Dynamics and Exact Bianchi I Spacetimes in Einstein–Æther Scalar ﬁeld Theory

Eur. Phys. J. C (2020) 80:589 https://doi.org/10.1140/epjc/s10052-020-8148-7

Regular Article - Theoretical Physics

Dynamics and exact Bianchi I spacetimes in Einstein–æther scalar ﬁeld theory

Andronikos Paliathanasis1,2,a, Genly Leon3,b 1 Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa 2 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, 5090000 Valdivia, Chile 3 Departamento de Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280, Antofagasta, Chile

Received: 20 April 2020 / Accepted: 15 June 2020 / Published online: 29 June 2020 © The Author(s) 2020

Abstract We determine exact and analytic solutions of the dominated period. These anisotropies are observed in the gravitational field equations in Einstein–aether scalar model cosmic microwave background, which support the idea that field with a Bianchi I background space. In particular, we the spacetimes become isotropic ones by evolving in time consider nonlinear interactions of the scalar field with the [3–5]. In addition a recent detailed study by using the X-ray aether field. For the model under consideration we can write clusters challenged the isotropic scenario [6] and supported the field equations by using the minisuperspace description. the anisotropic cosmological scenario. The point-like Lagrangian of the field equations depends on The spatial homogeneous but anisotropic spacetimes are three unknown functions. We derive conservation laws for the known as either Kantowski-Sachs or Bianchi cosmologies. field equations for specific forms of the unknown functions The isometry group of Kantowsky-Sachs spacetime is R × such that the field equations are Liouville integrable. Further- SO(3), and does not act simply transitively on spacetime, more, we study the evolution of the field equations and the nor does it possess a subgroup with simple transitive action. evolution of the anisotropies by determining the equilibrium This model isotropizes to close FLRW models [7–10]. On the points and analyzing their stability. other hand, Bianchi spacetimes contain many important cosmological models including the standard FLRW model in the limit of the isotropization, e.g., Bianchi III isotropizes 1 Introduction to open FLRW models, and Bianchi I isotropizes to flat FLRW models. In Bianchi models, the spacetime manifold According to the cosmological principle, the universe is is foliated along the time axis with three dimensional homo- homogeneous and isotropic in large scales. Indeed, the evo- geneous hypersurfaces. The Bianchi classification provides lution of the universe from the radiation dominant epoch till a list of all real 3-dimensional Lie algebras up to isomor- the present cosmic acceleration can be well-explained by phism. The classification contains eleven classes, nine of the homogeneous Friedmann–Lemaître–Robertson–Walker which contain a single Lie algebra and two of which con- (FLRW) model [1]. However, FLRW fails to explain the tain a continuum-sized family of Lie algebras, but two of early history of the universe starting from the origin and pre- the groups are often included in the infinite families, giv- inflation epoch where quantum effects should be taken into ing nine types of Bianchi spatially homogeneous spacetimes account. instead of eleven classes. Bianchi spacetimes contain several Inflation is the main mechanism to explain today important cosmological models that have been used for the isotropization of the observable universe. The mechanism of discussion of anisotropies of primordial universe and for its inflation is often based on the existence of a scalar field known evolution towards the observed isotropy of the present epoch as inflaton [2]. The scalar field energy density temporarily [11–14]. There is an interesting hierarchy of Bianchi models. dominates the dynamics and drives the universe towards a In particular, the LRS Bianchi I model naturally appears as a locally isotropic and homogeneous form that leaves only boundary subset of the LRS Bianchi III model. The last one very small residual anisotropies at the end of a brief inflaton- is an invariant boundary of the LRS Bianchi type VIII model as well. Additionally, LRS Bianchi type VIII can be viewed a e-mail: [email protected] as an invariant boundary of the LRS Bianchi type IX models b e-mail: [email protected] (corresponding author) [15–20]. 123 589 Page 2 of 16 Eur. Phys. J. C (2020) 80 :589

Bianchi spacetimes in the presence of a scalar field were any isometry in Einstein–æther theory with a matter source studied in [21]; where it has been found that an initial were derived recently in [52]. anisotropic universe can end into a FLRW universe (i.e., It has been proposed that a scalar field contributes to the it isotropizes) for specific initial conditions whenever the field equations of Einstein–aether theory where the scalar scalar field potential has a large positive value. For exponen- field can interact with the aether [55]. Such model can tial scalar field the exact solution of the field equations have describe the so-called Lorentz-violated inflation [56]. The been found for some particular Bianchi spacetimes [22–24]. dynamics of spatially homogeneous Einstein-aether cosmo- These exact solutions lead to isotropic homogeneous space- logical models with scalar field with generalized harmonic times as it was found in [25,26]. potential in which the scalar field is coupled to the aether An exact anisotropic solution of special interest is the Kas- field expansion and shear scalars were studied in [57,58], ner universe. The Kasner spacetime is the exact solution of with emphasis on homogeneous Kantowski–Sachs models the field equations in general relativity in the vacuum for the in [59–61]. A similar analysis on the equilibrium points of Bianchi I spacetime, where the space directions are isome- the field equations was performed for isotropic FLRW space- tries, that is, the three-dimensional space admits three trans- times in [62–64]. Exact and analytic solutions of isotropic lation symmetries. Kasner universe has various applications and homogeneous spacetimes in Einstein–aether scalar field in gravitation. One of the most important application is that cosmology are presented in [56,65–67]. Kasner solution can describe the evolution of the Mixmas- In the following we are interested on the exact solutions of ter universe when the contribution of the Ricci scalar of the Bianchi I spacetimes in Einstein–aether theory with a scalar three-dimensional spatial hypersurface in the field equations field interacting with the aether field. We consider a nonlin- is negligible [27]. Hence, Kasner solution is essential for the ear interaction, and we are able to write the field equations description of the BKL singularity. For other applications of by using the minisuperspace approach. The existence of a the Kasner universe and in general of the Bianchi I space- point-like Lagrangian which can describe the field equations times in gravitational physics we refer the reader to [28–35] is essential for our analysis because we can apply techniques and references therein. of Analytic Mechanics to study the dynamics and deter- In this work, we are interested on the study of the gravita- mine exact solutions for field equations. We are interested tional field equations in a Lorentz-violating theory known as on the dynamical systems analysis of the equilibrium points Einstein–aether theory [36–40]. Specifically, it is introduced for the gravitational field equations. From such analysis we a unit vector, the aether, in the gravitational action. The exis- can extract information for the evolution of the field equa- tence of the aether spontaneously breaks the boost sector of tions and for the main phases of the cosmological history. For the Lorentz symmetry by picking out a preferred frame at this analysis one can apply linearization around equilibrium each point in spacetime. The action for Einstein–aether the- points, monotonic principle [68], the invariant manifold theory is the most general covariant functional of the spacetime orem [69–73], the center manifold theorem [69–71,74], and a metric gab and aether field u involving no more than two normal forms theory [69–71]. derivatives, excluding total derivatives [41,42]. The plan of the paper is as follows. In Sect. 2, we present There are few known exact solutions of the field equations the model of our consideration which is the Einstein–æther in Einstein field equations. Exact solutions in the Vacuum for scalar field theory in Bianchi I spacetime. We write the the Bianchi I, the Bianchi III, the Bianchi V and the isotropic field equations and the specific form of the interaction term FLRW spacetime were derived recently in [43,44]. In [45] between scalar field and aether field. We write the point- the authors presented a generic static spherical symmetric like Lagrangian of the field equations. In Sect. 3, we present solution in Einstein–aether theory, where it has been shown analytic solutions of the field equations, the method that we that the Schwarzschild spacetime is recovered. Other inho- use to constraint the unknown functions of the model and mogeneous exact solutions have been studied previously in determine that analytic solutions is based on the existence of [46–48]. The spherical collapse in Einstein–aether theory is conservation laws. In particular, we investigate the Liouville studied in [49] where a comparison with the Hoˇrava grav- integrability of the field equations. In Sect. 4, we perform ity is presented. We remark that Einstein–aether theory can a detailed analysis of the equilibrium points for the gravita- be seen as the classical limit of Hoˇrava gravity. Moreover, tional field equations by using Hubble-normalized variables. Gödel-type spacetimes are investigated in [50,51]. Additionally, we use the center manifold theorem and the Furthermore, there are various studies of Einstein-aether normal forms calculations to analyze the stability of sets of models with a matter source. The general evolution in the nonhyperbolic equilibrium points. It is well-known that the presence of modified Chaplygin gas was studied in [53], procedure based on the formal series of polynomial changes while an analysis with the presence of a Maxwell field was of coordinates devised by Poincarè [75–79] to integrate lin- performed in [54]. Exact inhomogeneous spacetimes without earizable dynamical systems in the neighbourhood of a equilibrium point. It can also be used to normalize the system 123 Eur. Phys. J. C (2020) 80 :589 Page 3 of 16 589 √ in the neighborhood of a equilibrium point for nonlineariz- 2 2 2λ(t) 2β+(t) 2 − 1 ds =−N (t) dt + e e dx + e √ β+ (t) able dynamical systems, which are systems whose lineariza- 2 tion at the equilibrium point present resonances. This pro- 1 3 3 2 − √ β+(t)+ β−(t) 2 cedure is the basis of the normal forms calculations to be + β− (t) dy + e 2 2 dy , (4) implemented in Sect. 4.1.2. In Sect. 5, we use an alternative 2 dynamical system’s formulation which leads to the evolution of anisotropies decouples; and we study a reduced two- where eλ(t) is the radius of the three dimensional space, and dimensional dynamical system with local and with Poincarè β+ (t) ,β− (t) are the anisotropy parameters. In the limit variables. Section 6 is devoted to conclusions. β+ (t) → 0 and β− (t) → 0, the line element (4) reduces to that of the spatially flat FLRW spacetime. The Kasner spacetime as we discussed in the introduction 2 Einstein–aether scalar field model is an exact solution of the field equations of general relativity for the line element (4) where the anisotropic parameters β ( ) ,β ( ) In this work, we consider the Einstein–aether theory with a + t − t are always different from zero. While Kasner scalar field interacting with aether field, with action integral spacetime can describe the BKL singularity, in the presence [55]: of additional matter source the behaviour of the spacetime is different. √ 4 R Specifically, in the presence of a cosmological, the evolu- S = dx −g − Sφ − SAether , (1) 2 tion of the spacetime can describe an early anisotropic space μ where the isotropic de Sitter universe is a future attractor. SAether corresponding to the aether field u as follows: Bianchi I spacetime is the simplest anisotropic model and √ it is one the basic models to study small anisotropies in the = 4 − αβμν − λ c + , SAether d x g K uμ;αuν;β u uc 1 universe. It has been proposed that the small anisotropies of (2) the CMB can be related with the early anisotropies of the universe [80], while Bianchi models can reproduce anisotropies and Sφ to the action integral of the scalar field of that kind [81]. In [82] it has been show that Bianchi I models with anisotropic fluid sources can reproduce measurable √ 4 1 μν αβ α anisotropy in the CMB without effects upon the primordial Sφ = dx −g g φ;μφ;ν + V φ,g , u ;β , u . 2 a nucleosynthesis of helium-4. (3) In addition a detailed study of dynamics of Bianchi I universe with anisotropic source and an isotropic dark energy The interaction of the scalar field φ (xμ) with the aether component performed in [83], where it was found that the μ field u , is introduced in the potential function of the scalar asymptotic behaviour of the universe mimics the de Sitter αβ α field V = V φ,g , ua;β , u , function λ is a Lagrange universe, while anisotropic fluid source contributes in the multiplier which has been introduced to ensure the unitarity CMB quadrupole. In the presence of a homogeneous scalar μ μ of the aether field u , i.e. u uμ + 1 = 0. Moreover, tensor field with an exponential potential the late time attractor is K αβμν is defined by the metric tensor gμν as follows also the isotropic FLRW spacetime for a specific values of the exponent for the potential where the Kitada and Maeda αβμν αβ μν αμ βν αν βμ μν α β K ≡ c1g g + c2g g + c3g g + c4g u u , no-hair theorem is applied [26,84]. Bianchi I spacetime admits three isometries which are the in which c , c , c and c are the coupling constants of the 1 2 3 4 three translations of the Euclidean space, that is, the vector aether field with the gravitational field. Consequently, since fields ∂ ,∂ ,∂ . Furthermore, we assume that the scalar φ ( μ) x y z the scalar field x is interacting with the aether field, and field φ (xμ) = φ (t, x, y, z), inherits the symmetries of the the latter is interacting with the gravitational fields, we can spacetime which means that the scalar field is homogeneous φ ( μ) say that the scalar field x is not minimally coupled to and depends only on the variable t, that is, φ (xμ) = φ (t). gravity. However, our proposal is rather different from the However, the field equations of the Bianchi I spacetime so-called scalar tensor theory. can describe the evolution of the anisotropic parameters in inhomogeneous spacetimes, such are the Szekeres space- 2.1 Bianchi I spacetime times [85]. Specifically, in the case of silent universes, spacetimes with zero magnetic part of the Weyl tensor, and with For the underlying space in our consideration, we assume an inhomogeneous pressureless fluid source, the dynamics of the locally rotational symmetric Bianchi I spacetime with the anisotropic parameters of the inhomogeneous spacetime line element admit as past attractors two Kasner universes. However, in 123 589 Page 4 of 16 Eur. Phys. J. C (2020) 80 :589 the same model in the presence of additional isotropic fluid φ 1 αβ 2 Tμν = φ;μφ;ν − g φ;αφ;β + V φ,θ,σ gμν source, the anisotropic inhomogeneous Kasner-like attrac- 2 a tors reduce to anisotropic and homogeneous spacetimes [85– + θV,θ gμν + (Vθ );α u hμν 87]. + θ + a σ V,σ 2 V,σ 2 ; u μν For the aether field we choose the comoving observer: a μ = 1 δμ u ( ) t . For this selection, the aether field inherits the α 2 N t + V,σ 2 σμν;αu − 2σ uμuν . (11) symmetries of the spacetime, while the limit of the FLRW spacetime can be recovered [43]. Moreover, as we shall see in the following, with this specific selection for aether 2.2 Energy–momentum tensors field uμ the field equations can be derived by minisuperspace approach for a specific form of the potential function In [57,58] the dynamical analysis of the field equations for the αβ α V φ,g , ua;β , u . locally rotational Bianchi I spacetime studied for the potential For the Bianchi I spacetime the kinematic quantities of the form θ,σ2,ω ,αμ μν for aether field of our consideration, i.e. φ,θ,σ2 = (φ) + (φ) θ + (φ) σ, μ = 1 δμ, V V0 V1 V2 (12) u N(t) t are derived for specific functions of VI (φ). In particular, for exponential 3 ˙ 2 3 ˙ 2 ˙ 2 θ = λ, σ = β+ + β , (5) functions V (φ) in [58] or for power-law functions V (φ) in N 8N 2 _ I I [57]. In the case of FLRW spacetime, where σ = 0, scalar and field potentials with more general nonlinear dependence on θ μ parameter , have been proposed and studied in the literature ωμν = 0,α= 0. (6) [62–66]. In the case of FLRW spacetime, in [56] the authors pro- θ,σ2,ω ,αμ The kinematic quantities μν are the expan- posed an Einstein–aether scalar field where the interaction sion rate, the shear, the vorticity and the acceleration for the between the aether and the scalar fields is introduced in the uμ + aether field , as they are defined in the 1 3 decomposition; coupling coefficients of the aether field with the gravitational that is, field. That leads to an equivalent theory with that [55], where μ;ν ν 2 μν the scalar field potential is quadratic in the expansion rate θ = uμ;ν h , aμ = uμ;νu ,σ= σ σμν, (7) θ. The theory has been proposed as an alternative Lorentz in which violating inflationary model. In this theoretical framework the field equations can be described by a canonical point- κ λ 1 κλ σμν = u(κ;λ) hμhν − h hμν and like Lagrangian. Because of that property, various techniques 3 from analytic mechanics applied in [67] can be used to deter- κ λ ωμν = u[κ;λ]hμhν . (8) mine new exact solutions. Hence, in this work we consider the scalar field potential while hμν = gμν + uμuν. to be quadratic on θ and σ , that is, For a potential function of the form V = V φ,θ,σ2 , 2 2 2 variation with respect to the metric tensor of (1) produce V φ,θ,σ = V0 (φ) + V1 (φ) θ + V2 (φ) σ , (13) gravitational field equations, which are in order to be in agreement with the Einstein–aether scalar Aether φ Gμν = Tμν + Tμν, (9) field model proposed in [56]. For the line element (4) with N (t) = 1 and for the aether μν μν where G is the Einstein tensor, T is the energy– μ = 1 δμ Aether Aether field u N(t) t the energy–momentum tensor Tμν is momentum tensor of the aether field defined as [55]: diagonal with the following nonzero components: Aether α αβ tAether ˙ 2 Tμν = 2c1 u;μuα;ν − uμ;αuν;β g T = 3 (c1 + 3c2 + c3) λ t 3 + 2λuμuν + gμν + ( + ) β˙ 2 + β˙ 2 , u c1 c3 + − (14a) α α α 4 − 2 (u(μ J ν));α + (u J(μν));α − (u(μ Jν) );α xAether= ( + + ) λ¨ + λ˙ 2 α β Tx c1 3c2 c3 2 3 − 2c uμ;αu uν;β u , (10) 4 + √ √ c1 c3 ¨ ˙ 2 ˙ ˙ + 4 2β+ − 3 β+ + 12 2λβ+ μ μβ α αβ μ ν φ 4 with J ν =−K ανu;ν, u =−K μνu;αu;β , while Tμν ( + ) 3 c1 c2 ˙ 2 − β− , (14b) is the energy momentum tensor of the scalar field [57,58]: 4 123 Eur. Phys. J. C (2020) 80 :589 Page 5 of 16 589 √ yAether ¨ ˙ 2 Ty = (c1 + 3c2 + c3) 2λ + 3λ 6 ˙ ˙ − V2 (φ),φ β−φ. (15d) + √ √ 4 c1 c3 ¨ ˙ 2 ˙ ˙ − 2 2β+ + 3 β+ + 6 2λβ+ 4 + √ √ 2.3 Minisuperspace description c1 c3 ¨ ˙ 2 ˙ ˙ + 2 6β− − 3 β− + 6 6λβ+ , 4 Similarly with the case of FLRW in [56], the field equations (14c) of the gravitation Action Integral (1) can be derived from the zAether= ( + + ) λ¨ + λ˙ 2 Tz c1 3c2 c3 2 3 point-like Lagrangian of the form + √ √ c1 c3 ¨ ˙ 2 ˙ ˙ A A A A A A − 2 2β+ + 3 β+ + 6 2λβ+ L y , y˙ = LGR y , y˙ + Lφ y , y˙ 4 + √ √ c1 c3 ¨ ˙ 2 ˙ ˙ + A, ˙ A , − 2 6β− + 3 β− + 6 6λβ+ . L Aether y y (16) 4 (14d) A A ˙ ˙ ˙ where the vector fields y is y = N,λ,λ, β+, β+,β−, β−, φ,φ˙ , while a dot demotes total derivative with respect to the Similarly, for the potential (13) the energy-momentum A φ variable t, that is y˙ A = dy . tensor Tμν have the following nonzero components dt A A Function LGR y , y˙ describes the point-like Lagrangian of General Relativity, t φ = (φ) λ˙ 2 + 3 (φ) β˙ 2 + β˙ 2 Tt 9V1 V2 + − 8 3λ A A e ˙ 2 3 ˙ 2 ˙ 2 1 LGR y , y˙ = −3λ + β+ + β− , − φ˙2 − V (φ) , (15a) N 8 2 0 (17) x φ = (φ) λ¨ + λ˙ 2 + (φ) λ˙φ˙ Tx 3V1 2 3 6V1 ,φ that term Lφ y A, y˙ A describes the point-like Lagrangian of 1 ˙2 + φ − V0 (φ) + V2 (φ) the scalar field, that is, 2 √ 3λ A A e ˙ 2 3 ˙ 2 2 ¨ 3 ˙ 2 ˙ 2 Lφ y , y˙ =− 9V (φ) λ + V (φ) β+ × β+ − β+ + β− N 1 8 2 2 8 √ √ ˙ 2 1 ˙2 + β− + φ − NV (φ) , (18) 3 2 ˙ ˙ 2 ˙ ˙ 2N 0 + λβ+ + V (φ),φ β+φ, (15b) 2 2 2 A A while L Aether y , y˙ includes the terms which correspond y φ ¨ ˙ 2 ˙ ˙ μ Ty = 3V1 (φ) 2λ + 3λ + 6V1 (φ),φ λφ to the aether field u , which is given by the following expres- sion 1 ˙2 1 + φ − V0 (φ) − V2 (φ) 2 8 A A 3 3λ ˙ 2 √ √ L Aether y , y˙ =− e (c1 + 3c2 + c3) λ × β¨ + β˙ 2 + λ˙β˙ N 2 2 + 3 + 6 2 + (c1 + c3) 2 2 √ + β˙+ + β˙− . 2 ˙ ˙ 1 4 − V (φ),φ β+φ + V (φ) 4 2 8 2 (19) √ √ ¨ ˙ 2 ˙ ˙ × 2 6β− − 3 β− + 6 6λβ− √ Therefore, the point-like Lagrangian (16) is written as follows + 6 (φ) β˙ φ,˙ V2 ,φ − (15c) 3λ 4 A A e ˙ 2 3 ˙ 2 L y , y˙ = −3F (φ) λ + M (φ) β+ N 8 and ˙ 2 1 ˙2 3λ + β− + φ − Ne U (φ) , (20) z φ = (φ) λ¨ + λ˙ 2 + (φ) λ˙φ˙ 2 Tz 3V1 2 3 6V1 ,φ in which U (φ) = V (φ) , F (φ) = (1 + c + 3c + c 1 ˙2 1 0 1 2 3 + φ − V0 (φ) − V2 (φ) + (φ)) (φ) = ( − ( + ) − (φ)) 2 8 3V1 and M 1 2 c1 c3 V2 . √ √ ¨ ˙ 2 ˙ ˙ Variation with respect to the lapse function N gives the × 2 2β+ + 3 β+ + 6 2λβ+ √ constraint equation 2 ˙ ˙ 1 3 1 − V2 (φ),φ β+φ − V2 (φ) − (φ) λ˙ 2 + (φ) β˙ 2 + β˙ 2 + φ˙2 4 8 3F M + − √ √ 8 2 ¨ ˙ 2 ˙ ˙ × 2 6β− + 3 β− + 6 6λβ− +U (φ) = 0, (21) 123 589 Page 6 of 16 Eur. Phys. J. C (2020) 80 :589 where we have set N (t) = 1. Moreover, from the varia- U (φ) , F (φ) and M (φ). As we mentioned before, the point- tion with respect to the variables {λ, β+,β−,φ}, we find the like Lagrangian (20) describes the motion of a point in a four- second-order field equations: dimensional space with conservation laws: the quantities I , I , I and the constraint equation (21), which can be seen 1 2 3 ¨ ˙ 2 3 ˙ 2 ˙ 2 h λ, λ,˙ β+, β˙+,β−, β˙−,φ,φ˙ F (φ) 2λ + 3λ + M (φ) β+ + β− as the Hamiltonian function , 8 with Hamiltonian constraint h = 0. The four conservation 1 ˙2 ˙ ˙ laws are independent and not all, but only three of them, are + φ − U (φ) − 6F (φ),φ λφ = 0, (22a) 2 in involution. They are {I1, I2, h}. Therefore, in order to infer ¨ ˙ ˙ ˙ 2 φ + 3λφ + U (φ),φ + 3F (φ),φ λ about the integrability of the field equations and to be able to write an analytic solution we need to determine at least an 3 ˙ 2 ˙ 2 − M (φ),φ β+ + β− = 0, (22b) additional conservation law. 8 ¨ ˙ ˙ ˙ ˙ In order to specify the unknown functions U (φ) , F (φ) β+ + 3λβ+ + ln (M (φ)),φ β+φ = 0, (22c) and M (φ) such that the field equations admit additional con- ¨ ˙ ˙ ˙ ˙ β− + 3λβ− + ln (M (φ)),φ β−φ = 0. (22d) servation laws, we apply the analysis presented before in [88– 90]. We use the theory of point transformations to provide a The latter two equations can be integrated as follows geometric criteria to constrain the unknown functions of the 3λ ˙ 3λ ˙ gravitational theory and construct conservation laws. M (φ) e β+ − I1 = 0 , M (φ) e β− − I2 = 0, (23) We focus on the construction of conservation laws lin- where I1, I2 are integration constants. The first-order differ- ear in the momentum. In order to have the latter true, two ential equations (23) are two conservation laws for the field main requirements should be satisfied: the minisuperspace equations. (25) to admit isometries and the effective potential Vef f = In addition we can construct the third conservation law Ne3λU (φ) to be invariant under the action of a point trans- 3λ ˙ ˙ formation with generator and isometry of (25). M (φ) e β−β+ − β+β− − I3 = 0, (24) √ We define the new scalar field dφ = K (ψ)dψ, such which is the angular momentum in the plane {β+,β−}. that the minisuperspace (25) takes the form Lagrangian function (20) describes a singular dynamical ∂ L = 3λ system, because ∂ ˙ 0. However, without loss of gener- 2 e 2 3 2 N ds( ) = −3F (ψ) dλ + M (ψ) (dβ+) ality we can select N (t) = N (λ (t) ,β+ (t) ,β− (t) ,φ(t)), 4 N 8 such that Lagrangian (20) describes the equation of motion 2 1 2 + (dβ−) + K (ψ) dψ . (26) of a point particle which motion takes place into the four- 2 dimensional manifold with line element 3λ For arbitrary functions F (ψ) , M (ψ) the latter line element 2 e 2 3 2 ds( ) = −3F (φ) dλ + M (φ) (dβ+) 2 4 N 8 admits only three isometries, which form the E group in the plane {β+,β−}. The corresponding conservation laws are the 2 1 2 + (dβ−) + dφ , (25) I , I and I . There are two cases in which we classify the 2 1 2 3 existence of solutions. These are Case A: F (ψ) arbitrary and 3λ under the action of the potential function Vef f = Ne U (φ). Case B: M (ψ) arbitrary. The line element (25) is called the minisuperspace of the gravitational system. 3.1 Case A: F (ψ) arbitrary The minisuperspace description is very helpful because techniques and results from Analytic Mechanics can be Without loss of generality we assume K (ψ) = F (ψ) and applied to study the dynamics and the general evolution of ¯ (ψ) N = e3λ F (ψ) , M (ψ) = F (ψ) M¯ (ψ) , U (ψ) = U , the field equations; and also determine exact and analytic F(ψ) hence the point-like Lagrangian (20) is written solutions of the field equations. We proceed our analysis by constructing analytic solu- A A ˙ 2 3 ¯ ˙ 2 ˙ 2 1 ˙ 2 tions of the gravitational field equations. We assume N (t) = L y , y˙ = −3λ + M (ψ) β+ + β− + ψ 8 2 N (λ (t) ,β+ (t) ,β− (t) ,φ) . −e6λU¯ (ψ) . (27)

3 Analytic solutions The field equations which are derived from the point-like Lagrangian (27) admit additional conservation laws linear in ψ ¯ ψ = M0 , ¯ ψ = , In this Section, we present some analytic solutions of the the momentum when M ( ) M1e U ( ) U0 ¯ M ψ ¯ −6κφ field equations for specific forms of the unknown functions M (ψ) = M1e 0 , U (ψ) = U0e . 123 Eur. Phys. J. C (2020) 80 :589 Page 7 of 16 589 ⎛ ⎞ ψ √ ¯ (ψ) = M0 , ¯ (ψ) = λ 2 3.1.1 M M1e U U0 λ − 2 0 ( − ) ⎜ 8 0 1 tanh M t t1 ⎟ 1 ⎜ 2 0 ⎟ ψ ψ (t) =− ln M1 , ¯ (ψ) = M0 ¯ (ψ) = ⎝ 2 2 ⎠ For M M1e and U U0, the gravitational M0 3 (I1) + (I2) field equations admit the additional conservation laws (33b) M0ψ ˙ ˙ 1 = e β+β+ + β−β− , (28a) with 3 2 M φ 2 2 = M e 0 (β+) − (β−) β˙− − β+β−β˙+ 2 16 0 √ 8I1 M1 2λ0 + β˙ + β φ˙ , β+ ( ) = − M0 − (28b) t 2 2 3M0 (I1) + (I2) 3 φ = 2 M0 (β )2 − (β )2 β˙ − β β β˙ √ 2 2 M0 e + − + + − − 2λ 16 × 0 ( − ) + β , ˙ ˙ tanh M0 t t1 +0 (33c) − β+ + M β+φ , (28c) 2 0 √ 8I2 M1 2λ0 β+ ( ) = By using the conservation laws I1, I2 the field equations t 2 2 3M0 (I1) + (I2) are described by the point-like Lagrangian √ 2 2λ0 1 λ × tanh M (t − t ) + β− . (33d) L y A, y˙ A = −3λ˙ 2 + ψ˙ 2 − U e6 0 2 1 0 2 0 ( )2 + ( )2 3 I1 I2 − ψ = , λ ( ) = − e M0 , (29) In√ the latter solution if U0 0 it follows t λ 8 3 0 ( − ) 3 t t0 . where the reduced field equations are We remark that the line element of the underlying space has the following form λ λ¨ − U e6 = 0, (30a) 0 2 2 3λ(t) 2 2λ(t) 2 2 ds =− e F (ψ (t)) dt + e 3 (I1) + (I2) M0 − ψ ψ¨ − M0 = , e 0 (30b) √ 8 M1 β ( ) − √1 β+(t)+ 3 β−(t) × e 2 + t dx2 + e 2 2 dy2 with constraint equation − √1 β ( )− 3 β ( ) + t 2 − t 2 2 2 +e 2 dz , (34) 1 λ 3 (I1) + (I2) − ψ −3λ˙ 2 + ψ˙ 2 + U e6 + e M0 = 0. 2 0 8M 1 where F (ψ (t)) is an arbitrary function. (30c) ¯ Hence, the field equations are reduced to the following sys- 3.1.2 Analytic solution for arbitrary M (ψ) tem We observe that using the conservation law I1, I2 in (27) and ˙ 2 6λ ¯ (ψ) = 3λ − U0e = λ0, (31a) for U U0, the point-like Lagrangian of the reduced field equations is written 1 3 (I )2 + (I )2 ˙ 2 1 2 −M0ψ ψ + e = λ0. (31b) 1 λ 2 8M1 L y A, y˙ A = −3λ˙ 2 + ψ˙ 2 − U e6 − M (ψ) , (35) 2 0 We find that the latter two equations are conservation laws for ( )2+( )2 (ψ) = 3 I1 I2 ( (ψ))−1 the field equations, but they are nonlinear in the momentum where M 8 M . and are hidden symmetries [91–93]. The reduced gravitational field equations are For λ = 0 the analytic solution is 0 λ¨ − 6λ = , ψ¨ + (ψ) = , U0e 0 M ,ψ 0 (36) 1 λ (t) =− ln 3U (t − t )2 , 0 0 − λ˙ 2 + 1 ψ˙ 2 + 6λ + (ψ) = 6 with constraint 3 2 U0e M 0, and ( )2 + ( )2 hidden conservation laws 1 3 I1 I2 2 ψ (t) = ln − (t − t1) . (32) M 16M λ 1 0 1 3λ˙ 2 − U e6 = λ , ψ˙ 2 + M (ψ) = λ . (37) 0 0 2 0 On the other hand for λ0 = 0 the analytic solution is from which it follows 1 λ0 2 1 λ0 2 λ (t) = ln tanh 3λ0 (t − t0) − 1 , (33a) λ (t) = ln tanh 3λ0 (t − t0) − 1 or 6 U0 6 U0 123 589 Page 8 of 16 Eur. Phys. J. C (2020) 80 :589 √ λ 3 0 A, ˙ A = − ¯ (ζ ) λ˙ 2 + 3 β˙ 2 + β˙ 2 λ (t) = (t − t0) for U0 = 0, (38) L y y 3F + − 3 8 1 λ while ψ (t) is given in terms of quadratures. + ζ 2 − e6 U¯ (ζ ) , (44) Some functions of M (ψ) where ψ (t) is expressed in 2 closed form are presented in [94]. Recall that the conservation λ ¯ ˙ ¯ ˙ where we have set N = e3 M (ζ ) and the new functions are laws I , I are I = M (ψ) β+, I = M (ψ) β− . ¯ 1 2 1 2 (ζ ) = ¯ (ζ ) (ζ ) (ζ ) = U(ζ) defined as F F M , U M(ζ) . In addition, , ψ − κφ we apply the conservation-laws I1 I2 such that the remain- ¯ (ψ) = M0 , ¯ (ψ) = 6 3.1.3 M M1e U U0e ing field equations are simplified to ψ − κφ When M¯ (ψ) = M eM0 , U¯ (ψ) = U e 6 , then the grav- 1 0 λ¨ + λ˙φ˙ − 6λ (ζ ) = , itational field equations admit the additional conservation law F F,ζ e U 0 (45a) ¨ ˙ 2 6λ φ + 3F,ζ λ + e U,ζ = 0, (45b) φ˙ 3 M M ˙ 1 0 M0φ ˙ ˙ 4 =−6 λ + + e β+β+ + β−β− , ¯ ˙ 2 3 ˙ 2 ˙ 2 1 2 6κ 8 κ −3F (ζ ) λ + β+ + β− + ζ 8 2 (39) λ 3 + e6 U¯ (ζ ) + (I )2 + (I )2 = 0. (45c) 8 1 2 The five conservation laws {h, I1, I2, I3, 4} do not provide any set of four-conservation laws which are in involution ¯ We apply the same procedure as before, where we find except from the case where M0 = 0, that is M (ψ) = M1 i.e. ˙ F (ψ) M (ψ). Thus, the anisotropic parameters β+ and that the reduced dynamical system admits linear conserva- β˙ tion laws for the following sets of the unknown functions − are linear functions of t, that is − ζ { (ζ ) = , (ζ ) = } (ζ ) = F1 , (ζ ) = F F0 U U0 , F F0e U 0 − 6 β+ (t) = I1t + I+,β− (t) = I2t + I−, (40) 2 F and F (ζ ) = F0ζ , U (ζ ) = U0ζ 0 . The two first sets while the other field equations are generated by the point-like are covered in case A; therefore, we continue with the pre- Lagrangian sentation of the new analytic solution for the power-law functions. A A ˙ 2 1 ˙ 2 6(λ−κφ) L y , y˙ = −3λ + ψ − U0e 2 − 6 (ζ ) = ζ 2, (ζ ) = ζ F0 3 3.2.1 F F0 U U0 − (I )2 + (I )2 . (41) 8 1 2 − 6 √ (ζ ) = ζ 2, (ζ ) = ζ F0 1 For F F0 U U0 the reduced field We define the new scalars λ = u + v and ψ = κ u + 6v, equations (45a ), (45b√ ) and (45c) admit the extra conservation where the gravitational field equations are simplified to d − 6F λ law 5 = e 0 ζ . dt √ λ √ Using the new canonical variables x = e 6F0 ζ and 1 2 1 6 1− 6κ v √ 3 − u˙ + 6 1 − √ u˙v˙ − U e − λ x κ2 κ 0 y = e 6F0 ζ or ζ 2 = xy ,λ= √1 ln , the field 2 6 2 6F0 y equations are written as − 3 ( )2 + ( )2 = , I1 I2 0 (42) 8 √ √ 6 −1− 6 U κ 6κ − 6 ¨ − F0 , ¨ = , 0 6 1− 6κ v x 2 y y 0 u¨ + √ e = 0, F0 6κ − 1 − 6 2 √ 1 F 3 2 2 U0 6κ − 1 6 1− 6κ v x˙ y˙ + U0 y 0 + (I1) + (I2) = 0, (46) v¨ − √ e = 0. (43) 2 8 6κ − 1 where the conservation law 5 becomes 5 =˙y. The latter system can be easily integrated and written the Consequently, we find the analytic solution analytic solution by using closed-form functions. −1 2U0 6 − 6 x =− − ( (t − t )) F0 1 5 0 3.2 Case B: M (ψ) arbitrary 5 F0 √ +x1 (t − t0) + x0 , y = 5 (t − t0) , (47) We define a new field dζ = Mdφ, such that the point-like + 3 ( )2 + ( )2 = Lagrangian (20) to be written as with constraint equation x1 5 8 I1 I2 0. 123 Eur. Phys. J. C (2020) 80 :589 Page 9 of 16 589

Recalling that at this case, the line element is of the form has been widely applied before in various cosmological mod- √ els with many interesting results, for example, we refer the λ( ) 2 λ( ) β ( ) ds2 =− e3 t M (ζ (t)) dt2 + e2 t e 2 + t dx2 reader to [95–104] and references therein. The equilibrium points of a spherically symmetric cos- − √1 β+(t)+ 3 β−(t) − √1 β+(t)− 3 β−(t) +e 2 2 dy2 + e 2 2 dz2 , mology in Einstein–aether theory were studied before in [48]; specifically, non-comoving perfect fluid has been con- where M (ζ (t)) is an arbitrary function and for the latter sidered. Static gravitational models in Eintein–æther theory solution the anisotropic functions β+,β− are linear functions with a perfect fluid with a barotropic equations of state and on t and the scale factor λ (t) is expressed as a scalar field were studied in [60,61]. In addition in [52] √ √ it was performed a detailed study of the stability for inho- 2 6F 2 6F x 0 −1− √1 0 mogeneous and anisotropic models of generalized Szekeres exp (λ)= = U¯ (t − t ) 6F0 + x . y 1 0 1 spacetimes. Moreover, isotropic and homogeneous models (48) in Einstein–aether theory with scalar field were considered before in [65–67]. The equilibrium points of Einstein–aether ¯ = ¯ ( , , ) where U1 U1 U0 5 F0 . scalar field theory in Bianchi I spacetimes were studied in (ζ ( )) Considering now the case where M t is a constant [57,58]. ( − ) λ function, then for large values of t t we have e x1, We continue by defining new variables in the so-called from where we find the exact solution = θ = λ˙ H-normalization (recall H 3 N )[105]: √ − √1 + 3 2 6 2 2 2I1t 2 I1t 2 I2t 2 ds =−(x1) dt + x e dx + e 2 dy 1 φ˙ U M β+ = √ , 2 = ,= 1 , x ˙ y ˙ + ˙ √1 3 λ λ2 2 2F λ − I1t− I2 6F 3F +e 2 2 dz2 , (49) 1 M β− − = , (52) λ˙ the latter is an anisotropic solution with constant volume. On 2 2F the other hand, for small values of t √− t0 it follows that the − − With the use of the new variables the gravitational field equa- λ ¯ ( − ) 2 6F0 2 dominant term is e U1 t t0 from which we tions (21)–(22d) are written as follows write √ − − 6 √ 2 =− ¯ ( − ) 2 6F0 2 2 dx 3 2 1 2 ds U1 t t0 dt =− 1 − x x − 6μ + 3x y √ √ dλ 2 2 2 − − + U¯ (t − t ) 4 6F0 4 e 2I1t dx2 1 0 3 F,φ 2 − 1 − x √1 3 √1 3 2 F − I1t+ I2t − I1t− I2t √ +e 2 2 dy2 + e 2 2 dz2 , (50) M 3Mx + 6FM,φ 2 2 + (+) + (−) , 2 ( − ) τ 1 2F or under the√ change of coordinates t t0 K , where (53a) C − 1 =−6 6F0 − 6, the spacetime metric is written as √ dy 1 2 2 ⎛ = y + μx + x − y 1 3 6 3 √ 1 √1 3 λ 2(K −1) − I1+ I2 τ K d 2 2 2 ⎝ 2I τ K 2 2 2 2 ds =−dτ + τ 3K e 1 dx + e dy 2 3 F,φ M 2 2 + xy + 3 (+) + (−) , ⎞ 1 2 F F √1 3 − I1− I2 τ K +e 2 2 dz2⎠ , (51) (53b) d + 3 M 2 2 =− 1 − x + y + where we have removed the non-essential constants. dλ 2 F 5 2 3 M 2 2 + + (+) + (−) 2 F 4 Dynamical systems analysis √ √ F,φ √ M,φ + 6 M x+ − 6√ x+, (53c) F M We continue our study by performing a detailed analysis of d − 3 M 2 2 the equilibrium points for the gravitational field equations. =− 1 − x + y − From such analysis we can extract information of the evolu- dλ 2 F 5 2 tion of solutions of field equations and for the description of 3 M 2 2 + − (+) + (−) the main phases of the cosmological history. This approach 2 F 123 589 Page 10 of 16 Eur. Phys. J. C (2020) 80 :589 √ √ √ F,φ M,φ μ = . + 6 M x− − 6√ x−, (53d) where now const Using constraint (54) the system (59) F M becomes dx 1 √ along with the algebraic equation =− 3x + 2 6 y2, dλ 2 √ 2 2 2 2 dy 1 1 − x + y − (+) + (−) = 0. (54) = y 3x2 + 6(μ + 2)x − 6y2 + 6 , dλ 2 √ ,φ d + 2 d − 2 Given μ = F U we can express φ as a function of μ =−3+ y , =−3− y , (60) U dλ dλ through φ = φ (μ). The evolution equation for μ is given by the first-order ordinary differential equation where the evolution equation for − is decoupled, therefore the system’s dimensionality can be reduced in one- dμ 3 U,φφ F,φ dimension. We restrict the analysis to the reduced system in = x 2F + μ√ − 2μ2 . (55) dλ 2 U F the three dimensional manifold {x, y,+}, where the equilibrium points of (60), have the following coordinates For any equilibrium solution of the field equations, P = μ + − μ ( + μ) x , y ,+ ,− ,Eq.(22a) becomes 2 2 4 P p p p PA = (x A, 0,+A) , PB = − √ , , 0 . 6 6 ¨ √ λ 1 − 1 (61) =− , (λ ) 1 = 4 6x + 3x2 λ2 λ 0 p p 0 2 Point PB describes an isotropic FLRW universe, the exact + − 2 + 2 + 2 3 1 yp +p −p (56) solution at the equilibrium point. It is a scaling solution with w =− + (μ+2)2 − an equation of state parameter√ φ B 1 3 . Point PB λ (t) = λ (t − t ) (λ ) 1 = which means that 0 ln 0 for 0 0. exists when |μ + 2| < 6. On the other hand, P describes β ,β A Similarly, for the anisotropic parameters + − we find a two-dimensional surface, that is, a family of nonhyperbolic √ √ ¨ ˙ 2 equilibrium points, which in general describe an anisotropic β± =−σ±λ ,σ± = 2 3 2 + 4 3x p ±p (57) universe when +A = 0. At the family of points PA only the kinetic part of the scalar field contributes in the cosmological β ( ) = (λ )2 σ ( − ) + from which it follows ± t 0 ± ln t t0 solution. β ( − ) ±1 t t0± . We determine the eigenvalues of the linearized system Finally, the exact solution for the scalar field at the critical around the critical points. For the family of points PA the point P is eigenvalues are √ √ 1 1 (1+ 1−4φ0) (1− 1−4φ0) φ (t) = φ1t 2 + φ2t 2 , (58) 3 e1 (PA) = 3 + (μ + 2) x A, e2 (PA) = e3 (PA) = 0. √ 2 2 2 2 2 where φ0 = 3 (λ0) 6x + μy − 2 +p + −p − 1 . (62) ( ) Eigenvalues √ e1 PA√ can be negative when√ (φ) = φ2, (φ) = φ2 (φ) = φμ 6 4.1 F M and U U0 μ<−2 − 6, − μ+ < x A ≤ 1 or μ>−2 + 6, √ 2 −1 ≤ x − 6 . However, because two of the eigenval- We proceed our analysis by considering F (φ) = φ2, A μ+2 2 μ ues are zero the center manifold theorem should be applied M (φ) = φ and U (φ) = U0φ , where the system (53) is simplified as (see Sect. 4.1.1). For point PB the three eigenvalues are √ dx 1 2 2 2 1 = 2 6 + 3x x + (+) + (−) − 1 , e1 (PB) = μ (4 + μ) − 2, e2 (PB) = e3 (PB) = e1 (PB) . dλ 2 2 (59a) (63)

dy 1 √ = y x2 + (μ + ) x + x2 Consequently, point PB, whenever it exists, is always an λ 3 6 2 3 d 2 attractor. 2 2 2 +3 1 − y + (+) + (−) , (59b) 4.1.1 Center manifold theorem for PA d + 3 2 2 2 2 = + x + (+) + (−) − y − 1 , (59c) dλ 2 Introducing the new variables d − 3 2 2 2 2 = − x + (+) + (−) − y − 1 , (59d) dλ 2 x1 = x − x A, x2 = + − +A, x3 = y, (64) 123 Eur. Phys. J. C (2020) 80 :589 Page 11 of 16 589 we obtain the evolution equations whose general solutions are μ + √ −3τ 2 −3τ dx1 1 2 x (τ) = c e − √ − x , x (τ) = c e − + , =− 6(μ + 2) + 6x + 6x A x , (65a) 1 1 A 2 2 A dλ 2 1 3 6 √ dx2 2 −3τ −6τ 2 =− (x + + ) x , 2 6c1(μ + 2)e + 6c3e − μ − 4μ + 2 λ 3 2 A 3 (65b) d x3(τ) = √ . √ √ 6 dx3 1 2 = 6(μ + 2)x − 6x + 6(μ + 2)x A + 6 x . dλ 2 1 3 3 (72) (65c) μ√+2 Hence, x (τ) →− −x A, x (τ) →−+A, x (τ) → √ 1 6 2 3 The center manifold is therefore given by the graph −μ2−4μ+2 √ ,asτ →∞. 6 3 (x2, x1, x3)∈R : x3 =h(x1, x2), Dh = 0, h(0, 0) = 0, 4.1.2 Normal forms 2 2 x1 + x2 ≤ δ , (66) In this section we show normal form of expansions for the where h satisfies the partial differential equation vector field (65) defined in a vicinity of PA, expressed in the form of Proposition 1. In general, let X : Rn → Rn be a √ ∂h h h (μ + ) + x + x smooth vector field satisfying X(0) = 0. We can formally 6 2 6 1 6 A ∂ x1 construct the Taylor expansion of x about 0, namely, X = ∂ k+1 r h X1 + X2 +···+Xk + O(|x| ), where Xr ∈ H , the real +6(+A + x2) − 6h ∂x2 vector space of vector fields whose components are homo- √ √ geneous polynomials of degree r, X1 = DX(0)x ≡ Ax, i.e., + 6(μ + 2)x1 + 6(μ + 2)x A + 6 = 0. (67) the matrix of derivatives. For r = 1tok we write The above equation admits the three solutions: r r n m Xr (x) = ··· Xm, j x e j , mi = r, (73) h(x1, x2) = 0, m1=1 mn=1 j=1 i ( , ) =± h x1 x2 = m := m1 m2 √ Let denote the vector space Br x ei x1 x2 m3 r 6 (x + + )2 c (ξ) − μ2 − 4μ + 2 6(μ + 2)ξ (x + + ) + 2 x e |m ∈ N, m = r, i, j = 1, 2, 3 ⊂ H . 2 A 1 √ 2 A , 3 i j j (r) ( ) ∈ r 6 Let LA be the linear operator that assigns to h y H (68) the Lie bracket of the vector fields Ay and h(y): √ (r) r r (r) (μ+ )+ + L : H → H , h → L h(y) = Dh(y)Ay − Ah(y). where c is an arbitrary function of ξ = 6 2 6x1 6x A . A A 1 6(x2++A) Only the first solution satisfies Dh = 0, h(0, 0) = 0. There- (74) m fore, the center manifold is given locally by Applying this operator to monomials x ei , where m is a multiindex of order r and e basis vector of R3, we find 3 2 2 i (x1, x2, x3) ∈ R : w = 0, x1 + x2 ≤ δ . (69) (r) m = {( · λ) − λ } m . LJ x ei m i x ei (75) The evolution on the center manifold is given by The eigenvectors in Br for which m,i ≡ (m · λ) − λi = 0 form a basis of Br = L (Hr ) and those such that , = 0, = , = . J m i x1 0 x2 0 (70) associated to the resonant eigenvalues, form a basis for the complementary subspace, Gr , of Br in Hr . Obtaining the That is x1 and x2 are constants at the center manifold. normal form we must look for resonant terms, i.e., those df = 1 df m Introducing the time rescaling τ 2 λ , the equations termsoftheformx e with m and i such that , = 0for d x3 d i m i become the available m, i. √ 1 Theorem 1 (Theorem 2.3.1 in [69]) Giving a smooth vector x (τ) = − 6(μ + 2) − 6x1(τ) − 6x A , (71a) ( ) Rn ( ) = , 1 2 field X x on with X 0 0 there is a polynomial trans- (τ) =− (τ) − , formation to new coordinates, y, such that the differential x2 3x2 3 +A (71b) √ √ equation x = X(x) takes the form y = Jy + N w (y) + 2 r=1 r 6(μ + 2)x (τ) − 6x (τ) + 6(μ + 2)x + 6 N+1 x (τ) = 1 3 A , O(|y| ), where J is the real Jordan form of A = DX(0) 3 (τ) r r r 2x3 and wr ∈ G , a complementary subspace of H on B = (71c) r (r) LA(H ), where LA is the linear operator that assigns to 123 589 Page 12 of 16 Eur. Phys. J. C (2020) 80 :589 √ √ r h(y) ∈ H the Lie bracket of the vector fields Ay and h(y), 6(μ+2)x A +6, (0,0,2),2 = 6(μ+2)x A +6, (1,0,1),3 = (r) : r → r , → (r) ( ) = ( ) − ( ). LA H H h LA h y Dh y Ay Ah y 0. Eliminating the non-resonant quadratic terms, we imple- ∗ = ( , , )T ∈ . Let x x A +A 0 PA By taking the linear trans- ment the quadratic transformation formation x1 = x − x A, x2 = + − +A, x3 = y, 2 2 as in Sect. 4.1.1, we obtain the vector field X given by x → y + h2(y), h2 : H → H , ∞ ⎛ √ ⎞ (65) which is C in a neighborhood of the origin. Let 2 (μ+ )+ √ y3 6 2 6x A λ = 0,λ = 0,λ = 1 6(μ + 2)x + 6 ∈/ Z. ⎜ − √ ⎟ 1 2 3 2 A ⎜ 2 6(μ+2)x +6 ⎟ ( ) = ⎜ A ⎟ , h2 y ⎜ 3+ y2 ⎟ (79) Proposition 1 (Leon and Paliathanasis 2020) Let be the vec- ⎝ − √ A 3 ⎠ 6(μ+2)x A+6 tor field X given by (65). Then, there exist a transformation to 0 new coordinates x → z, such that (65), defined in a vicinity of 0, has normal form such that the vector field (77) transforms into

√ (2) 2 2 = − ( ) + ( ) + ˜ ( )+ ˜ ( ) + (| |5), 3 6(μ + 2)(μ(μ + 4) − 2)z z y Jy L h2 y X2 y X3 y X4 y O y =− 1 3 J z1 √ 2 6(μ + 2)x A + 6 (80) √ √ 4 (μ+ )(μ(μ+ )+ )+ (μ+ ) 2 + (μ(μ+ )+ ) 3z3 6 2 4 10 12 6 2 x A 18 4 6 x A − where √ 2 2 6(μ+2)x +6 A ⎛ ⎞ 2 3(μ(μ+4)−2)y1 y + O(|z|5), (76a) √ 3 ⎜ √ 6(μ+2)x A+6 ⎟ ⎜ (μ+ ) 2 ⎟ 2 2 2 2 3 6 2 +A y1 y 2 18(μ + 2) +A z z 3(μ + 2)z1z2z ˜ ( ) = ⎜ √ 3 − 3y y ⎟ , =− 1 3 + √3 X3 y ⎜ (μ+ )x + 2 3 ⎟ z2 √ 6 2 A √6 2 (μ + ) + ⎝ 3 ⎠ (μ + ) + 2 x A 6 3y μ(μ+4)+3 6(μ+2)x A+16 6 2 x A 6 − 3 √ (μ+ ) + √ 2 6 2 x A 12 4 ⎛ √ √ ⎞ 9+ z μ(μ + 4) + 2 6(μ + 2)x + 10 4 2 A 3 A 3y 6(μ+2)(μ(μ+4)+10)+12 6(μ+2)x +18(μ(μ+4)+6)x A − ⎜ − 3 A ⎟ √ 2 ⎜ √ 2 ⎟ (μ + ) + 2 6(μ+2)x A+6 6 2 x A 6 ⎜ √ ⎟ ˜ ⎜ 4 μ(μ+ )+ (μ+ ) + ⎟ X4(y) = ⎜ 9 +A y 4 2 6 2 x A 10 ⎟ , 5 − 3 + O(|z| ), (76b) ⎜ √ 2 ⎟ ⎝ (μ+ ) + ⎠ √ 6 2 x A 6 1 z = z 6(μ + 2)(x A + z ) + 6 0 3 2 3 1 √ (μ + ) 3 (μ(μ + ) + ) + (μ + ) (81) 3 2 z1z3 6 4 7 9 2 x A + √ 2 (μ + ) + 6 2 x A 6 such as + O(|z|5). (76c) √ − (2) ( ) + ( ) = 6(μ + ) ⇒ = LJ h2 y X2 y 2 y1 y3e3 y Jy Proof The system (65) can be written as √ 2 6 ˜ ˜ 5 x = Jx + X (x) + X (x) (77) + (μ + 2)y1 y3e3 + X3(y) + X4(y) + O(|y| ). (82) 2 3 2 where x stands for the phase vector x = (x , x , x )T , and 1 2 3 (3) : ⎛ ⎞ Simplifying the cubic part The linear operator LJ 00 0 3 → 3 m = ⎜ ⎟ H H has eigenvectors x ei with eigenvalues m,i = 00 0 , J ⎝ ⎠ λ3m3 −λi , i = 1, 2, 3, m1, m2, m3 ≥ 0, m1 +m2 +m3 = 3. 3 00 x A(μ + 2) + 3 The eigenvalues m,i for the allowed√m, i are (1,0,2),1 = ⎛ 2 ⎞ (1,0,2),2 = (0,1,2),2 = (0,0,3),3 = 6(μ + 2)x A + 6. 2 3 ⎜ x − (μ + 2) − 3x A ⎟ Eliminating the non-resonant terms of third order, we ⎜ 3 2 ⎟ X2(x) = ⎜ − 2 ⎟ , implement the coordinate transformation ⎝ 3 +Ax3 ⎠ 3 (μ + ) 3 3 2 x1x3 y = z + h3(z), h3 : H → H , ⎛ 2⎞ ⎛ ⎞ 3(μ(μ+4)−2)z z2 −3x x2 1 3 1 3 ⎜ √ 2 ⎟ ( ) = ⎝ − 2 ⎠ . 6(μ+2)x A+6 X3 x 3x2x3 (78) ⎜ √ √ ⎟ 3 ⎜ 2 (μ+ ) + − (μ+ ) ⎟ −3x ⎜ 3z3 z2 6 2 x A 6 6 2 +Az1 ⎟ 3 − ( ) = ⎜ √ 2 ⎟ , h3 z ⎜ (μ+ ) + ⎟ (83) (2) ⎜ 6 2 x A 6 ⎟ Simplifying the quadratic part The linear operator L : ⎜ √ ⎟ J 3 μ(μ+ )+ (μ+ ) + 2 2 m ⎝ 3z3 4 3 6 2 x A 16 ⎠ H → H has eigenvectors x e with eigenvalues , = − i m i √ 2 (μ+ ) + λ3m3 −λi , i = 1, 2, 3, m1, m2, m3 ≥ 0, m1 +m2 +m3 = 2. 2 6 2 x A 6 The eigenvalues m,i for the allowed m, i are (0,0,2),1 = 123 Eur. Phys. J. C (2020) 80 :589 Page 13 of 16 589 such as 5 Alternative dynamical system’s formulation

(3) ˜ Using the alternative variables and time derivative −L h3(z) + X3(z) = 0 ⇒ z = Jz J√ ˙ ˙ φ 3Fλ2 1 M β+ + 6(μ + ) + ( ) + (| |5), x = √ , z = ,+ = , 2 z1z3e3 X4 z O z (84) λ˙ U 2 2F λ˙ 2 6F 1 M β− df df where − = , = z , (91) 2 2F λ˙ dτ dλ ⎛ √ √ √ ⎞ 4 (μ+ )(μ(μ+ )+ )+ (μ+ ) 2 + (μ(μ+ )+ ) 3z3 6 2 4 10 12 6 2 x A 18 4 6 x A 3 6(μ+2)(μ(μ+4)−2)z2z2 − − 1 3 ⎜ √ 2 √ 2 ⎟ ⎜ (μ+ ) + (μ+ ) + ⎟ ⎜ 2 6 2 x A 6 6 2 x A 6 ⎟ ⎜ √ ⎟ 2 2 2 4 μ(μ+ )+ (μ+ ) + ⎜ 18(μ+2)2+ z z 3(μ+2)z z z 9 +Az3 4 2 6 2 x A 10 ⎟ − A 13 + 1 2√3 − X4(z) = ⎜ √ 2 √ 2 ⎟ . (85) ⎜ (μ+2)x A+ 6 ⎟ 6(μ+2)x A+6 6(μ+2)x A+6 ⎜ √ ⎟ ⎜ 3 ⎟ ⎝ 3(μ+2)z1z 6(μ(μ+4)+7)+9(μ+2)x A ⎠ 3 √ 2 6(μ+2)x A+6

Then, the result of the proposition follows. we obtain the dynamical system Finally, the fourth order terms, which all are non-resonant, 3 x =− (μ + 2) − 3x,+ =−3+, can be removed under the quartic transformation 2 ! √ " z =−z z 6(μ + 2)x + 6 − 6 . (92) √ 3 6(μ + 2)(μ(μ + 4) − 2)w2w2 z → w − 1 3 1 1 √ 3 It is worth noticing that the system (92) is integrable with 6(μ + 2)x A + 6 √ √ general solution w4 (μ+ )(μ(μ+ )+ )+ (μ+ ) 2 + (μ(μ+ )+ ) 3 3 6 2 4 10 12 6 2 x A 18 4 6 x A − , μ + √ 2 √ −3τ 2 −3τ (μ+ ) + (μ+ ) + x(τ) = c1e − √ ,+(τ) = c2e , 2 6 2 x A 6 2 6 2 x A 12 6 (86) 6 2 2 2 z(τ) = √ . 18(μ + 2) +Aw w − τ − τ → w − 1 3 (μ + ) 3 + 6 − μ2 − μ + z2 2 √ 3 2 6c1 2 e 6c3e 4 2 (μ + ) + 6 2 x A 6 (93) 3(μ + 2)w w w2 + 1 2 3 √ √ The equilibrium points are (μ + 2)x A + 6 6(μ + 2)x A + 6 √ 9+ w4 μ(μ + 4) + 2 6(μ + 2)x + 10 μ + 2 6 A 3 A : ( , , ) = − √ , , − , − , (87) PB x + z 0 √ 2 √ 6 μ(μ + 4) − 2 6(μ + 2)x A + 6 2 6(μ + 2)x A + 12 √ μ + 2 (μ + )w w3 (μ(μ + ) + ) + (μ + ) PC : (x,+, z) = − √ , 0, 0 . (94) 3 2 1 3 6 4 7 9 2 x A → w + . 6 z3 3 √ 3 (88) 6(μ + 2)x A + 6 The eigenvalues of PB are {−6, −3, −3}, therefore, it is a sink. On the other hand, PC has eigenvalues {6, −3, −3}, Neglecting the higher order terms we obtain the integrable and it is a saddle. Interestingly, the equation for + decou- system ples, and we can study a reduced dynamical system for the variables (x, z). √ 1 In Fig. 1 are presented some orbits of system (92)forsome z = 0, z = 0, z = z 6(μ + 2)(x A + z ) + 6 , 1 2 3 2 3 1 values of parameter μ in the Poincarè variables (U, V ).The (89) point PB, whenever it exists, it is a sink. Point PC is always a saddle. The system admits three conﬁgurations at inﬁnity with general solution Q1 : (U, V ) = (1, 0), Q2 : (U, V ) = (0, 1), and Q3 : (U, V ) = (−1, 0), whose dynamics is shown in the plots. In : ( , ,+) = ( , ,+ ) z1(λ) = z10, z2(λ) = z20, this coordinates the set PA x y x A 0 A is √ =∞, ( , ) = 1 λ (μ+ )( + )+ translated to Q2 due to limy→0 z limz→∞ U V 2 6 2 x A z10 6 z3(λ) = z30e . (90) (0, 1). 123 589 Page 14 of 16 Eur. Phys. J. C (2020) 80 :589

(a) (b)

Fig. 1 Orbits of the system (92) for some values of parameter μ in system admits three conﬁgurations at inﬁnity Q1 : (U, V ) = (1, 0), : ( , ) = ( , ) : ( , ) = (− , ) the Poincarè variables (U, V ) = √ x , √ z . The point Q2 U V 0 1 ,andQ3 U V 1 0 , whose dynamics is 1+x2+z2 1+x2+z2 shown in the plots PB , whenever it exists, it is a sink. Point PC is always a saddle. The

6 Conclusions consider the homogeneous but anisotropic Bianchi I spacetime. In this paper we have investigated a Lorentz violating We have extended previous analyses on the subject by Einstein-aether theory which contains a scalar field nonmini- considering an interacting function between the scalar and the mally coupled with the aether field. For the physical space we aether fields, which is nonlinear on the kinematic quantities of the time-like aether field. In particular we assume that

123 Eur. Phys. J. C (2020) 80 :589 Page 15 of 16 589 the interacting function is quadratic on the expansion rate θ Data Availability Statement This manuscript has no associated data or and on the shear σ , while in the generic scenario has three the data will not be deposited. [Authors’ comment: This is a Theoretical unknown functions of the scalar field, as expressed by Eq. Research Project.] (13). Open Access This article is licensed under a Creative Commons Attri- The novelty of the interacting function under consider- bution 4.0 International License, which permits use, sharing, adaptation, ation is that we can determine a point-like Lagrangian and distribution and reproduction in any medium or format, as long as you write the field equations by using the minisuperspace descrip- give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes tion. Indeed, the field equations can be seen as the motion of were made. The images or other third party material in this article a point-like particle in a four-dimensional Riemannian space are included in the article’s Creative Commons licence, unless indi- wich coordinates the three scalars of the Bianchi I space- cated otherwise in a credit line to the material. If material is not time and the field φ, under the action of a potential function. included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permit- By using this property, we are able to apply methods from ted use, you will need to obtain permission directly from the copy- analytic mechanics and study the integrability properties of right holder. To view a copy of this licence, visit http://creativecomm the field equations. We use Ansätze for conservation laws ons.org/licenses/by/4.0/. 3 which are linear in the momentum, such that it is possible to Funded by SCOAP . specify the unknown functions of the field equations, which allows for exact or analytic solutions of the field equations by using closed-form functions. Hence, the field equations References are Liouville integrable. In order to study the dynamics and the evolution of the 1. E.W. Kolb, M.S. Turner, The early universe (Addison-Wesley, anisotropies, we determine the equilibrium points for the New York, 1990) field equations. These points describe some specific physical 2. A. Guth, Phys. Rev. D 23, 347 (1981) solutions for the model of our consideration. We perform our 3. C.W. Misner, The Isotropy of the universe. Ap. J. 151, 431 (1968) 4. O. Hrycyna, M. Szydlowski, Dynamics of the Bianchi I model analysis by using the Hubble-normalized variables, also by with non-minimally coupled scalar field near the singularity. 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