Wheeler-DeWitt equation and Lie symmetries in Bianchi scalar-field cosmology

Lie symmetries are discussed for the Wheeler-De Witt equation in Bianchi Class A cosmologies. In particular, we consider General Relativity, minimally coupled scalar field gravity and Hybrid Gravity as paradigmatic examples of the approach. Several invariant solutions are determined and classified according to the form of the scalar field potential. The approach gives rise to a suitable method to select classical solutions and it is based on the first principle of the existence of symmetries.


INTRODUCTION
Nowadays astronomical observations have shown that if we consider our Universe on a large scale, its visible structure is accelerating, homogeneous and isotropic and, essentially, filled with pressureless dust. The simplest cosmic model which describes a Universe with the above properties is the Friedmann-Lemaître-Robertson-Walker (FLRW) model [1]. The evolution of the Universe from the radiation dominant epoch till the present cosmic acceleration can be well-explained by the FLRW model with a cosmological constant (the so-called ΛCDM model). However, it fails if one tries to address the whole early and late history of the Universe starting from the origin and the inflation epoch where quantum effects should be taken into account.
Anisotropies observed in the cosmic microwave background (CMB) are small enough to suggest that anisotropic models of spacetimes become isotropic ones by evolving in time [2][3][4]. One can expect that pre-inflationary anisotropies played an important role (for example they can be responsible for the coupling between the gravitational field and the inflaton field) so if inflationary models are considered, one should understand the dynamics of anisotropies. Models describing anisotropic but homogeneous universes are the so-called Bianchi cosmologies. They can be considered in standard General Relativity and in its extensions containing scalar fields.
In this paper we will consider the Lie symmetries of the Wheeler-DeWitt equation (WDW) in General Relativity and in scalar field cosmology assuming Bianchi spatially homogeneous spacetimes. We will use the Lie symmetries in order to define the unknown potential and derive exact solutions for the WDW equation and for the field equations. Symmetries are considered to play a central role in physical problems because they provide first integrals which can be utilized in order to simplify a given system of differential equations and thus to determine the integrability of the system. Indeed, in [5,7] it has been shown that the Lie symmetries of a dynamical system are related to the geometry of the underlying space where dynamics occurs.
In this work we will not apply the Noether symmetries of the field equations but the Lie symmetries of the WDW equation. It has been proved in [7] that the Lie symmetries of the WDW equation could form a greater Lie algebra than the Noether symmetries of the Lagrangian of the field equations. Hence, it is possible to determine new cases where the field equations are integrable. This method was applied in a scalar tensor cosmological models adopting a FLRW geometry with a perfect fluid and new integrable models, cosmologically viable, raised [33,34]. Recently a similar method has been applied to some axisymmetric quantum cosmologies with scalar fields [35].
The layout of the paper is the following. In Sec. 2 we give the basic definition of the Bianchi classification while in Sec. 3 we study the Lie symmetries of the WDW equation of class A Bianchi spacetimes in the case of General Relativity. In Sec. 4, the previous results are applied in order to reduce the WDW equation by using the Lie invariants and determine invariant solutions. Secs. 5 and 6 are devoted to the same analysis in the case of a minimally coupled scalar tensor gravity and we use the Lie symmetries in order to determine the field potential by using the Lie symmetries of the WDW equation as a geometric criterion. We show that, in scalar tensor cosmology, there exist invariant solutions of the WDW equation: in Bianchi I spacetime, for constant potential and for exponential potential and, in Bianchi II spacetime, for a kination scalar field. For convenience of the reader, we present the Lie symmetry classification for each model in tables. Furthermore in Sec. 7, we show how these results are related with the so-called Hybrid Gravity and conformal transformations. Finally, in Sec. 8, we discuss our results and draw conclusions. Appendix A completes our presentation. Here the basic theory of Lie symmetries is briefly discussed.

THE CLASS A OF BIANCHI SPACETIMES
The class of Bianchi spatially homogeneous cosmologies contains several important cosmological models which have been used for the discussion of anisotropies of primordial Universe and for its evolution towards the observed isotropy of the present epoch [2,[36][37][38][39]. In these models, the spacetime manifold is foliated along the time axis, with three dimensional homogeneous hypersurfaces which admit a group of motions G 3 . Bianchi classified all three dimensional real Lie algebras and has shown that there are nine possible G 3 groups. This results in nine types of Bianchi spatially homogeneous spacetimes. The importance of Bianchi cosmological models is that, in these models, the physical variables depend on the time only, reducing the Einstein and other governing equations to ordinary differential equations.
In the (3 + 1) decomposition of spacetime manifold (Arnowitt-Deser-Misner (ADM) formalism), the line element of the Bianchi models can be written in the following form [40,41] where N (t) is the lapse function and {ω i } denotes the canonical basis 1-forms satisfying the Lie algebra where C i jk are the structure constants of the algebra. The spatial metricḡ ij is diagonal (following the notation of [5,30] and references therein) and can be factorized as follows where e λ(t) is the scale factor of the universe and the matrix β ij is diagonal and traceless. The matrix β ij depends on two independent quantities β 1 , β 2 which are called the anisotropy parameters. The matrix β ij can be selected as 1 and, in these variables, it is √ḡ = e 3λ . The structure constants of the Lie algebra G 3 can be expressed in terms of a three dimensional vector field a i and a symmetric 3 × 3 tensor m ij as follow [42] C i jk = ǫ jks m si + δ i k a j − δ i j a k , and the Bianchi models are grouped in two classes: class A for a i = 0 and Class B for a i = 0. Each class is divided into several types according to the rank and the signature of the tensor m ij . Specifically, the Bianchi models are divided into two subclasses A (a i = 0) and B (a i = 0) containing Bianchi types corresponding to the form of the metric m ij . In this paper, we are interested in the class A models for which there exists a Lagrangian of the field equations.
For the line element (1) with the definitions (3) and (4), the Ricci scalar of the Bianchi class A spacetimes can be written as where and R * = R * (λ, β 1 , β 2 ) is the component of the three dimensional hypersurface which depends on the structure constants of the algebra N 1−3 of the Killing algebra of the Bianchi algebras [43]. The general form of R * is and the special forms for the class A spacetimes can be found in Table I. In the case of General Relativity, when the action of the field equations is the Einstein-Hilbert action (we consider that the spacetime is empty) the field equations for the Bianchi class A spacetimes follow from the Lagrangian and the corresponding field equations are the Euler-Lagrange equations with respect to the variables {N, λ, β 1 , β 2 }. The Euler-Lagrange equations for the variables β 1 , β 2 are: for the variable λ: and for the variable N we have the G 0 0 = 0 Einstein equation Lagrangian (10) is singular since ∂L ∂Ṅ = 0, however, if we consider that then Lagrangian (10) becomes a regular time independent Lagrangian which admits always, as a Noether integral, the Hamiltonian constant. Hence, equation (13) can be seen as the energy constrain of the field equations. In the following we will quantize equation (13) in order to write the Wheeler-DeWitt (WDW) equation and to perform a symmetry analysis using the Lie point symmetries in the case of General Relativity and minimally coupled scalar tensor cosmology.

SYMMETRIES OF THE WDW EQUATION IN GENERAL RELATIVITY
In order to simplify equation (13), we consider N (t) =N (t) e −3λ in the line element (1). Furthermore, we consider the following change of the variables (λ, 3 z , then the Lagrangian (10) becomes Therefore, we define the momentum p (x,y,z) = ∂L ∂(ẋ,ẏ,ż) and equation (13) has now the form Equation (15) can be seen as the Hamiltonian of a particle moving in the space M 3 with potential V (x, y, z) = −e √ 3x R * . Furthermore, the field equations are the Hamiltonian constraint (15) and the Hamilton equationṡ Since the minisuperspace is flat; that is the Ricci scalar vanishes, the WDW equation can be achieved by a standard quantization, assuming the conjugate momenta p i = ∂L ∂x i . Hence from (15), we have the WDW equation of the form which is nothing else but the Klein Gordon equation in the M 3 space. In order to determine the Lie symmetries of (19), we will use the geometric results in Ref. [7]. The M 3 spacetime admits a ten dimensional conformal algebra. In particular, it admits a six dimensional Killing algebra which is the T 3 ∪ SO (3) with elements: one gradient homothetic Killing vector (HV) and three special conformal Killing vectors (CKVs) which are See Ref. [7] for details. Furthermore, by applying the results in [7], we find that the WDW equation (19) admits: i) for the Bianchi I model, eleven Lie symmetries, ii) for the Bianchi II model, five Lie symmetries, iii) two Lie symmetries for the models VI 0 /VII 0 and iv) one Lie symmetry, the linear one, for the models VIII and IX. In Table  II, we give the corresponding Lie symmetries of the WDW equation (19) for each Bianchi model.

INVARIANT SOLUTIONS OF THE WDW EQUATION IN GENERAL RELATIVITY
In this section, we will apply the zero th order invariants of the Lie point symmetries to reduce the order of the WDW equation (19) and, if possible, to determine invariant solutions. From the results of Table II, we can see that it is possible to find invariant solutions for the WDW equation for Bianchi I and II spacetimes.

Bianchi I cosmology
Since for Bianchi I spacetimes hold the property that R * = 0, the WDW equation (19) is the (1+2) wave equation. The reduction of the (1+2) wave equation and the invariant solutions have been studied in [44] and recently in [45]. However, we want to give a concrete example of application of the Lie symmetries.
Let us consider the Lie algebra X I (x) , X I (y) with commutators X I (x) , X I (y) = 0 where hence the solution of equation (19) is One can also consider the Lie algebra X I (x) , X I (yz) = R (yz) + µΨ∂ Ψ for which the invariant solution is where I iµ , K iµ are the modified Bessel functions of the first and the second kinds. Similarly, for the others Lie subalgebras, we can construct invariant solutions. We want also to recall that the WDW equation is a linear second order partial differential equation and any linear combination of the solutions is also a solution. which is the null Hamilton-Jacobi equation of the Hamiltonian system (15) Hence the Hamilton equations (16)- (18) reduce to the following system (whereN = 1) and therefore we have the solutions and, by the coordinate transformation dτ = e 3λ dt in the line element (1), we obtain a Kasner spacetime.

Bianchi II cosmology
For the Bianchi II spacetime, using Table II, we have that if we use the Lie symmetries where and I c0 , K c0 are the modified Bessel functions of the first and the second kinds, respectively. For the Lie algebra we have the invariant solution The last solution Ψ 2 II is included in solution Ψ 1 II for µ 12 = µ 3 = 0.

The WKB approximation and the classical solution
One can also apply the WKB approximation for the Bianchi II model to the equation (19): Let us adopt, the coordinate transformation y = w − x 2 in the Hamiltonian system. Hence, the new Hamilton-Jacobi equation (29) becomes with the solution where the functions S 1,2,3 follow from the system The Hamilton function S (x, w, z) is where c 12 = 3c 2 2 − 4c 2 1 and M (w) = 3e 4 √ 3 3 w − c 12 . Therefore the reduced Hamiltonian system is (recall thatN = 1) Hence the solution of the system (35), (36) is which is the solution of the empty Bianchi II spacetime in General Relativity.
In the following section we will apply the same procedure in the case of minimally coupled scalar-tensor cosmology. Furthermore, we will use the Lie symmetries of the WDW equation in order to determine the unknown potential of the scalar field.

SYMMETRIES OF THE WDW EQUATION IN MINIMALLY COUPLED SCALAR-TENSOR COSMOLOGY
Let us continue the Lie symmetry analysis of the WDW equation for cosmological containing a minimally coupled scalar field in the gravitational action. The Noether symmetry classification of the field equations has been studied in [30][31][32] . Noether symmetries have been adopted in the jet space in [5]. In Ref. [6], a detailed study of integrable cosmological models with non-minimally coupled scalar fields is presented.
Let us now take into account the following action: From the line element (1) and equations (6), (7) we find the Lagrangian [36] L N, λ, β 1 , β 2 , φ,λ,β 1 , Hence, by applying the Euler-Lagrange vector in (41), we find four field equations which are the two equations (11) and The last corresponds to the the G 0 0 = T 0 0 Einstein equation. Furthermore, from the Euler-Lagrange equation d dt , we obtain the field equation for the scalar field which can also be derived by the Bianchi identity T µν ;ν = 0, where T µν is the energy-momentum tensor for scalar field. As in the case of General Relativity, the coordinate transformations (λ, β 1 , 3 z and N (t) = N (t) e −3λ , can be adopted. Hence, equation (43) becomes Moreover, by using the momentum p (x,y,z,φ) = ∂L ∂x i , equation (45) becomes and hence, from (46), we have the following WDW equation since the minisuperspace of equation (46) is the flat space M 4 . We note that equation (47) is the Klein Gordon equation in M 4 . Since we want to use the geometric approach in Ref. [7], we need the conformal algebra of the M 4 spacetime. The four dimensional flat space M 4 admits ten Killing vectors (KVs) which are one gradient HVH = x∂ x + y∂ y + z∂ z + φ∂ φ , and four special CKVsC One can find that the WDW equation (47) admits Lie symmetries for an arbitrary potential V (φ); in the case of Bianchi I, spacetime equation (47) admits four Lie symmetries; for Bianchi II, one has two Lie symmetries and one Lie symmetry for the rest of the Bianchi models. Therefore, for special forms of potentials V (φ), it is possible that the WDW equation (47) Tables III and IV. We will continue our analysis using the results of Tables III and IV in order to determine invariant solutions of the WDW equation (47) for cases where it is possible.

INVARIANT SOLUTIONS OF THE WDW EQUATION IN SCALAR FIELD COSMOLOGY
From the symmetries in Tables III and IV, we observe that invariant solutions for the WDW equation can be determined for the Bianchi type I model for (a) V (φ) = 0, (b) V (φ) = V 0 , (c) V (φ) = V 0 e µφ ; and, for the Bianchi II model, for zero potential. It is worth noticing that, in the following, we will chooseN (t) = 1.

Bianchi I cosmology
For the Bianchi I spacetime, the WDW equation (47) becomes If the potential V (φ) = 0, then (48) becomes the (1+3) wave equation in E 3 [44], hence we will omit this case. When V (φ) = V 0 , V 0 = 0, the field equations are equivalent to the case of General Relativity with stiff matter and a cosmological constant. In this case, we use zero order invariants of the Lie symmetries which form a closed Lie algebra. In this case, equation (48) reduces to the linear second order ordinary differential equation Therefore, the solution of equation (50) is where J c , Y c are the Bessel functions of the first and second kind and the constant is c = 2 . For the exponential potential, we apply the zero order invariants of the Lie symmetries and the WDW equation (48) becomes Hence, for various values of the constant µ from (53), we have that

The WKB approximation and the classical solutions
In the WKB approximation the WDW equation (48) becomes the Hamilton-Jacobi equation where S = S (x, y, z, φ) describes a motion of a particle in the M 4 space. The solution of the Hamilton Jacobi function leads us to the following reduced Hamiltonian systeṁ For V (φ) = 0, from equation (56) we have that Then from (57) and (58) we have Similarly, for constant potential, i.e. V (φ) = V 0 , we have Hence the reduced Hamilton equations (57) arė and the exact solution of the field equations is For the exponential potential, V (φ) = V 0 e µφ , as we saw previously, there exist different solutions of the WDW equation for different values of the constant µ. Hence, the solution of the Hamilton-Jacobi equation (56) is determined by the various values of µ. Let us set µ = − √ 3. Applying the coordinate transformation φ = ψ + x in the Hamiltonian system, the new Hamilton-Jacobi equation is and the reduced Hamiltonian system iṡ Therefore from (65) the Hamilton action is and the field equations reduce to the systeṁ with the solution and Furthermore, for |µ| = √ 3, we apply the coordinate transformation φ = ψ − √ 3 µ x, hence from the Hamilton-Jacobi equation (56) we have where dγ dψ The reduced Hamiltonian system (57) in the new coordinates becomeṡ and therefore from (72) and (73) the last becomeṡ However, if one wants to write an analytical solution of this system, we have to perform another transformation that is dt → dτ . The exact solution of the exponential potential in Bianchi I scalar field cosmology was found in [5] so we will omit the derivation it in this work.

Bianchi II cosmology
One can observe from Tables III and IV that, for Bianchi type II spacetimes, we can determine invariant solution of the WDW equation only for zero potential, i.e. the scalar field is a kination fluid acting as stiff matter p φ = ρ φ . In this case, the WDW equation (47) becomes By applying the zero order invariants of the following three dimensional closed Lie algebra with vanishing commutators, we find the invariant solution where λ = 1 3 12ν 2 − 9 µ 2 (z) + µ 2 (φ) and u (x, y) = exp √ 3 3 (2y + x) .
In WKB approximation, where the WDW equation (79) reduces to the Hamilton-Jacobi equation, we apply the same approach as for the case of General Relativity, Sec. 4.2.1, hence we will omit it. However, we would like to note that the solution for the kination scalar field is φ (t) = c φ t, where c φ is a constant. In the following section, we discuss how these solutions can be applied, under a conformal transformation, in the case of f (R) Hybrid Gravity. This means that the approach can be easily extended to higher-order gravity and non-minimally coupled cases.

HYBRID GRAVITY IN BIANCHI COSMOLOGY
In this section we consider the action of the Hybrid metric-Palatini Gravity with the action of the form [46][47][48] where R is the metric Ricci curvature scalar and f (R) is a function of the Palatini curvature scalar which is constructed by an independent connectionΓ. A variation of the above action with respect to the metric gives the gravitational field equations where G µν is the Einstein tensor for metric g ij while R µν is the Ricci tensor constructed by the conformally related metric h µν = f ′ (R)g µν [49]. It is well known that Hybrid Gravity is equivalent to a non-minimally coupled scalar tensor theory [48]. In particular, if we consider a new scalar field ψ = f ′ (R) by using a Lagrange multiplier and the relation between the two Ricci scalars R and R, action (85) can be written in the following form where is a Clairaut equation with the singular solution Furthermore, from the action (87) and for the Bianchi spacetimes (1), we have that the Lagrangian of the field equations is: As it is shown in [29], the action (87) of the Hybrid Gravity is equivalent to a phantom minimally coupled scalar field under the conformal transformationḡ ij = (1 + ψ) g ij , and action (87) becomes whereR is the Ricci scalar of the conformal metricḡ ij ; therefore under the transformation dφ = i 6ψ+9 2(1+ψ)ψ dψ and V (φ) = − 1 (1+ψ) 2 V (ψ), we have the action of the form of (40). From the transformation ψ → φ , we find that the only potential which admits Lie point symmetries has the following form which is the exponential potential in the case of minimally coupled scalar field cosmology with κ = κ (µ) and κ (0) = 0, i.e. V (ψ) = V 0 (1 + ψ) 2 . This potential transforms to the constant potential in the case of minimally coupled scalar field [54].

DISCUSSION AND CONCLUSIONS
In this work we studied the Lie symmetries of the WDW equation in the Bianchi Class A spacetimes for General Relativity and scalar field cosmologies, considering minimally coupled scalar tensor gravity and non-minimally coupled gravity coming from Hybrid Gravity. We applied the Lie invariants in order to construct solutions of the WDW equation. In the case of General Relativity, we found exact solutions of the WDW equation for the Bianchi I and Bianchi II spacetimes. In scalar field cosmology, we applied the Lie symmetries as a criterion for the selection of the unknown potential of the scalar field and we were able to construct exact solutions for the Bianchi I spacetime for zero potential V (φ) = 0, constant potential V (φ) = V 0 , and exponential potential V (φ) = e µφ . For the Bianchi II spacetime we obtained solutions for the zero potential case. In each case, we show that when the WDW equation is invariant under the action of the three dimensional Lie algebra with zero commutators, the Hamilton-Jacobi equation of the Hamiltonian system which was defined by the field equations, can be solved by the method of separation of variables; that means that the field equations are Liouville integrable. It is important to note that, in the case of FLRW scalar cosmology, we have more potentials where the WDW admits Lie symmetries. However, since the Lie symmetries are connected to the conformal algebra of the minisuperspace, in the case of FLRW scalar field cosmology, the dimension of the minisuperspace is two, which means that the last admits an infinite number of conformal killing vectors, whereas, for the Bianchi models, the minisuperspace has dimension four and admits a fifteen dimensional conformal algebra, i.e. less possible generators for the Lie symmetries of the WDW equation.
Finally, we studied the case of the Hybrid Gravity in the Bianchi Class A spacetimes. Since the Hybrid Gravity is equivalent to a scalar tensor theory, we were able to related all the potentials we found in the case of minimally coupled scalar field to that of Hybrid Gravity.
This analysis is important in the sense that can be used in order to construct solutions of the wave function of the Universe and, at the same time, conservation laws, and classical solutions for the field equations. Following the discussion in [55], the presence of symmetries gives rise to a straightforward interpretation of the Hartle criterion: the symmetries generates oscillatory behaviors in the wave function of the Universe and then allow correlations among physical variables. This give rise to classically observable cosmological solutions. Here we generalized this result considering Bianchi models. On the other hand, other general selection rules can be identified in Quantum cosmology, as discussed in [56]. This will be the topic of forthcoming papers.

Lie point symmetries and invariant functions
A second order DE is a function H = H(x i , u A , u A ,i , u A ,ij ) in the jet space B M , where x i are the independent variables and u A are the dependent ones. Let be the generator of the infinitesimal point transformation The function H = 0, is invariant under the action of the infinitesimal point transformation (A2),(A3) if there exists a function λ such that [50] X [2] (H) = λH (A4) The vector field X is called Lie point symmetry of the function H and X [n] is the second prolongation of X in the jet space B M where and The importance of Lie point symmetries is that the last one can be used in order to reduce an order of a differential equation. When a reduction is possible, one can determine invariant solutions or transform them to another ones [51]. From condition (A4) one defines the Lagrange system whose solution provides the characteristic functions x k , u , Z [1]i x k , u, u i , Z [2] x k , u, u ,i , u ij .
The solution Z [k] is called the kth order invariant of the Lie symmetry vector (A1). By writing the DE in terms of the invariants Z [k] , we can reduce the order of the DE, for details see for instance [51,52]. Below we discuss the application of the Lie symmetries and of the Lie invariants for the WDW equation.

Reduction and invariant solutions of the WDW equation by Lie point symmetries
In order to determine the Lie symmetries of the WDW equation we apply a geometric method which is established by Paliathanasis & Tsamparlis [7]. The method relates the Lie symmetries of the Klein Gordon equation to the conformal algebra of the underlying geometry. Hence, in the following we will not present the construction of the Lie symmetries of the WDW equation but we will give the results.
In particular, the general form of a Lie symmetry of the WDW equation is where ξ i x k is a CKV of the metric which defines the conformal Laplace operator, (in our consideration the minisuperspace) and ψ x k is the conformal factor of the CKV, recall that since ξ i is a CKV of g ij , it means that, L ξ g ij = 2ψg ij . Furthermore, it is possible to consider a coordinate transformation x i →x i so that ξ i x k ∂ i → ∂ J (these are called normal coordinates). In the normal coordinates the symmetry vector takes the following simple form, where now either with the method of Lie invariants, or with the method of linear differential operators (see [33] for details) we find the following expression for the solution of Ψ, from which follows again that the coordinate x J is factored out from the solution Ψ. Furthermore, in [7] it was also shown that the symmetries of the WDW equation can be used in order to find Noether symmetries for classical particles 2 . The exact relation among the Lie symmetries of the WDW equation, the Noetherian conservation laws of the field equations and the interpretability conditions are given in [33].