforementioned equations become where the new field ψ is defined as dψ = V (φ)dφ. ForN (t) = 1, these equations can be integrated resulting in the scale factor and the scalar field ψ (t) while the line element (3) reads We observe that the potential functions V (φ) has been included in the line element and consequently it affects all the geometric and physical quantities. The solution that we constructed here is known as algebraic solution because the physical quantities are given by algebraic equations. This kind of solution has been before derived for the quintessence field in [52] with various physical applications [53,54].

CONCLUSIONS
In this piece of work we considered an Einstein-aether scalar field cosmological theory proposed as a Lorentz violating inflationary model. The scalar and aether fields are interacting due to the assumption that the constants of the aether part of the action are taken to be functions of the scalar field. A useful and critical property of this theory is that, for the assumed geometry and the consequent assumptions for the fields, the reduced field equations are correctly described by the corresponding minisuperspace Lagrangian inferred by the reduced action. This occurrence is not at all automatic for arbitrary reductions, and is certainly not common in Einstein-aether theories. Yet, it is an important facilitation in order to study the quantization process.
For the spatially flat FLRW geometry considered, the metric of the two-dimensional minisuperspace (spanned by a, φ) depends on two unknown functions, the scalar field potential V (φ) and the collective gravitational coupling function for the aether field, B (φ). For a specific relation of the two unknown functions, for which the configuration manifold becomes flat, we were able to: (A) write the general algebraic classical solution to the simplified cosmological field equations; and (B) present the quantization of the model which is carried out in the flat coordinates of the configuration space.
Surprisingly enough, the Wheeler-DeWitt equation is revealed as that of a free particle in a two dimensional flat space of hyperbolic signature, while the constant appearing due to spatial integration needs to assume discrete values in order for the states to be orthogonal.
In a future work we plan to apply the classical solution in order to study the physical applications of the model. Also, to investigate the general case of unrelated V (φ), B (φ) as well as different geometries, such as Bianchi I, V .