Anisotropic chiral cosmology: exact solutions
Luis Rey Díaz-Barrón, Abraham Espinoza-García, S. Pérez-Payán, J. Socorro
AAnisotropic chiral cosmology: exact solutions
J. Socorro, ∗ S. P´erez-Pay´an, † Abraham Espinoza-Garc´ıa, ‡ and Luis Rey D´ıaz-Barr´on § Departamento de F´ısica, DCeI, Universidad de Guanajuato-Campus Le´on, C.P. 37150, Le´on, Guanajuato, M´exico Unidad Profesional Interdisciplinaria de Ingenier´ıa,Campus Guanajuato del Instituto Polit´ecnico Nacional.Av. Mineral de Valenciana
In this work we investigate the anisotropic Bianchi type I cosmological model in the chiral setup, ina twofold manner. Firstly, we consider a quintessence plus a K-essence like model, where two scalarfields but only one potential term is considered. Secondly, we look at a model where in addition tothe two scalar fields the two potential terms are taken into account as well as the standard kineticenergy and the mixed term. Regarding this second model, it is shown that two possible cases canbe studied: a quintom like case and a quintessence like case. In each of the models we were able tofind both classical and quantum analytical solutions.
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I. INTRODUCTION
The incorporation of scalar fields into cosmological models has helped explain different phenomena of our Universe.For example, single scalar field cosmological models have been used to describe the inflationary phase and the late timeacceleration of the Universe, also the dark matter component of the Universe can be accounted as well as unificationof the early inflation to the late acceleration [1–14], to mention a few.In recent years multi-scalar field cosmological models have drawn tremendous attention from the scientific commu-nity. In this cosmological scenario two or more scalar fields are considered to describe adequately the evolution of theUniverse, and generally, the interaction of these scalar fields occur in the potential, in the kinetic terms or both. Theadvantages of multi-field cosmological models (over single field ones) is the introduction of new degrees of freedomallowing the explanation of several physical phenomena. In this setup, an inflationary stage can also be achieved[15, 16], even in the case where the fields are non interacting [17]. Moreover, multi-scalar fields models can also beused to explain the primordial inflation perturbations analysis [18, 19] or the assisted inflation [16, 20]. Anotherappealing reason to work this models is that when two scalar fields are considered the crossing of the cosmologicalconstant boundary “-1” can be described, in the litearature this models are known as quintom models [21–24] (singlescalar field models do not have this malleability, since they only describe either the phantom or quintessence regime).Furthermore, this multi-field models can also tackle the hybrid inflation of the Universe, which gives an alternativegraceful exit in comparison to the standard inflationary picture [25–32]. From a phenomenological point of view, themost successful models are those that have incorporated quintessence scalar fields, slow-roll inflation, chiral cosmologyconnected with f ( R ) theories and the nonlinear sigma model [7, 17, 33–54].On the other hand, the hypothesis of primordial anisotropy at early stages of the Universe (even predating in-flation), is a tempting proposal that can point in the right direction regarding the anomalies found in the cosmic ∗ Electronic address: socorro@fisica.ugto.mx † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] a r X i v : . [ g r- q c ] J a n microwave background (CMB) anisotropies on large angular scales. Following these guidelines, some attempts havebeen presented in [55–58], where anisotropic cosmological models, mostly the Bianchi I model, have been used as abackground space-time in an early anisotropic but homogeneous Universe that develops isotropization at the onset ofinflation, however, the imprints of such anisotropy would lead to the anomalies found in the thermal maps of the CMB;and after the inflationary period as a consequence of this isotropization, the Universe acquires a Friedmann-Lamaˆitre-Robertson-Walker geometry recovering the standard picture of the evolution of the Universe. Hence, anisotropiccosmological models represent an attractive arena to test the early stages of the Universe, even if no conclusiveevidence that a primordial anisotropy is needed.On this regard, multi-field anisotropic cosmological models of inflation have been explored. In [59] the authordelves into the study of the Bianchi type I cosmology considering two interacting scalar fields and a potential of theform V ( φ, χ ) ∼ φ + χ , founding numerical solutions as well as the asymptotically isotropic Friedmann case. Otherinteresting works are presented in [29, 60], where the potential, with structure V ( φ, σ ) ∼ e φ + σ it is shown to be a goodviable candidate to address the inflationary era in both flat isotropic and anisotropic space-times. More recently, in[61] the authors present the case of the anisotropic Bianchi type I cosmology in the multi-field setup with a potential ofthe form V e − ( λ φ + ··· + λ n φ n ) , founding inflationary exact solutions in a quintessence framework. Additional researchregarding multi-filed anisotropic cosmological models can be found in [44, 62–71].In the present work we present the anisotropic Bianchi type I cosmological model with two scalar fields in a twofoldmanner. Following closely the developments introduced in [72, 73], first, we put forward a simple quintessence plusa K-essence model which arises from considering the interaction of the two scalar fields but only one term in thepotential. And second, a chiral approach is studied, here in addition to the two scalar fields we also consider the twopotential terms as well as the standard kinetic energy and the mixed term. For each model, classical and quantumanalytical solutions are found.This paper is arrange as follows. In section II we introduce the first model, where the Einstein-Klein-Gordon(EKG) equations are calculated and the Lagrangian and Hamiltonian approach is implemented in order to find thecorresponding solutions as well as the anisotropic parameters. In section III, the second model presented, here, afterobtaining the Hamiltonian density we can distinguish two possible scenarios: a quintom like epoch and a quintessenceepoch. For both scenarios the corresponding solutions are obtained. Section IV is devoted two implement the quantumversions of the previous two models and the corresponding solutions are obtained. Finally, section V is left for thefinal remarks. II. FIRST MODEL: QUINTESSENCE PLUS K-ESSENCE
As mention above, we are going to start by analyzing the quintessence plus K-essence model. For this purpose weare going to consider the multi-field Bianchi type I model where the two scalar fields are taken into account but onlyone potential term. The Lagrangian density for such a model is L = √− g (cid:18) R −
12 g µν ∇ µ φ ∇ ν φ −
12 g µν ∇ µ φ ∇ ν φ + V( φ ) (cid:19) , (1)where R is the Ricci scalar, V( φ ) is the corresponding scalar field potential, and the reduced Planck mass M P =1 / πG = 1. The corresponding variations of Eq.(1), with respect to the metric and the scalar fields give the EKGfield equations G αβ = − (cid:18) ∇ α φ ∇ β φ −
12 g αβ g µν ∇ µ φ ∇ ν φ (cid:19) + 12 g αβ V( φ ) − (cid:18) ∇ α φ ∇ β φ −
12 g αβ g µν ∇ µ φ ∇ ν φ (cid:19) , (2) (cid:3) φ − ∂ V ∂φ = g µν φ ,µν − g αβ Γ ναβ ∇ ν φ − ∂ V ∂φ = 0 , (3)g µν φ ,µν − g αβ Γ ναβ ∇ ν φ = 0 . (4)The line element for the anisotropic cosmological Bianchi type I model in the Misner’s parametrization isds = − N dt + a dx + a dy + a dz , = − N dt + e (cid:104) e β + +2 √ β − dx + e β + − √ β − dy + e − β + dz (cid:105) , (5)where a i (i = 1 , ,
3) are the scale factor on directions (x , y , z), respectively, and N is the lapse function. Forconvenience, and in order to carry out the analytical calculations, we consider the following representation for the lineelement (5) ds = − N dt + η (cid:2) m dx + m dy + m dz (cid:3) , (6)where the relations between both representations (5) and (6) are given by η = e Ω , m = e β + + √ β − , ˙m m = ˙ β + + √ β − , m = e β + −√ β − , ˙m m = ˙ β + − √ β − , (7)m = e − β + , ˙m m = − β + , where η is a function that has information regarding the isotropic scenario and the m i are dimensionless functionsthat have information about the anisotropic behavior of the Universe, such that (cid:89) i=1 m i = 1 , (cid:89) i=1 a i = η , (cid:88) i=1 ˙m i m i = 0 , (8)act as constraint equations for the model. A. General Solutions to the Field Equations
In this subsection we present the solutions of the field equations for the anisotropic cosmological model, consideringthe temporal evolution of the scale factors with barotropic fluid and standard matter. The solutions obtained alreadyconsider the particular choice of the Misner-like transformation discussed lines above. Using the metric (6) and aco-moving fluid, equations (2) take the following form (cid:18) (cid:19) ˙m Nm ˙m Nm + ˙m Nm ˙m Nm + ˙m Nm ˙m Nm + 3 (cid:18) ˙ η N η (cid:19) − π G ρ − (cid:32)
12 ˙ φ N + V( φ ) (cid:33) −
14 ˙ φ N = 0 , (9) (cid:18) (cid:19) − ˙NN (cid:20) ˙m Nm + ˙m Nm + 2 ˙ η N η (cid:21) + ¨m N m + ¨m N m + ˙m Nm ˙m Nm + 2 ¨ η N η + (cid:18) ˙ η N η (cid:19) + 3 ˙ η N η (cid:20) ˙m Nm + ˙m Nm (cid:21) + 12 (cid:32)
12 ˙ φ N − V( φ ) (cid:33) + 14 ˙ φ N + 8 π GP = 0 , (10) (cid:18) (cid:19) − ˙NN (cid:20) ˙m Nm + ˙m Nm + 2 ˙ η N η (cid:21) + ¨m N m + ¨m N m + ˙m Nm ˙m Nm + 2 ¨ η N η + (cid:18) ˙ η N η (cid:19) + 3 ˙ η N η (cid:20) ˙m Nm + ˙m Nm (cid:21) + 12 (cid:32)
12 ˙ φ N − V( φ ) (cid:33) + 14 ˙ φ N + 8 π GP = 0 , (11) (cid:18) (cid:19) − ˙NN (cid:20) ˙m Nm + ˙m Nm + 2 ˙ η N η (cid:21) + ¨m N m + ¨m N m + ˙m Nm ˙m Nm + 2 ¨ η N η + (cid:18) ˙ η N η (cid:19) + 3 ˙ η N η (cid:20) ˙m Nm + ˙m Nm (cid:21) + 12 (cid:32)
12 ˙ φ N − V( φ ) (cid:33) + 14 ˙ φ N + 8 π GP = 0 , (12)here a dot ( ˙ ) represents a time derivative. The corresponding Klein-Gordon (KG) equations are given by˙NN ˙ φ N − ˙ φ ¨ φ N − ηη ˙ φ N − ˙V = 0 , → ddt Ln (cid:18) N η ˙ φ (cid:19) = N ˙V˙ φ , ˙NN ˙ φ N − ˙ φ ¨ φ N − ηη ˙ φ N = 0 , → ddt Ln (cid:18) N η ˙ φ (cid:19) = 0 , (13)where from the last equation in (13) it is easy to see that the solution for the scalar field φ (in quadrature form) isgiven by ∆ φ = φ (cid:90) N η dt , (14)with φ an integration constant.Now, subtracting (10) from the component (11) we obtain˙NN (cid:20) ˙m m − ˙m m (cid:21) − ˙m Nm ˙m Nm + 1N (cid:20) ¨m m − ¨m m (cid:21) + ˙m Nm ˙m Nm + 3 ˙ η N η (cid:20) ˙m m − ˙m m (cid:21) = 0 , (15)noticing that 1N (cid:20) ˙m Nm − ˙m Nm (cid:21) • = 1N (cid:20) ¨m m − ¨m m (cid:21) − (cid:34)(cid:18) ˙m m (cid:19) − (cid:18) ˙m m (cid:19) (cid:35) + ˙NN (cid:20) ˙m m − ˙m m (cid:21) , (16)equation (15) can be rearranged and written as (where • also denotes a time derivative)1N (cid:20) ˙m Nm − ˙m Nm (cid:21) • + 3 ˙ η N η (cid:20) ˙m Nm − ˙m Nm (cid:21) = 0 , (17)finally, defining R = ˙m Nm − ˙m Nm , the last equation can be casted as ˙R R + 3 ˙ ηη = 0 , whose solution is given byR = (cid:96) η , (18)where (cid:96) is an integration constant.When we perform the same procedure with the other pair of equations, namely, subtracting (11) from the component(12) one obtains˙NN (cid:20) ˙m m − ˙m m (cid:21) − ˙m Nm ˙m Nm + 1N (cid:20) ¨m m − ¨m m (cid:21) + ˙m Nm ˙m Nm + 3 ˙ η N η (cid:20) ˙m m − ˙m m (cid:21) = 0 , (19)which has the same structure as equation (15). Proceeding in the same manner as we did above, we define R = ˙m Nm − ˙m Nm , obtaining a differential equation whose solution is analogous to (18),R = (cid:96) η , (20)where (cid:96) is an integration constant. And lastly, subtracting (12) from (10) we getR = (cid:96) η , (21) (cid:96) is also a constant that comes from integration, these three constants satisfy (cid:96) + (cid:96) + (cid:96) = 0.From equation (18) we have that 2 ˙m Nm − ˙m Nm − ˙m Nm = (cid:96) η , (22)and using the constraints from (8), the last equation reduces to2 ˙m Nm + ˙m Nm = (cid:96) η , (23)finally, as a last step, utilizing (20) we get 3 ˙m Nm + (cid:96) η = (cid:96) η . (24)In order to investigate the solution for the last equation we cast it in the following form˙m Nm = (cid:96) − (cid:96) η = (cid:96) η , (25)where (cid:96) = (cid:96) − (cid:96) . The other components can be obtained in a similar fashion, which read˙m Nm = (cid:96) η , (26)˙m Nm = (cid:96) η , (27)the constants being (cid:96) = (cid:96) − (cid:96) and (cid:96) = (cid:96) − (cid:96) , also, these constants satisfy (cid:80) j =1 (cid:96) j = 0. Now that equations(25)-(27) are written in a more manageable way, obtaining the solutions is straightforward, these are given bym i (t) = δ i Exp (cid:20) (cid:96) i (cid:90) Ndt η (cid:21) , (28)where Π δ j = 1, setting the gauge N → η , the solution becomesm i (t) → α i Exp [ (cid:96) i ∆t] . (29)Unfortunately under this approach we could not find analytical solution for η , because we need to know the solutionfor the scalar field φ (see equation (9)). To be able to reach a solution we are going to resort to the Hamiltonianformalism. To this end, we employ equation (1) and the line element (6), now the Lagrangian density with the scalarfield V( φ ) = V e − λ φ becomes L = η (cid:32) (cid:18) ˙ ηη (cid:19) − (cid:34)(cid:18) ˙m m (cid:19) + (cid:18) ˙m m (cid:19) + (cid:18) ˙m m (cid:19) (cid:35) − ˙ φ − ˙ φ
2N + NV e − λ φ (cid:33) , (30)where the momenta are Π η = 12 η N ˙ η Π φ = − η N ˙ φ Π φ = − η N ˙ φ , Π = − η N (cid:18) ˙m m (cid:19) , Π = − η N (cid:18) ˙m m (cid:19) , Π = − η N (cid:18) ˙m m (cid:19) , ˙ η = N12 η Π η , ˙ φ = − N η Π φ ˙ φ = − N η Π φ , ˙m = − Nm Π η , ˙m = − Nm Π η , ˙m = − Nm Π η , (31)leading to the Hamiltonian density, which takes the form H = 124 η Π η − η m Π − η m Π − η m Π − η Π φ − η Π φ − V η e − λ φ . (32)Making the transformation Π η = ∂ S ∂η , and Π i = ∂ S ∂ m i and choosing η = e u and m i = e u i , where P i = ∂ S ∂ u i and π u = ∂S∂u ,the Hamiltonian density becomes H = e − (cid:2) π − − − − φ − φ − U(u , φ ) (cid:3) , (33)where U(u , φ ) = 24V e − λ φ is the potential function. In the gauge N = 24e , the Hamilton equations are˙u = 2 π u , ˙ π u = 6U , ˙ φ = − φ , ˙ φ = − φ , ˙Π φ = − λ U , ˙Π φ = 0 , ˙u i = − i , ˙P i = 0 . (34)From the Hamilton equations Eq.(34), we can find relations between the scale factor and the scalar fields, such as˙ φ = − φ = 4 λ π u + 24p φ = 2 λ ˙u + 24p φ , ˙ φ = − φ = 24p φ , (35)˙u i = − i , where p φ , p φ and p i are integration constants to be determined by suitable conditions. The solutions of equations(35) read ∆ φ = 2 λ ∆u + 24p φ ∆t , (36)∆ φ = 24p φ ∆t , (37)∆u i = 12p i ∆t , (38)m i = β i e − i ∆t , (39)with the constraints between the constants p i that must be fulfil: (cid:80) p i = 0 and (cid:80) p = 2(p + p p + p ).Equations (36-39) are expressions similar to the solution found by algebraic manipulation to (13) for the scalar field φ and the Einstein equation (29) for the m i functions. These expressions are indeed general relations, since theysatisfy the EKG equations Eqs.(9-13).On the other hand, taking into account the constraint H = 0, we obtain the temporal dependence for π u (t) whichallows us to construct a master equation: d π u α π − α π u − α = dt , (40)where the parameters α i with i = 1 , ,
3, are α = 2(3 − λ ) = 2 β , α = 24 λ p φ , α = 72 (cid:2) p φ + c (cid:3) , and c = p φ + p + p p + p . (41)In the next subsections, we present solutions for three different values of the parameter λ and also are able toconstruct the anisotropic parameters. B. Case α > and λ < √ For this case, we have that the solution for π u (t) is given by π u = 14 β (cid:104) α − α Coth (cid:16) α (cid:17)(cid:105) , (42)where α = 24 ω with ω = (cid:113) p φ + c β . The solutions of the set of variables (u , u i , φ , φ ) and (Π φ , Π φ , P i ) are:u = u + 12 λ p φ β t + ln [Csch(12 ω t)] /β , (43) φ = p φ + 72 p φ β t − Ln[Sinh(12 ω t)] λ /β , (44) φ = p φ + 24p φ t , (45)u i = − i ∆t , (46)Π φ = − φ β + λ α β coth (cid:16) α (cid:17) , (47)Π φ = − p π , (48)P i = p i , (49)here (u , p φ , p φ , p i ) are integration constants. Finally the scale factor η = e u and the anisotropic parameters takethe form η = η Exp (cid:20) λ p φ β t (cid:21) Csch β (cid:16) (cid:113) φ + p φ β t (cid:17) , m i (t) = β i Exp [ − i ∆t] , (50)where η = e u , (cid:80) p i = 0 and Π i =1 β i = 1. C. Case α < and λ > √ In this instance its appropriate to take the relation between the momentaΠ φ = − λ π u + p φ , and Π φ = − p φ = constant , (51)then the constant − α = 24 λ p φ , allowing us to obtain the temporal dependence for π u (t) with which a masterequation can be constructed d π u − α π + α π u − α = dt , (52)where we have included the minus sign such the constant α = 2( λ −
3) = 2 β >
0. Then, defining ω = α − βα =576 ω with ω = 3p φ − ( λ − , we can rewrite (52) as8 β dπ u ω − (4 βπ u − λ p φ ) = dt , (53)where the constraint over the parameters p φ > c (cid:114)(cid:16) λ √ (cid:17) − βπ u − λ p φ , thus, the solution for the momenta π u (t) becomes π u = 6 λ p φ β + 6 ω β Tanh (12 ω (t − t )) . (54)Using the relations from Eq.(34) and after some algebra, the solutions for the set of variables (u , φ , φ ) and (Π φ , Π φ )are: u = u + 12 λ p φ β (t − t ) + 1 β Ln [Cosh (12 ω (t − t ))] , (55)u i = − i ∆t , (56) φ = φ + 72 p φ β (t − t ) + 2 λ β Ln [Cosh (12 ω (t − t ))] , (57) φ = φ + 24p φ (t − t ) , (58)Π φ = − φ β − λ ω β Tanh (12 ω (t − t )) , (59)Π φ = − p φ , (60)where (u , φ , φ , p i ) are all integration constants. Finally the scale factor becomes η = η Exp (cid:20) λ p φ β (t − t ) (cid:21) Cosh β (12 ω (t − t )) , (61)with η = e u and the anisotropic dimensionless function ism i (t) = β i Exp [ − i ∆t] , (62)we can see that Eq.(62) has the same functional form as before (Eq.(50)). D. Case α = 0 and λ = 3 . For this case the coefficient α = 0 and the master equation to solve is reduced to (cid:90) d π u α π u − α = (cid:90) dt , (63)thus the solution for π u (t) can be obtained relatively easily, which read π u (t) = α α + pe α (t − t ) , (64)where p is an integration constant. As before, we can use relations from Eq.(34) and after some manipulation, thesolutions for (u , φ , φ ) and (Π φ , Π φ ) are:u = u + 2 √ φ + c p φ (t − t ) + √ φ e √ φ (t − t ) , (65)u i = − i ∆t , (66) φ = φ + 12 c − p φ +p φ (t − t ) + p6 e √ φ (t − t ) , (67) φ = φ + 24p φ (t − t ) , (68)Π φ = 12 p φ − c +p φ − √ √ φ (t − t ) , (69)Π φ = − p φ , (70)again (u , φ , φ , p i ) are all integration constants. Finally the scale factor η (t) for this case is η = η Exp (cid:34) √ φ + c p φ (t − t ) (cid:35) Exp (cid:34) √ φ e √ φ (t − t ) (cid:35) , (71)where η = e u , and as before, the anisotropic dimensionless function m i (t) is the same as in (50). E. Anisotropic Parameters
In anisotropic cosmology, the Hubble parameter H is defined in analogy with the FRW cosmology, that isH = ˙aa = ˙ ηη = 13 (H x + H y + H z ) , (72)where H x = ˙a a , H y = ˙a a , and H z = ˙a a .The scalar expansion θ , the shear scalar σ and the average anisotropic parameter A m are defined as θ = (cid:88) i=1 ˙a i a i = 3H , σ = 12 (cid:32) (cid:88) i=1 H − θ (cid:33) , A m = 13 (cid:88) i=1 (cid:18) H i − HH (cid:19) , (73)respectively.Following [74], we consider the volume deceleration parameter,q(t) = − v¨v˙v , (74)where v = η = a a a is the (isotropic) volume function of the Bianchi type I model, which is in this case given bythe exact solution presented for each case in the λ parameter, givingq(t) = − − βω ( λ p φ Sinh(12 ω ∆t) − ω Cosh(12 ω ∆t) ) , for λ < √ − − βω ( λ p φ Cosh(12 ω ∆t)+ ω Sinh(12 ω ∆t) ) , for λ > √ − − √ pp φ e √ pφ t (cid:16) pp φ e √ pφ t + √ (cid:104) p φ + c (cid:105)(cid:17) , for λ = √ q , q and q stand for the solutions0 Out[ ! ]= q q q Figure 1:
Deceleration parameter for the three classical solutions. Here we have taken λ = 0 . λ (cid:48) = 2, p φ = 0 .
4, p φ = 0 . p = p = 0 .
01 and p=0.7. λ < √ λ > √ λ = √
3, respectively. Using the results for the average scale factor η and the dimensionlessanisotropic functions m i , the average anisotropic parameter isA m = β ( (cid:96) + (cid:96) (cid:96) + (cid:96) )Cosh (12 ω ∆t) ( ω Cosh(12 ω ∆t) − λ p φ Sinh(12 ω ∆t) ) , for λ < √ β ( (cid:96) + (cid:96) (cid:96) + (cid:96) ) Cosh (12 ω ∆t) ( ω Sinh(12 ω ∆t)+ λ p φ Cosh(12 ω ∆t) ) , for λ > √ p φ ( (cid:96) + (cid:96) (cid:96) + (cid:96) ) (cid:16) pp φ e √ pφ t + √ (cid:104) p φ + c (cid:105)(cid:17) . for λ = √ θ = β ( λ p φ − ω Ctgh(12 ω ∆t)), for λ < √ β [ λ p φ + ω Tanh(12 ω ∆t)], for λ > √ (cid:18) p e √ φ ∆t + √ p φ +c p φ (cid:19) , for λ = √ σ = ( (cid:96) + (cid:96) (cid:96) + (cid:96) ) Sinh (12 ω ∆t) − ( ω Cosh(12 ω ∆t) − λ p φ Sinh(12 ω ∆t) ) β Sinh (12 ω ∆t) , for λ < √ ( (cid:96) + (cid:96) (cid:96) + (cid:96) ) Cosh (12 ω ∆t) − ( ω Sinh(12 ω ∆t)+ λ p φ Cosh(12 ω ∆t) ) β Cosh (12 ω ∆t) , for λ > √ (cid:40) (cid:0) (cid:96) + (cid:96) (cid:96) + (cid:96) (cid:1) − (cid:20) p p φ e √ φ + √ ( p φ +c ) p φ (cid:21) (cid:41) , for λ = √ σ/θ ≤ . σ to Hubble constant H in the neighborhood of our Galaxy today in order to have a sufficiently isotropic1cosmological model, in this regard we obtain σ θ = − + β ( (cid:96) + (cid:96) (cid:96) + (cid:96) ) Sinh (12 ω ∆t) ( ω Cosh(12 ω ∆t) − λ p φ Sinh(12 ω ∆t) ) , for λ < √ − + β ( (cid:96) + (cid:96) (cid:96) + (cid:96) ) Cosh (12 ω ∆)t ( λ p φ Cosh(12 ω ∆t)+ ω Sinh(12 ω ∆t) ) , for λ > √ − + φ ( (cid:96) + (cid:96) (cid:96) + (cid:96) ) (cid:104) p p φ e √ φ + √ (cid:16) p φ +c (cid:17)(cid:105) , for λ = √ m to the following value for both λ > √ λ = √
3: A m ≤ .
54, signaling that the anisotropic phase still continues.
III. SECOND MODEL: CHIRAL ANISOTROPIC MODEL
Now we turn our attention to the second model to be considered. In this case in addition to the two scalar fieldswe also consider the two potential terms as well as the standard kinetic energy and the mixed term. The action forsuch a Universe is L = √− g (cid:18) R −
12 g µν m ab ∇ µ φ a ∇ ν φ b + V( φ , φ ) (cid:19) , (80)where R is the Ricci scalar, V( φ , φ ) is the corresponding scalar field potential, and m ab is a 2 × = m . The EKG equations are obtained varying Eq.(80) with respect to the metric and the scalar fields,resulting in G αβ = −
12 m ab (cid:18) ∇ α φ a ∇ β φ b −
12 g αβ g µν ∇ µ φ a ∇ ν φ b (cid:19) + 12 g αβ V( φ , φ ) , (81)m ab (cid:3) φ b − ∂ V ∂φ a = m ab g µν φ b ,µν − m ab g αβ Γ ναβ ∇ ν φ b − ∂ V ∂φ a = 0 , a , b = 1 , . (82)Consequently the Klein-Gordon equations arem φ (cid:48)(cid:48) φ (cid:48) + m φ (cid:48)(cid:48) φ (cid:48) + 3 η (cid:48) η (cid:16) m φ (cid:48) + m φ (cid:48) φ (cid:48) (cid:17) + (cid:16) ˙V (cid:17) φ = 0 , (83)m φ (cid:48)(cid:48) φ (cid:48) + m φ (cid:48)(cid:48) φ (cid:48) + 3 η (cid:48) η (cid:16) m φ (cid:48) + m φ (cid:48) φ (cid:48) (cid:17) + (cid:16) ˙V (cid:17) φ = 0 , (84)here (cid:48) = d/dτ , d τ = Ndt and (cid:16) ˙V (cid:17) φ i (with i=1,2) means that the derivative is calculated maintaining φ i constant. Anequivalent form to write equations (83) and (84) ism ˙ φ ddt Ln (cid:18) N η ˙ φ (cid:19) + m ˙ φ ddt Ln (cid:18) N η ˙ φ (cid:19) = N (cid:16) ˙V (cid:17) φ ˙ φ , (85)m ˙ φ ddt Ln (cid:18) N η ˙ φ (cid:19) + m ˙ φ ddt Ln (cid:18) N η ˙ φ (cid:19) = N (cid:16) ˙V (cid:17) φ ˙ φ . (86)Taking the metric (5) and pluging it into (80), the Lagrangian density becomes L = η (cid:32) (cid:18) ˙ ηη (cid:19) − (cid:34)(cid:18) ˙m m (cid:19) + (cid:18) ˙m m (cid:19) + (cid:18) ˙m m (cid:19) (cid:35) − m ˙ φ − m ˙ φ − m ˙ φ ˙ φ N + N (cid:2) V e − λ φ + V e − λ φ (cid:3)(cid:33) , (87)2the momenta are Π η = 12 η N ˙ η, Π φ = − η N (cid:16) m ˙ φ + m ˙ φ (cid:17) , Π φ = − η N (cid:16) m ˙ φ + m ˙ φ (cid:17) , Π = − η N (cid:18) ˙m m (cid:19) , Π = − η N (cid:18) ˙m m (cid:19) , Π = − η N (cid:18) ˙m m (cid:19) , ˙ η = N η Π η , ˙ φ = N η (cid:52) (cid:0) − m Π φ + m Π φ (cid:1) , ˙ φ = Nη ∆ (cid:0) m Π φ − m Π φ (cid:1) , ˙ m = − N m Π η , ˙ m = − N m Π η , ˙ m = − N m Π η , (88)where (cid:52) = m m − (m ) . Writing (87) in a canonical form, i.e. L can = Π q ˙ q − N H , we can perform the variationof this canonical Lagrangian with respect to the lapse function N , δ L can /δN = 0, resulting in the constraint H = 0,and making the same transformation as in (33), the Hamiltonian density is H = e − (cid:20) Π − − − − (cid:52) Π φ − (cid:52) Π φ + 24m (cid:52) Π φ Π φ − e − λ φ +6u − e − λ φ +6u (cid:21) . (89) Proposing the following canonical transformation on the variables ( η, φ , φ , u i ) ↔ ( ξ , ξ , ξ , u i ) ξ = − u + λ φ ,ξ = − u + λ φ ,ξ = − u + λ φ + λ φ , u i = u i , ←→ u = ξ + ξ − ξ ,φ = 3 ξ + ξ − ξ λ ,φ = ξ + 3 ξ − ξ λ , (90)and setting the gauge N = 24e , allows us to find a new set of conjugate momenta ( π , π , π )Π u = − π − π − π , Π φ = λ π + λ π , (91)Π φ = λ π + λ π , which finally leads us to the Hamiltonian density H = 12 (cid:18) − λ m (cid:52) (cid:19) π + 12 (cid:18) − λ m (cid:52) (cid:19) π + (cid:18)
16 + − λ m + 2 λ λ m − λ m (cid:52) (cid:19) π + 12 (cid:20)(cid:18) λ λ m − λ m (cid:52) (cid:19) π + (cid:18) λ λ m − λ m (cid:52) (cid:19) π (cid:21) π − − − + 24 (cid:18) λ λ m (cid:52) (cid:19) π π − (cid:0) V e − ξ + V e − ξ (cid:1) , (92)the parameter (cid:52) is the same that was defined after equations (88). The form that the Hamiltonian density (92)acquires after applying the transformation (90) into (89) will, in the end, allows us to obtain the solutions for this3model. First, let’s compute Hamilton’s equations, which read˙ ξ = 24 (cid:18) − λ m (cid:52) (cid:19) π + 24 (cid:18) λ λ m (cid:52) (cid:19) π + 12 (cid:18) λ λ m − λ m (cid:52) (cid:19) π , ˙ ξ = 24 (cid:18) − λ m (cid:52) (cid:19) π + 24 (cid:18) λ λ m (cid:52) (cid:19) π + 12 (cid:18) λ λ m − λ m (cid:52) (cid:19) π , ˙ ξ = 12 (cid:20)(cid:18) λ λ m − λ m (cid:52) (cid:19) π + (cid:18) λ λ m − λ m (cid:52) (cid:19) π (cid:21) + 2 (cid:18)
16 + − λ m + 2 λ λ m − λ m (cid:52) (cid:19) π , ˙ π = − e − ξ , (93)˙ π = − e − ξ , ˙ π = 0 , ˙P i = 0 , ˙u i = − P i , from this last set of equations is straightforward to see that π = p and P i = n i are constants and the solutions tou i = u i − i ∆t. Taking the time derivative of the first equation in (93), we obtain¨ ξ = − (cid:18) − λ m (cid:52) (cid:19) e − ξ − (cid:18) λ λ m (cid:52) (cid:19) e − ξ . (94)The main purpose of introducing the transformation (90) was to be able to separate the set of equations arising fromthe Hamiltonian density (92). To reach to a solution for our problem we set to zero the coefficient that is multiplyingthe mixed momenta term in (92), which sets the following constraint on the matrix element m m = λ λ (cid:32) ± (cid:115) m λ λ (cid:33) , (95)which implies that the second term in the square root of (95) is a real number, say (cid:96) = 36(m m /λ λ ) ∈ R + , givingthe same weight to the matrix elements m and m , whose values are m = √ (cid:96)λ and m = √ (cid:96)λ . We are goingto distinguish two possible scenarios for m as: m = λ λ (cid:0) √ (cid:96) (cid:1) > − = − λ λ (cid:0) √ (cid:96) − (cid:1) < enables us to have a quintom like case and quintessence like case, respectively. With thesetwo possible values for the matrix element m we can see that (cid:52) + = − λ λ (cid:0) √ (cid:96) (cid:1) < and (cid:52) − = λ λ (cid:0) √ (cid:96) − (cid:1) > − . A. Quintom like case
We begin by analyzing the quintom like case, for which the matrix element m − = − ( √ (cid:96) − λ λ , theHamiltonian density is rewritten as, H = − π µ (cid:96) − π µ (cid:96) + (cid:18) − (cid:96) (cid:19) ( π + π ) π + (cid:18) − (cid:96) (cid:19) π − (cid:0) P + P + P (cid:1) − e − ξ − e − ξ , (96)also we have define the parameters µ (cid:96) = √ (cid:96)/ (cid:16) √ (cid:96) − √ (cid:96) (cid:17) and c (cid:96) = √ (cid:96)/ (cid:104)(cid:0) √ (cid:96) (cid:1) + √ (cid:96) (cid:105) . Thus,Hamilton equations for the new simplified coordinate ξ i are˙ ξ = − π µ (cid:96) + (cid:18) − (cid:96) (cid:19) π , ˙ ξ = − π µ (cid:96) + (cid:18) − (cid:96) (cid:19) π , (97)˙ ξ = + (cid:18) − (cid:96) (cid:19) ( π + π ) + 2 (cid:18) − (cid:96) (cid:19) π , π i remain the same as in (93). Taking the derivative of the first equation of (97) yields¨ ξ = 48V µ (cid:96) e − ξ , (98)which has a solution of the form e − ξ = µ (cid:96) r Sech (r t − q ) . (99)From (97) we can see that ˙ ξ has the same functional structure as ˙ ξ , therefore its solution will be of the same formas (99), so we have e − ξ = µ (cid:96) r Sech (r t − q ) , (100)where r i and q i (with i = 1 ,
2) are integration constants, both at (99) and (100). Reinserting these solutions intoHamilton equations for the momenta, we obtain π = α − µ (cid:96) r Tanh (r t − q ) , (101) π = α − µ (cid:96) r Tanh (r t − q ) . (102)With (101) and (102), it can be easily check that the Hamiltonian is identically null when α = α = 72 µ (cid:96) −
16 p , p = µ (cid:96) (r + r ) + 6n µ (cid:96) + 1) , n = n + n + n , (103)where n belongs to the contribution on the anisotropic functions. Now we are in position write the solutions for the ξ i coordinates, which read ξ = β + Ln (cid:2) Cosh (r t − q ) (cid:3) , (104) ξ = β + Ln (cid:2) Cosh (r t − q ) (cid:3) , (105) ξ = β + p (cid:20)
16 (1 + 72 µ (cid:96) ) − µ (cid:96) c (cid:96) (cid:21) ∆t − (cid:18) − (cid:96) (cid:19) µ (cid:96) Ln [Cosh (r t − q ) Cosh (r t − q )] , (106)here the β i , with i = 1 , ,
3, terms are constants coming from integration. Applying the inverse canonical transforma-tion we obtain the solutions in the original variables ( η, φ , φ ) as η = η + 112 Ln (cid:2) Cosh (r t − q ) Cosh (r t − q ) (cid:3) −
12 p (cid:20)
16 (1 + 72 µ (cid:96) ) − µ (cid:96) c (cid:96) (cid:21) ∆t+ 12 µ (cid:96) (cid:18) − (cid:96) (cid:19) Ln [Cosh (r t − q ) Cosh (r t − q )] ,φ = φ + 12 λ Ln (cid:2) Cosh (r t − q ) Cosh (r t − q ) (cid:3) − λ p (cid:20)
16 (1 + 72 µ (cid:96) ) − µ (cid:96) c (cid:96) (cid:21) ∆t+ 3 λ µ (cid:96) (cid:18) − (cid:96) (cid:19) Ln [Cosh (r t − q ) Cosh (r t − q )] ,φ = φ + 12 λ Ln (cid:2) Cosh (r t − q ) Cosh (r t − q ) (cid:3) − λ p (cid:20)
16 (1 + 72 µ (cid:96) ) − µ (cid:96) c (cid:96) (cid:21) ∆t+ 3 λ µ (cid:96) (cid:18) − c (cid:96) (cid:19) Ln [Cosh (r t − q ) Cosh (r t − q )] , (107)where η , φ and φ are given in terms of the β i constants as η = β + β − β , φ = 3 β + β − β λ , φ = β + 3 β − β λ . (108)5 B. Quintessence like case
Now we turn our attention to the quintessence like case, for which the matrix element m = (cid:0) √ (cid:96) (cid:1) λ λ ,then the Hamiltonian density describing this quintessence model is rewritten as H = π ν (cid:96) + π ν (cid:96) + (cid:18) − (cid:96) (cid:19) ( π + π ) π + (cid:18) − (cid:96) (cid:19) π − (cid:0) P + P + P (cid:1) − e − ξ − e − ξ , (109)here we define the parameter ν (cid:96) = √ (cid:96)/ (cid:16) √ (cid:96) + √ (cid:96) − (cid:17) and c (cid:96) = √ (cid:96)/ (cid:16) + √ (cid:96) + 1 − √ (cid:96) (cid:17) .From (109) we can calculate Hamilton equations for the phase space spanned by ( ξ i , π i ), given by˙ ξ = 2 π ν (cid:96) + (cid:18) − (cid:96) (cid:19) π , ˙ ξ = 2 π ν (cid:96) + (cid:18) − (cid:96) (cid:19) π , (110)˙ ξ = (cid:18) − (cid:96) (cid:19) ( π + π ) + 2 (cid:18) − (cid:96) (cid:19) π , P i = n i = constant , as in the quintom case ˙ π i remain the same as in (93). Proceeding in a similar way as in the previous case, we takethe derivative of the first equation in (110), obtaining¨ ξ = − ν (cid:96) e − ξ , (111)which the corresponding solution is e − ξ = ν (cid:96) r Csch (r t − q ) . (112)Also in this quintessence like setting, the ˙ ξ functional form is the same as ˙ ξ , indicating that the solution is of thesame type as (112), that is e − ξ = ν (cid:96) r Csch (r t − q ) , (113)in (112) and (113) the r i and q i (with i = 1 ,
2) are constants coming from integration. With (112) and (113) at hand,we can reinsert them into Hamilton equations for the momenta, giving π = − a + ν (cid:96) r Coth (r t − q ) , (114) π = − a + ν (cid:96) r Coth (r t − q ) , (115)where it can be easily verify that with (114) and (115) at hand the Hamiltonian is identically zero whena = a = 72 ν (cid:96) + 16 p , p = ν (cid:96) (r + r ) + 6n ν (cid:96) − , n = n + n + n . (116)So, the solutions for the ξ i coordinates become ξ = β + Ln (cid:2) Sinh (r t − q ) (cid:3) , (117) ξ = β + Ln (cid:2) Sinh (r t − q ) (cid:3) , (118) ξ = β − p (cid:20)
16 (72 ν (cid:96) − − ν (cid:96) c (cid:96) (cid:21) ∆t + (cid:18) − (cid:96) (cid:19) ν (cid:96) Ln [Sinh (r t − q ) Sinh (r t − q )] , (119)6where β i are integration constants. After applying the inverse canonical transformation we get the solutions in termsof the original variables (Ω , φ , φ ) as η = η + 112 Ln (cid:2) Sinh (r t − q ) Sinh (r t − q ) (cid:3) + 12 p (cid:20)
16 (72 ν (cid:96) − − ν (cid:96) c (cid:96) (cid:21) ∆t − (cid:18) − (cid:96) (cid:19) ν (cid:96) Ln [Sinh (r t − q ) Sinh (r t − q )] ,φ = φ + 12 λ (cid:20) Ln (cid:2) Sinh (r t − q ) Sinh (r t − q ) (cid:3) − (cid:18) − (cid:96) (cid:19) ν (cid:96) × Ln [Sinh (r t − q ) Sinh (r t − q )] (cid:21) + 3 λ p (cid:20)
16 (72 ν (cid:96) − − ν (cid:96) c (cid:96) (cid:21) ∆t ,φ = φ + 12 λ (cid:20) Ln (cid:2) Sinh (r t − q ) Sinh (r t − q ) (cid:3) − (cid:18) − (cid:96) (cid:19) ν (cid:96) × Ln [Sinh (r t − q ) Sinh (r t − q )] (cid:21) + 3 λ p (cid:20)
16 (72 ν (cid:96) − − ν (cid:96) c (cid:96) (cid:21) ∆t , (120)where η , φ and φ are given in terms of the β i constants as η = β + β − β , φ = 3 β + β − β λ , φ = β + 3 β − β λ . (121)It is clear that the standard quintessence model with two scalar fields cannot be reproduce under this approach,because when we set m = 0, this imply that parameter (cid:96) is equal to zero, then, the matrix elements m = m arezero too, this was the challenge to resolve. IV. QUANTUM APPROACH
On the Wheeler-DeWitt (WDW) equation there are a lot of papers dealing with different problems, for examplein [76], they asked the question of what a typical wave function for the universe is. In Ref. [77] there appears anexcellent summary of a paper on quantum cosmology where the problem of how the universe emerged from big bangsingularity can no longer be neglected in the GUT epoch. On the other hand, the best candidates for quantumsolutions become those that have a damping behavior with respect to the scale factor, since these allow to obtaingood classical solutions when using the WKB approximation for any scenario in the evolution of our universe [78, 79].In this section we present the quantum version of the classical anisotropic cosmological models studied above alongwith its solutions. Since we already have the classical Hamiltonian density, the quantum counterpart can be obtainedmaking the usual replacement Π q µ = − i (cid:126) ∂ q µ . First we modified the classical Hamiltonian density (33) in order toconsider the factor ordering problem between the function e − and its moment π u , introducing the linear term ase − π → e − (cid:2) π + Qi (cid:126) π u (cid:3) where Q is a real number that measure the ambiguity in the factor ordering. A. Quantum Anisotropic Quintessence-K-essence Model
The quantum version for the first cosmological model we employ the modified Hamiltonian density, H = π + Qi (cid:126) π u − φ − φ − − − − e − λ φ , (122)7in order to obtain the WDW equation, we implement the following change of variables (u , φ , φ , u i ) ↔ ( ξ , ξ , ξ ) ξ = 6u − λ φ ,ξ = u ,ξ = φ , ←→ u = ξ ,φ = − ξ + 6 ξ λ ,φ = ξ , u i = u i , (123)where u i are the conjugate coordinate to momenta P i , and also, obtaining a new set of conjugate momenta (in thesame manner as (31)), of the variables ( ξ , ξ , ξ ), namely ( π , π , π ), which read π u = 6 π + π , Π φ = − λ π , Π φ = π , (124)which in turn transforms the Hamiltonian density (122) as H = 12 (cid:0) − λ (cid:1) π + π + 12 π π − π + i (cid:126) Q(6 π + π ) − − − − e ξ . (125)Introducing the replacement π q µ = − i (cid:126) ∂ q µ , the WDW equation becomes H Ψ = − (cid:126) (cid:0) − λ (cid:1) ∂ Ψ ∂ξ − (cid:126) ∂ Ψ ∂ξ − (cid:126) ∂ Ψ ∂ξ ∂ξ + 12 (cid:126) ∂ Ψ ∂ξ + Q (cid:126) (cid:18) ∂ Ψ ∂ξ + ∂ Ψ ξ (cid:19) +6 (cid:126) (cid:18) ∂ Ψ ∂u + ∂ Ψ ∂u + ∂ Ψ ∂u (cid:19) − V e ξ Ψ = 0 , (126)due that the scalar potential does not depend on the coordinates ( ξ , ξ , u i ), we propose the following ansatz for thewave function Ψ( ξ , ξ , ξ , u i ) = e − (a ξ +a ξ +a u +a u +a u ) / (cid:126) G( ξ ) where the a i are arbitrary constants. Introducingthe mentioned ansatz in (126) we have that − (cid:126) (cid:0) − λ (cid:1)
1G d Gd ξ + 6 (cid:126) (2a + (cid:126) Q) 1G dGd ξ − a (a + (cid:126) Q) + 12a + 6a − e ξ = 0 , identifying a = a + a + a , where we also divided the whole equation by the ansatz; this in turn leads us to thefollowing differential equationd Gd ξ − + (cid:126) Q2 (cid:126) (3 − λ ) dGd ξ + 112 (cid:126) (3 − λ ) (cid:2) e ξ + η (cid:3) G = 0 , (127)here η = a ( a + (cid:126) Q) − − . The last equation can be casted as y (cid:48)(cid:48) + ay (cid:48) + (be κ x + c) y = 0 (and whose solutionswill depend on the value of λ ) [80], wherey = Exp (cid:16) − ax2 (cid:17) Z ν (cid:32) √ b κ e κ x2 (cid:33) , (128)here Z ν is the Bessel function and ν = √ a − c/κ being the order. The corresponding relations between thecoefficients of (127) and a , b , c and κ are a = + (cid:126) Q2 (cid:126) ( λ − , when λ > − + (cid:126) Q2 (cid:126) (3 − λ ) , when λ < − (cid:126) ( λ − , when λ > (cid:126) (3 − λ ) , when λ < − η (cid:126) ( λ − ) , when λ > η (cid:126) ( − λ ) , when λ < κ = 1 , (132)according to the constant b, the solution to the function G becomesG( ξ ) = Exp (cid:18) − + (cid:126) Q4 (cid:126) ( λ − ξ (cid:19) K ν (cid:32) (cid:126) (cid:114) λ − ξ (cid:33) , λ > ξ ) = Exp (cid:18) + (cid:126) Q4 (cid:126) (3 − λ ) ξ (cid:19) J ν (cid:32) (cid:126) (cid:115) − λ e ξ (cid:33) , λ < ν = Exp (cid:18) − + (cid:126) Q4 (cid:126) ( λ − ξ − a ξ + a ξ (cid:126) − a u + a u + a u (cid:126) (cid:19) K ν (cid:32) (cid:126) (cid:115) λ − ξ (cid:33) , λ > ν = Exp (cid:18) + (cid:126) Q4 (cid:126) (3 − λ ) ξ − a ξ + a ξ (cid:126) − a u + a u + a u (cid:126) (cid:19) J ν (cid:32) (cid:126) (cid:114) − λ e ξ (cid:33) , λ < . (136)where ν = (cid:114)(cid:16) − + (cid:126) Q4 (cid:126) ( λ − (cid:17) + η (cid:126) ( λ − and ν = (cid:114)(cid:16) + (cid:126) Q4 (cid:126) (3 − λ ) (cid:17) − η (cid:126) (3 − λ ) are the corresponding orders of thewave function. Applying the inverse transformation on the variables ξ i , we can write the wave function in terms ofthe original variables (A = e Ω , φ i , m i = e u i ), which readΨ ν = m − a4 (cid:126) m − a5 (cid:126) m − a6 (cid:126) A − α Exp (cid:18) + (cid:126) Q4 (cid:126) ( λ − λ φ − a (cid:126) φ (cid:19) K ν (cid:32) (cid:126) (cid:115) λ − e λ φ (cid:33) , λ > ν = m − a4 (cid:126) m − a5 (cid:126) m − a6 (cid:126) A − α Exp (cid:18) − + (cid:126) Q4 (cid:126) ( λ − λ φ − a (cid:126) φ (cid:19) J ν (cid:32) (cid:126) (cid:114) − λ A e λ φ (cid:33) , λ < . (138)with α = (cid:126) (cid:16) a +
32 2 a + (cid:126) Qλ − (cid:17) and α = (cid:126) (cid:16) a −
32 2 a + (cid:126) Q − λ (cid:17) . In Fig(2) we can see the behavior of the probability (cid:1)(cid:2)(cid:3)(cid:4) (cid:1) (cid:5)(cid:6) Figure 2:
Behavior of the probability density for λ < √
3, for Q = 1, λ , a = 0 . a = 1 and a = a = a = 0 . density of the wave function for the solution λ < √
3. It is observed that the evolution of the wave function withrespect of the scale factor is damped, which is a good characteristic and this kind of behavior also have been reported9 (cid:1)(cid:2)(cid:3)(cid:4) (cid:1) (cid:5)(cid:6) (cid:1)(cid:2)(cid:3)(cid:4) (cid:1) (cid:5)(cid:6)
Figure 3:
Behavior of the probability density for λ > √
3. For both figures λ = 6, a = 2, a = 1, a = a = a = 0 . − − in [61, 73, 81]. In comparison with isotropic model [73], we can see that the anisotropies shrink the probability densityof the wave function.In Fig.(3) we can observed the evolution of the wave function for the solution λ > √
3. In this particular casethe values of Q act as a retarder (for negative values) for the wave function and compresses the length over theaxis were the scalar field evolves (this should also retard the inflation epoch), but still having the damped behavior.Contrasting this results with those of the isotropic treatment [73], we can see that the anisotropies shrink the theprobability density of the wave function along the evolution of the scalar field.Finally, for the particular case of λ = √ ξ ) becomesG( ξ ) = G Exp (cid:20) η (cid:126) (2a + (cid:126) Q) ξ (cid:21) Exp (cid:18) (cid:126) (2a + (cid:126) Q) e ξ (cid:19) , and the wave function isΨ(A , φ i , m i ) = Ψ m − a4 (cid:126) m − a5 (cid:126) m − a6 (cid:126) A r Exp (cid:18) − a (cid:126) φ − λ (cid:126) (2a + (cid:126) Q) φ (cid:19) Exp (cid:20) (cid:126) (2a + (cid:126) Q) A e − λ φ ) (cid:21) . (139)where the constant r = − a (cid:126) + η (cid:126) (2a + (cid:126) Q) . B. Quantum Anisotropic Quintom Case
For the second cosmological model, the quintom like case, the quantum version of this model is obtained applying,again, the recipe Π q µ = − i (cid:126) ∂ q µ to the Hamiltonian density (96), hence (cid:20) (cid:126) µ (cid:96) ∂ ∂ξ + (cid:126) µ (cid:96) ∂ ∂ξ − (cid:126) (cid:18) − (cid:96) (cid:19) (cid:18) ∂ ∂ξ ∂ξ + ∂ ∂ξ ∂ξ (cid:19) − (cid:126) (cid:18) − (cid:96) (cid:19) ∂ ∂ξ +6 (cid:126) (cid:18) ∂ Ψ ∂u + ∂ Ψ ∂u + ∂ Ψ ∂u (cid:19) − V e − ξ − V e − ξ (cid:21) Ψ = 0 , (140)because the scalar potential does not depend on the coordinate ξ , we propose the following ansatz for the wavefunction Ψ( ξ , ξ , ξ ) = e (a ξ +a u +a u +a u ) / (cid:126) A ( ξ ) B ( ξ ) where a i (with i = 3 , , ,
6) are an arbitrary constants.Substituting and dividing by the ansatz in (140), we obtain (cid:126) µ (cid:96) A d A d ξ + (cid:126) µ (cid:96) B d B d ξ − a (cid:126) (cid:18) − (cid:96) (cid:19) (cid:18) A d A d ξ + 1 B d B d ξ (cid:19) − a (cid:18) − (cid:96) (cid:19) + 6a − e − ξ − e − ξ = 0 , (141)0with a = a + a + a , where we can separate the equations asd A d ξ − a µ (cid:96) (cid:126) (cid:18) − (cid:96) (cid:19) d A d ξ − µ (cid:96) (cid:126) (cid:18) a (cid:18) − (cid:96) (cid:19) − − α + 24V e − ξ (cid:19) A = 0 , (142)d B d ξ − a µ (cid:96) (cid:126) (cid:18) − (cid:96) (cid:19) d B d ξ − µ (cid:96) (cid:126) (cid:18) a (cid:18) − (cid:96) (cid:19) − + α + 24V e − ξ (cid:19) B = 0 , (143)with α being the separation constant. The corresponding solutions of (142) and (143) have the following form [80]Y(x) = Exp (cid:16) − ax2 (cid:17) Z ν (cid:32) √ b λ e λ x2 (cid:33) , (144)here Z ν are the generic Bessel function with order ν = √ a − /λ . If √ b is real, Z ν are the ordinary Bessel function,otherwise the solution will be given by the modified Bessel function. Making the following identifications λ = − , (145)a = − a µ (cid:96) (cid:126) (cid:18) − c (cid:96) (cid:19) , (146)b , = − µ (cid:96) (cid:126) V , , (147)c ∓ = − µ (cid:96) (cid:126) (cid:18) a (cid:18) − c (cid:96) (cid:19) − a ∓ α (cid:19) , (148) ν ∓ = (cid:115) a µ (cid:96) + 4c ∓ , (149)we can check that the value for √ b is imaginary, which as already mentioned, gives a solution in terms of the modifiedBessel function Z ν = K ν whose order lies in the reals. Thus, the wave function isΨ ν ± = Exp (cid:20)(cid:18) µ (cid:96) (cid:126) (cid:18) − (cid:96) (cid:19) ( ξ + ξ ) + ξ (cid:126) + a u + a u + a u (cid:126) (cid:19) a (cid:21) K ν − (cid:18) (cid:126) (cid:112) µ (cid:96) e − ξ (cid:19) × K ν + (cid:18) (cid:126) (cid:112) V µ (cid:96) e − ξ (cid:19) . (150) C. Quantum Anisotropic Quintessence Case
Lastly, we are going to consider the quantum version of the anisotropic quintessence like case. As in the previoustwo subsections, what we want is to obtain an equation of the form H Ψ( ξ i ) = 0, to achieve this we introduce thestandard prescription Π µq = − i (cid:126) ∂ q µ in (109), obtaining (cid:20) − (cid:126) ν (cid:96) ∂ ∂ξ − (cid:126) ν (cid:96) ∂ ∂ξ − (cid:126) (cid:18)
24 + 13 ν (cid:96) (cid:19) (cid:18) ∂ ∂ξ ∂ξ + ∂ ∂ξ ∂ξ (cid:19) − (cid:126) (cid:18)
12 + 118 ν (cid:96) (cid:19) ∂ ∂ξ +6 (cid:126) (cid:18) ∂ Ψ ∂ u + ∂ Ψ ∂ u + ∂ Ψ ∂ u (cid:19) − e − ξ − e − ξ (cid:21) Ψ = 0 , (151)we can see that the scalar potential does not depend on the coordinates ξ , u i , consequently we propose the followingansatz for the wave function Ψ( ξ , ξ , ξ ) = e (b ξ +b u +b u +b u ) / (cid:126) A ( ξ ) B ( ξ ) where b i (with i = 3 , , ,
6) are anarbitrary constant. Applying and dividing by the ansatz in (151) we get − (cid:126) ν (cid:96) A d A d ξ − (cid:126) ν (cid:96) B d B d ξ − b (cid:126) (cid:18) − (cid:96) (cid:19) (cid:18) A d A d ξ + 1 B d B d ξ (cid:19) − b (cid:18) − (cid:96) (cid:19) +6b − e − ξ − e − ξ = 0 , (152)1with b = b + b + b , separating the equations we have thatd A d ξ + b ν (cid:96) (cid:126) (cid:18) − (cid:96) (cid:19) d A d ξ + ν (cid:96) (cid:126) (cid:18) b (cid:18) − (cid:96) (cid:19) − − α + 24V e − ξ (cid:19) A = 0 , (153)d B d ξ + b ν (cid:96) (cid:126) (cid:18) − (cid:96) (cid:19) d B d ξ + µ (cid:96) (cid:126) (cid:18) b (cid:18) − (cid:96) (cid:19) − + α + 24V e − ξ (cid:19) B = 0 , (154)where α is the separation constant. These last two equations are similar to the quantum quintom like case (142) and(143). Proceeding in a similar fashion as the previous subsection (IV B), we make the following identifications λ = − , (155)a = b ν (cid:96) (cid:126) (cid:18) − c (cid:96) (cid:19) , (156)b , = ν (cid:96) (cid:126) V , , (157)c ∓ = ν (cid:96) (cid:126) (cid:18) b (cid:18) − c (cid:96) (cid:19) − a ∓ α (cid:19) , (158)(159)and conclude that the solutions are given by the ordinary Bessel function J ν with order ν ∓ = (cid:112) (a /ν (cid:96) ) + 4c ∓ . Thus,the wave function becomesΨ ν ± = Exp (cid:20)(cid:18) ν (cid:96) (cid:126) (cid:18) − (cid:96) (cid:19) ( − ξ − ξ ) + ξ (cid:126) (cid:19) b + b u + b u + b u (cid:126) (cid:21) J ν − (cid:18) (cid:126) (cid:112) ν (cid:96) e − ξ (cid:19) × J ν + (cid:18) (cid:126) (cid:112) V ν (cid:96) e − ξ (cid:19) . (160) V. FINAL REMARKS
In this work we have studied the anisotropic Bianchi type model in the chiral cosmology setup in a twofold way. Inthe first model we consider two scalars fields but only one potential term. In the second one, additionally to the twoscalar fields, we also consider both terms in the potential as well as the standard kinetic energy and the mixed term.For both models we did a classical and quantum treatment, obtaining exact analytical solutions for both scenarios.In the first model, which can be thought as a quintessence plus K-essence model, our findings show that the volumeof the Universe grows in a accelerated manner for each of the three exact solutions that were found. This featurecan be seen from Fig.(1), where solutions for λ < √ λ = √ λ > √ σ/θ ≤ . m for two of the solutions, however the anisotropy continue, because for the cases λ > √ λ = √ m ≤ .
54. In the quantum regime we were also able tofind exact solutions. For the particular case of λ < √ λ > √
3, it is found that the damped behavior still exist, but the parameter Q acts as a retarder(for negative values) for the wave function and the length over the axis were the field evolves is compressed as shownin Fig.(3), signaling that the inflation epoch should also be retarded in time. In this case the anisotropies shrink theprobability density along the evolution of the scalar field. Finally, equation (139) depicts the quantum solution forthe case λ = √ K ± ν and the ordinary Bessel function J ± ν , as depicted in Eq.(150) andEq.(160), respectively. Acknowledgments
This work was partially supported by PROMEP grants UGTO-CA-3. J.S. were partially supported SNI-CONACYT.This work is part of the collaboration within the Instituto Avanzado de Cosmolog´ıa and Red PROMEP: Gravitationand Mathematical Physics under project
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