Generalized Scalar Field Cosmologies: a Perturbative Analysis
GGeneralized Scalar Field Cosmologies: a Perturbative Analysis
Genly Leon
Departamento de Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280 Antofagasta, Chile.E-mail: [email protected]
Felipe Orlando Franz Silva
Departamento de Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, Casilla 1280 Antofagasta, Chile.E-mail: [email protected]
Abstract.
Scalar field cosmologies with generalized harmonic potentials in flat or negatively curved Friedmann-Lemaitre Robertson-Walker metrics and in Bianchi I metric are investigated. Asymptotic methods and averaging theoryare used to obtain relevant information about the space of solutions. An interaction with the background matter withstrength Q = λρ m arising from the coupling function χ ( φ ) = χ exp (cid:16) λφ − γ (cid:17) is considered. Using averagingmethods for periodic functions of a given period T , it can be concluded that regardless of whether the scalar field isminimally or not minimally coupled to the matter field (at least for interactions of the type Q = λρ m ) and irrespectivethe geometry there is not difference in dynamics when performing the averaging process. The stability results arequalitatively the same as for the averaged systems, indicating that the asymptotic behavior when H → of the averagedmodel is independent of the coupling function, and is independent of the geometry.PACS numbers: 98.80.-k, 98.80.Jk, 95.36.+x a r X i v : . [ g r- q c ] N ov eneralized Scalar Field Cosmologies: a Perturbative Analysis
1. Introduction
This research is focused on perturbation problems consisting on the study of the phase portrait of a differentialsystem: ˙ x = X ( x ; ε ) , x ∈ R k , ε ∼ (1)near the zero of X ( x ; 0) [106, 107, 108, 109, 110, 111, 112]. In general, perturbation problems are expressed inFenichel’s normal form. That is, given ( x, y ) ∈ R n + m , and f, g smooth functions, equations that can be writtenas: ˙ x = f ( x, y ; ε ) , ˙ y = εg ( x, y ; ε ) , x = x ( t ) , y = y ( t ) . (2)The system (2) is called “fast system” as a difference with εx (cid:48) = f ( x, y ; ε ) , y (cid:48) = g ( x, y ; ε ) , x = x ( τ ) , y = y ( τ ) , (3)obtained after the re scaling τ = εt , that is called the “slow system”. Notice that for ε > , the phase portraitsof (2) and (3) coincide. it follows two problems that manifestly depend on two scales. The problem in termsof the “slow time” variable, the solution of which can be treated analogously to the outer solution in a boundarylayer problem. The fast system: a change of scale of the system that describes the rapid evolution that occurs inshorter times, analogous to the inner solution of a boundary layer problem. The solution of each subsystem willbe sought in the form of a regular perturbation expansion. For singularly perturbed problems, the subsystems willhave simpler structures than the full problem; allowing slow and fast dynamics to potentially be characterized interms of reduced phase line or phase plane dynamics. For ε > , let S denotes the singular points of (2). Defining S , the critical set of the singular perturbation problem, equations (3) define a dynamical system on S called thereduced problem. The implicit equation f ( x, y ; 0) = 0 is called the slow manifold or “slow solution curve”. Veryoften, the solution is pushed out of the slow manifold, at which point the solution is no longer described by thedynamics of the slow system; all out the slow manifold in the phase plane is part of the fast problem.Combining results of these two limiting problems some information of the dynamics for small values of ε is obtained. This technique is used to construct uniformly valid approximations to the solutions of perturbationproblems in which the solutions depend simultaneously on quite different scales by using seed solutions whichsatisfies some version of the original equations in the limit of ε → . One approach used to construct thatasymptotic expansions is to introduce the two time scales t = t and t = εt . For this reason, the method issometimes called two-timing, and t is said to be the fast time scale and t the slow scale. The list of possiblescales includes the following possibilities [110]:(i) Several time scales like t = t/ε, t = t, t = εt, t = ε t . . . may be needed.(ii) More complex dependence on ε , for example, t = (cid:0) ω ε + ω ε + . . . (cid:1) t and t = εt where the ω n aredetermined while solving the problem (Poincarè–Lindstedt’s method).(iii) The correct scaling may not be immediately apparent, and one starts off with something like t = ε α t and t = ε β t , where α < β .(iv) Nonlinear time dependence: for example, one may have to assume t = f ( t, ε ) and t = εt , where thefunction f ( t, ε ) is determined from the problem.Like the Poincarè–Lindstedt’s method, the method of multiple time-scales determines solutions to perturbedoscillators by suppressing resonant forcing terms that would yield spurious secular terms in the asymptoticexpansions. The method of multiple time-scales makes a less restrictive assumption on the form of the solutionthan employed by the Poincarè–Lindstedt’s method; it assumes that the solution can be expressed as a function ofmultiple (for our purposes, just two) time variables, which are introduced to keep the expansion well ordered, x ( t ) = X ( t, τ ) , (4)where t is the regular (or “fast”) time variable and τ is a new variable describing the “slow-time” dependence ofthe solution. The idea is to use whatever freedom there is in the dependence on τ to minimize the error in theapproximation and, whenever is possible, to remove unbounded or secular terms. Some examples to illustrate the use of perturbation methods are the following: eneralized Scalar Field Cosmologies: a Perturbative Analysis Considering the initial value problem with t > : d ydt = − εy ) , y (0) = 0 , y (cid:48) (0) = 0 . (5)Assuming the solution has an asymptotic expansion of the form: y ( t ) ∼ y ( t ) + ε α y ( t ) + . . . , α > , (6)and considering a very small z , (1 + z ) − ∼ − z , the original problem becomes: y (cid:48)(cid:48) ( t ) + ε α y (cid:48)(cid:48) ( t ) + . . . = − ε ( y ( t ) + . . . )] ∼ − εy ( t ) + . . . , (7)with initial conditions: y (0) + ε α y (0) + . . . = 0 , and y (cid:48) (0) + ε α y (cid:48) (0) + . . . = 1 .Taking α = 1 , the following systems are obtained:To order O (1) : y (cid:48)(cid:48) ( t ) = − , y (cid:48) (0) = 0 , y (0) = 1 has solution y ( t ) = − t + t .To order O ( ε ) : y (cid:48)(cid:48) ( t ) = 2 y ( t ) , y (cid:48) (0) = 0 , y (0) = 1 has solution y ( t ) = t − t . Finally, thesolution is given by y ( t ) ∼ t (cid:0) − t (cid:1) + εt (cid:0) − t (cid:1) . Considering the classical example [110] given by the ordinary differential equation: y (cid:48)(cid:48) + εy (cid:48) + y = 0 , t > , y (0) = 0 , y (cid:48) (0) = 1 . (8)This equation admits an exact solution: y ( t ) = 2 e − tε sin (cid:0) t √ − ε (cid:1) √ − ε . (9)This solution has an oscillatory component running on the scale of order O (1) , as well as a slow variation of order O ( ε − ) . Two time scales t, τ = εt are introduced and treated as independent variables. Using the chain rule: dfdt = ∂f∂t + ε ∂f∂τ , d fdt = ∂ f∂t + 2 ε ∂ f∂t∂τ + ε ∂ f∂τ (10)the initial value problem of a scalar differential equation: y tt + 2 εy t y τ + ε y ττ + ε ( y t + εy tτ ) + y = 0 , (11) y = 0 , y t + εy τ = 1 for t = 0 = τ, (12)is acquired, where the subscripts y t , y τ , . . . , denote the partial derivatives. Now, using series expansion of theform y ∼ y ( t, τ ) + εy ( t, τ ) + . . . (13)the following equation: y tt + y + ε ( y tt + y + 2 y tτ + y t ) + O ( ε ) + . . . (14)is obtained. Now, collecting the terms of ε order the following equation are obtained: O (1) : y tt + y = 0 (15)with general solution y ( t, τ ) = A ( τ ) sin( t ) + B ( τ ) cos( t ) (16) O ( ε ) : y tt + y = − (2 y tτ + y t ) (17) = sin( t ) (2 B (cid:48) ( τ ) + B ( τ )) − cos( t ) (2 A (cid:48) ( τ ) + A ( τ )) . (18)Then, the secular terms are (2 B (cid:48) ( τ ) + B ( τ )) = 0 , (2 A (cid:48) ( τ ) + A ( τ )) = 0 . (19)After imposing the initial conditions it follows B ( τ ) = 0 and A ( τ ) = e − τ and the solution y ∼ e − τ sin( t ) , (20)valid up to the first order of ε is obtained, which gives a good approximation for the problem. Indeed, the previousapproximation holds up to εt = O (1) , that is, it holds for ≤ εt ≤ T , where T is fixed. eneralized Scalar Field Cosmologies: a Perturbative Analysis The so-called induced gravity model has the action [113, 114]: S IG = (cid:90) √− g (cid:18) σ ω R − g µν ∂ µ σ∂ ν σ − γ U σ − γ (cid:19) , (21) ω > , γ ≥ . A massless scalar field is added to the action in [115]: S IGφ = S IG + (cid:90) √− g (cid:18) − g µν ∂ µ φ∂ ν φ (cid:19) . (22)The equation of motion for a massless scalar field is given by ¨ φ + 3 ˙ aa ˙ φ = 0 , (23)and admits the solution ˙ φ = εa − , where ε is an integration constant. Using the parametrization [113]: a = σ − exp( u + v ) , (24a) σ = exp( A ( u − v )) , (24b)with A = (cid:113) γ, the Raychaudhuri equation and the equation of motion for σ lead to: (cid:0) √ γ + 2 (cid:1) (cid:0) γ − (cid:1) ε exp (cid:0) √ γ ( u − v ) − u − v (cid:1) γ + 12 (cid:16) √ γ − (cid:17) ˙ u + (cid:16) √ γ + 6 (cid:17) U − u = 0 , (25a) − (cid:0) √ γ − (cid:1) (cid:0) γ − (cid:1) ε exp (cid:0) √ γ ( u − v ) − u − v (cid:1) γ − (cid:16) √ γ + 6 (cid:17) ˙ v + (cid:16) − √ γ (cid:17) U − v = 0 . (25b)where the Friedmann equation ˙ u ˙ v = 112 (cid:32) (cid:0) − γ (cid:1) ε exp (cid:0) √ γ ( u − v ) − u − v (cid:1) γ + U (cid:33) (26)is used to eliminate the mixed terms ∝ ˙ u ˙ v .Using series expansion of the form: u ∼ u ( t, τ ) + εu ( t, τ ) + O ( ε ) + . . . , v ∼ v ( t, τ ) + εv ( t, τ ) + O ( ε ) + . . . , (27)where the time variables t, τ = εt , are introduced and treated as independent variables. Collecting the terms oforder ε (see reference [115]) are found: O (1) : (cid:0) √ γ − (cid:1) u t ( t, τ ) − u tt ( t, τ ) + (cid:0) √ γ + 6 (cid:1) U = 0 , − (cid:0) √ γ + 6 (cid:1) v t ( t, τ ) − v tt ( t, τ ) + (cid:0) − √ γ (cid:1) U = 0 u t ( t, τ ) v t ( t, τ ) − U = 0 (see, e.g., similar equations (28) in [113], (2.24) in [114]) , (28) O ( ε ) : − u tτ ( t, τ ) + (cid:0) √ γ − (cid:1) u t ( t, τ ) ( u τ ( t, τ ) + u t ( t, τ )) − u tt ( t, τ ) = 0 , (2 v tτ ( t, τ ) + (cid:0) √ γ + 6 (cid:1) v t ( t, τ ) ( v τ ( t, τ ) + v t ( t, τ )) + v tt ( t, τ ) = 0 ,v t ( t, τ ) ( u τ ( t, τ ) + u t ( t, τ )) + u t ( t, τ ) ( v τ ( t, τ ) + v t ( t, τ )) = 0 . (29) eneralized Scalar Field Cosmologies: a Perturbative Analysis O (1) , the following is obtained: u ( t, τ ) = (cid:40) c ( τ ) − √ γ − , γ < c ( τ ) − √ γ − , γ ≥ , (30a)and v ( t, τ ) = (cid:40) c ( τ ) + √ γ +6 , γ < c ( τ ) + √ γ +6 , γ ≥ , (30b)where c ( τ ) , c ( τ ) and c ( τ ) are integration functions, and ∆ := ∆( t, τ ) = (cid:112) | γ − |√ U (24 c ( τ ) + t )2 √ . (31)Substituting (30) into the equations at order O ( ε ) , the following is obtained: u tt = − √ U (cid:16) √ U ( γ − ) c (cid:48) ( τ )+ ( √ γ − √ ) √ − γ tanh(∆) ( c (cid:48) ( τ )+ u t ) (cid:17) √ γ − , γ < √ U (cid:16) ( √ γ − √ ) √ γ − ( c (cid:48) ( τ )+ u t ) − √ U ( γ − ) c (cid:48) ( τ ) (cid:17) √ γ − , γ ≥ , (32) v tt = − √ U csch (∆) (cid:16) ( √ γ − ) √ − γ cosh (∆) ( u t + c (cid:48) ( τ ) ) +12 √ ( γ − ) √ U (5 sinh(∆)+sinh(3∆)) c (cid:48) ( τ ) (cid:17) ( √ γ +3 √ ) , γ < √ U csc (∆) (cid:16) ( √ γ − ) √ γ − (∆) ( u t + c (cid:48) ( τ ) ) +12 √ ( γ − ) √ U (5 sin(∆)+sin(3∆)) c (cid:48) ( τ ) (cid:17) ( √ γ +3 √ ) , γ ≥ , (33) v t = ( √ γ − ) c (cid:48) ( τ ) coth (∆)+ √ γu t coth (∆) − u t coth (∆) − √ √ U √ − γ c (cid:48) ( τ ) coth(∆) −√ γc (cid:48) ( τ ) − c (cid:48) ( τ ) √ γ +6 , γ < − ( √ γ − ) c (cid:48) ( τ ) cot (∆)+ √ γu t cot (∆) − u t cot (∆)+24 √ √ U √ γ − c (cid:48) ( τ ) cot(∆)+ √ γc (cid:48) ( τ )+6 c (cid:48) ( τ ) √ γ +6 , γ ≥ . (34) Integrating (32), is obtained: u = c ( τ )+ ( − t − c ( τ )) c (cid:48) ( τ ) + (( √ γ − √ ) c ( τ ) − √ U ( γ − ) ( t +24 c ( τ )) c (cid:48) ( τ )+ ( √ γ − √ ) c (cid:48) ( τ ) ) √ U ( √ γ − ) √ − γ , γ < − t − c ( τ )) c (cid:48) ( τ ) + (( √ γ − √ ) c ( τ ) − √ U ( γ − ) ( t +24 c ( τ )) c (cid:48) ( τ )+ ( √ γ − √ ) c (cid:48) ( τ ) ) √ U ( √ γ − ) √ γ − , γ ≥ . (35) To avoid the two secular terms ∝ t , conditions c (cid:48) ( τ ) = c (cid:48) ( τ ) = 0 are imposed, i.e., c and c are constants.Hence, u = c ( τ ) + √ c tanh(∆) √ U (6 − γ ) , γ < √ c tan(∆) √ U ( γ − , γ ≥ , (36)where ∆ := ∆( t ) = (24 c + t ) √ U √ | γ − | √ . Then, v tt = ( −√ γ ) √ U (6 − γ ) c coth(∆) csch (∆) √ ( √ γ +6 ) , γ < √ γ − ) √ U ( γ − c cot(∆) csc (∆) √ ( √ γ +6 ) , γ ≥ , (37) v t = ( √ γ − ) c csch (∆) √ γ +6 − c (cid:48) ( τ ) , γ < − ( √ γ − ) c csc (∆) √ γ +6 − c (cid:48) ( τ ) , γ ≥ . (38)Solving the second equation the following is obtained: v = − ( √ γ − √ ) c coth(∆) ( √ γ +6 ) √ U (6 − γ ) − tc (cid:48) ( τ ) + c ( τ ) , γ < ( √ γ − √ ) c cot(∆) ( √ γ +6 ) √ U ( γ − − tc (cid:48) ( τ ) + c ( τ ) , γ ≥ , (39) eneralized Scalar Field Cosmologies: a Perturbative Analysis v are identically satisfied. To avoid the secular terms ∝ t , the condition c (cid:48) ( τ ) = 0 is imposed, i.e., c is a constant. It is settled c = c = 0 for simplicity. Therefore, it follows: u ( t ; ε ) = c − (cid:40) √ γ − , γ < √ γ − , γ ≥ . + ε √ c tanh(∆) √ U (6 − γ ) , γ < √ c tan(∆) √ U ( γ − , γ ≥ O ( ε ) , (40a) v ( t ; ε ) = c + (cid:40) √ γ +6 , γ < √ γ +6 , γ ≥ ε − ( √ γ − √ ) c coth(∆) ( √ γ +6 ) √ U (6 − γ ) , γ < ( √ γ − √ ) c cot(∆) ( √ γ +6 ) √ U ( γ − , γ ≥ O ( ε ) . (40b)The relative errors in the approximation of (40) by u = u ( t ; 0) , v = v ( t ; 0) are: E r ( u ) := u ( t ; ε ) − u ( t ; 0) u ( t ; ε ) = ε √ c tanh(∆) √ U (6 − γ ) , γ < √ c tan(∆) √ U ( γ − , γ ≥ c − (cid:40) √ γ − , γ < √ γ − , γ ≥ . + O (cid:0) ε (cid:1) , (41a) E r ( v ) := v ( t ; ε ) − v ( t ; 0) v ( t ; ε ) = ε − ( √ γ − √ ) c coth(∆) ( √ γ +6 ) √ U (6 − γ ) , γ < ( √ γ − √ ) c cot(∆) ( √ γ +6 ) √ U ( γ − , γ ≥ c + (cid:40) √ γ +6 , γ < √ γ +6 , γ ≥ O (cid:0) ε (cid:1) . (41b)Taking the limit t → + ∞ it follows that the above relative errors tend to zero. Thus, the linear terms in ε in theequation (40) can be made a small percent of the contribution of the zeroth-solutions by taking τ large enough.Henceforth, this shows that the behavior of the solutions for the induced gravity model does not change abruptlywhen a massless scalar field, φ , with small kinetic term is added to the setup. Given a differential equation ˙ x = f ( x, t, ε ) with f periodic in t . One approximation schemewhich can be used consists in solving the problem for ε = 0 (unperturbed problem) and then, use this approximatedunperturbed solution to formulate variational equations in standard form which can be averaged.Take the simple equation ¨ φ + φ = ε ( − φ ) , (42)with φ (0) , ˙ φ (0) given.The unperturbed problem: ¨ φ + φ = 0 , (43)have as solution: ˙ φ ( t ) = r cos( t − ϕ ) , φ ( t ) = r sin( t − ϕ ) , (44)where r and ϕ are constants depending on the initial conditions. Then, using the amplitude-phase variables (alsocalled variation of constants, see [112], chapter 11) which are defined as: ˙ φ ( t ) = r ( t ) cos( t − ϕ ( t )) , φ ( t ) = r ( t ) sin( t − ϕ ( t )) , (45)such that r = (cid:113) ˙ φ ( t ) + φ ( t ) , ϕ = t − tan − (cid:18) φ ( t )˙ φ ( t ) (cid:19) . (46)Then, under the coordinate transformation ( ˙ φ, φ ) → ( r, ϕ ) , ¨ φ + φ = ε ( − φ ) , (47)becomes: ˙ r = − rε cos ( t − ϕ ) , ˙ ϕ = − ϕε sin(2( t − ϕ )) . (48) eneralized Scalar Field Cosmologies: a Perturbative Analysis r and ϕ are varying slowly with time, and system is in the form ˙ y = εf ( y ) . The idea isto consider only the nonzero average of the right-hand-sides, keeping r and ϕ fixed, and leave out the terms withaverage zero ignoring the slow-varying dependence of r and ϕ on t in the averaging process. Now, replacing r, ϕ by their averaged approximations ¯ r, ¯ ϕ : ˙¯ r = − ε π (cid:90) π r cos ( t − ϕ ) dt = − ε ¯ r, ˙¯ ϕ = − ε π (cid:90) π sin(2( t − ϕ )) dt = 0 . (49)Solving (49) with initial conditions ¯ r (0) = r , ¯ ϕ (0) = ϕ , the approximation takes the form: ¯ φ = r e − εt sin( t − ϕ ) , (50)which coincides with the result that would be obtained using the two-timing expansion procedure. These twoprocedures alleviate the failure of the regular asymptotic expansion that would yield spurious secular terms in theasymptotic expansions, say, on the regular asymptotic expansion, x ( t, ε ) = sin t − εt sin t + O ( ε ) , (51)the “next to leading term” εt sin t is dominant on scales εt = O (1) . In this paper there will be studied some perturbation problems in scalar field cosmologies in a vacuum and withmatter using similar procedures as in the previous examples. Relevant information about the solution’s space forscalar field cosmologies in Friedmann-Lemaitre Robertson-Walker metrics and in Bianchi I metrics is expected tobe obtained using qualitative techniques, asymptotic methods, and averaging theory. In this regard, this paper isa continuation of [118, 119]. There were reviewed some well-known results and there were proved new theoremsin the context of scalar field cosmologies with arbitrary potential (and with an arbitrary coupling to matter). Inparticular, there were incorporated cosine-like corrections with small phase to the harmonic potential for FLRWmetric and Bianchi I metrics inspired in [105].The paper is organized as follows. Sections 3.1 and 3.2 are devoted to applications of the Perturbation Theoryto analyze the periodic solutions of a scalar field minimally coupled to matter with self-interacting potentials V ( φ ) = f cos (cid:16) φf (cid:17) + φ and V ( φ ) = − f cos (cid:16) φf (cid:17) + f + φ , respectively. In sections 3.5 and 3.6 the previouspotentials V ( φ ) and V ( φ ) are taken, but it is assumed that matter and the scalar field interacts through the couplingstrength Q = λρ m that arises from the coupling function χ ( φ ) = χ e λφ − γ . Using averaging methods for periodicfunctions of a given period T , the focus is to study the imprint of the coupling function, as well as the influence ofthe metric on the dynamics of the averaged problem.
2. Scalar field cosmologies
There are a number of gravitational theories which included scalar fields that can be studied using local and globalvariables, providing a qualitative description of the space of solutions. In addition, it is possible to provide preciseschemes to find analytical approximations of the solutions, as well as exact solutions or solutions in quadrature bychoosing various approaches, e.g., [1, 3, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 68, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 75, 76, 77, 71, 72, 73, 74, 79, 80, 81, 78, 82, 83, 84]. Inparticular, relevant information about the properties of the flow associated with an autonomous system of ordinarydifferential equations can be obtained by using qualitative techniques of dynamical systems. See textbooks relatedto qualitative theory of differential equations [85, 86, 87, 88, 89, 90, 91, 92, 93, 94] and with some applications incosmology [95, 96, 97, 98, 99, 100, 101].Perturbations methods and averaging methods were used for example, in [102], in investigations of theoscillating behavior in scalar field cosmologies with harmonic potential using amplitude-phase variables. In [103],these techniques were used to prove statements about how the relationship between the equation of state of thefluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic late timebehavior. Slow-fast methods were used for example in GUP theories, say in [104]. In [119] averaging over anangle ϑ by using an amplitude-angle transformation of the form ˙ φ ( t ) = r ( t ) sin ϑ ( t ) and φ ( t ) = r ( t ) cos ϑ ( t ) wasused to study oscillations of the scalar field driven by generalized harmonic potentials. In the reference [116] wasapplied the averaging theory of first order to study the periodic orbits of Hamiltonian systems describing a universe eneralized Scalar Field Cosmologies: a Perturbative Analysis C . These techniques can be applied to Hamiltoniansystems with an arbitrary number of degrees of freedom.The field equations of a scalar field with self-interacting potential V ( φ ) in vacuum are given by: ¨ φ + 3 H ˙ φ + dV ( φ ) dφ = 0 , (52a) ˙ H = −
12 ˙ φ , (52b) H = 12 ˙ φ + V ( φ ) , (52c)where a ( t ) denotes the scale factor of the universe, H = ˙ aa denotes de Hubble scalar, a dot denotes derivative withrespect to t , φ is the scalar field and V ( φ ) the scalar field self-interacting potential which is assume to be of class C . On the other hand, the equations for a scalar field cosmology in the presence of matter for FLRW metrics andfor Bianchi I metrics are [117, 98]: ¨ φ + 3 H ˙ φ + dV ( φ ) dφ = 0 , (53a) ˙ ρ m + 3 γHρ m = 0 , (53b) ˙ a = aH, (53c) ˙ H = − (cid:16) γρ m + ˙ φ (cid:17) + 16 aG (cid:48) ( a ) , (53d) H = 12 ˙ φ + V ( φ ) + ρ m + G ( a ) , (53e)where ρ m corresponds to the energy density of matter with Equation of State (EoS) parameter w m = p m ρ m := γ − where ≤ γ ≤ denotes the barotropic index. An auxiliary function is used to include FLRW and Bianchi Igeometry defined by: G ( a ) = (cid:40) − ka , k = 0 , ± , spatial curvature of FLRW metrics, σ a , anisotropies of Bianchi I metric. (54)Integrating equation (53b) it follows ρ m = ρ m, a γ .The more general situation that can be considered here is when the scalar field is nonminimally coupled tomatter, say, for FLRW metrics and Bianchi I metric. The equations for a scalar field cosmology nonminimallycoupled to matter for FLRW metrics and for Bianchi I metrics are [74, 67]: ¨ φ + 3 H ˙ φ + dV ( φ ) dφ = 12 (4 − γ ) ρ m d ln χdφ , (55a) ˙ ρ m + 3 γHρ m = −
12 (4 − γ ) ρ m ˙ φ d ln χdφ , (55b) ˙ a = aH, (55c) ˙ H = − (cid:16) γρ m + ˙ φ (cid:17) + 16 aG (cid:48) ( a ) , (55d) H = ρ m + 12 ˙ φ + V ( φ ) + G ( a ) , (55e)Integrating (55b), the following is obtained: ρ m = ρ m, a γ χ ( a ) − γ . (56)In [118], some theorems related to the asymptotic behavior of a very general cosmological model given bysystem (55) were presented. It was examined to which extent the hypotheses of the theorems proved in[118] can be relaxed in order to obtain the same conclusions, or to provide a counterexample by means of eneralized Scalar Field Cosmologies: a Perturbative Analysis V ( φ ) = µ (cid:104) φ µ + bf cos (cid:16) δ + φf (cid:17)(cid:105) , b (cid:54) = 0 and V ( φ ) = µ (cid:104) bf (cid:16) cos( δ ) − cos (cid:16) δ + φf (cid:17)(cid:17) + φ µ (cid:105) , b (cid:54) = 0 . Harmonic potentials plus cosine corrections were introduced in thecontext of inflation in loop-quantum cosmology in [105]. Finally, using the Hubble-normalized formulation for ascalar field non-minimally coupled to matter with generalized harmonic potential V ( φ ) = φ + f (cid:104) − cos (cid:16) φf (cid:17)(cid:105) , f > , and with coupling function χ ( φ ) = χ e λφ − γ where λ is a constant and ≤ γ ≤ , γ (cid:54) = , were foundthe late time attractors correspond to the non zero local minimums of the potential for FLRW metrics and for theBianchi I metric. These equilibrium points are related with de Sitter solutions. The global minimum of V ( φ ) at φ = 0 is unstable to curvature perturbations for γ > in the case of a negatively curved FLRW model. Thisconfirms the result in [82] that in a non-degenerated minimum with zero critical value, the curvature will eventuallydominate both the perfect fluid and the scalar field densities on the late evolution of the universe for γ > / . Forthe Bianchi I model the global minimum V (0) = 0 is unstable to shear perturbations.In [119], a local dynamical systems analysis for arbitrary V ( φ ) and χ ( φ ) using Hubble normalized equationswas provided. The analysis relies on two arbitrary functions f ( λ ) and g ( λ ) which encode a potential and a couplingfunction through the quadrature: φ ( λ ) = φ (1) − (cid:90) λ f ( s ) ds, V ( λ ) + Λ = W (1) e (cid:82) λ sf ( s ) ds , χ ( λ ) = χ (1) e (cid:82) λ g ( s ) f ( s ) ds . Afterwards, a global dynamical systems formulation using the Alho & Uggla’s approach [84] was implemented.The equilibrium points that represent some solutions of cosmological interest were obtained. In particular, severalscaling solutions are found, as well as stiff solutions, and a solution dominated by the effective energy density ofthe geometric term G ( a ) , a quintessence scalar field dominated solution, the vacuum de Sitter solution associatedto the minimum of the potential and a non-interacting matter dominated solution. All of which reveal a very richcosmological behavior.
3. Applications of perturbation and averaging methods in cosmology
It is worth noticing that when Hubble-normalized quantities are used, more often the evolution equation for H ,which is given by the Raychaudhuri equation, decouples. The asymptotic of the remaining reduced system isthen typically given by the equilibrium points and often it can be determined by a dynamical system analysis[98, 120, 121]. In particular, this is always the case for a scalar field with exponential potential. This is due tothe fact the exponential potential has symmetry such that its derivative is also an exponential function. For otherpotentials that do not satisfy the above symmetry, like the harmonic potential V ( φ ) = µ φ , the Raychaudhuriequation fails to decouple [84]. Hubble-normalized equations often are very difficult to be analyzed using thestandard dynamical systems approach due to oscillations entering the system via the Klein-Gordon equation [122].In reference [122], oscillations and future asymptotics of locally rotationally symetric Bianchi type III cosmologieswith a massive scalar field with potential V ( φ ) = φ were studied.The preliminary analysis of oscillations in scalar-field cosmologies with generalized harmonic potentials oftype V ( φ ) = µ φ + cosine corrections is extended here using averaging techniques similar to those used in [122]for a family of generalized harmonic potentials when H monotonically tends to zero. In this approach, the Hubblescalar plays a role of a time dependent perturbation parameter which controls the magnitude of the error betweenfull-system and time-averaged solutions. These oscillations can be viewed as perturbations that can be smoothedout with the benefit that the averaged Raychaudhuri equation decouples in the averaged system. At the end, theanalysis of the system is reduced to the study of corresponding averaged equations. V ( φ ) = f cos (cid:16) φf (cid:17) + φ . In this section perturbation methods for analyzing the dynamic of a scalar field in vacuum with generalizedharmonic potential: V ( φ ) = f cos (cid:18) φf (cid:19) + φ f + ( f − φ f + O (cid:0) φ (cid:1) , (57)are applied. In Fig. 1 it is represented the generalized harmonic potential V ( φ ) = f cos (cid:16) φf (cid:17) + φ and its derivativefor b = 0 . , b = 0 . and b = 1 . In the first row of figure 1 the potential admits four local minimums and three eneralized Scalar Field Cosmologies: a Perturbative Analysis (cid:45) (cid:45) V (cid:72) Φ (cid:76) (a) f = 0 . (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) V' (cid:72) Φ (cid:76) (b) f = 0 . (cid:45) (cid:45) V (cid:72) Φ (cid:76) (c) f = 0 . (cid:45) (cid:45) (cid:45) (cid:45) V' (cid:72) Φ (cid:76) (d) f = 0 . (cid:45) (cid:45) V (cid:72) Φ (cid:76) (e) f = 1 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) V' (cid:72) Φ (cid:76) (f) f = 1 Figure 1. (a) Generalized harmonic potential V ( φ ) = f cos (cid:16) φf (cid:17) + φ and (b) its derivative. local maximums. In the second row the potential admits a local maximum and two local minimums, and in the lastone, the potential have a degenerated minimum at the origin.Using the amplitude-phase variables [112]: ˙ φ = r cos (cid:18) t − f t − ϕ (cid:19) , φ = r sin (cid:18) t − f t − ϕ (cid:19) , (58) ϕ = t − f t − tan − (cid:18) φ ˙ φ (cid:19) , (59)the following equations are obtained: ˙ r = cos (cid:18) t − t f − ϕ (cid:19) sin r sin (cid:16) t − t f − ϕ (cid:17) f − rH cos (cid:18) t − t f − ϕ (cid:19) , (60a) ˙ ϕ = sin (cid:16) t − t f − ϕ (cid:17) sin (cid:18) r sin ( − t f − ϕ + t ) f (cid:19) r + 32 H sin (cid:18)(cid:18) f − (cid:19) t + 2 ϕ (cid:19) − f , (60b)with restriction: f cos r sin (cid:16) t − t f − ϕ (cid:17) f + r = 6 H . (60c) eneralized Scalar Field Cosmologies: a Perturbative Analysis r → Ω = r H , the system P ( H ) : ˙ H = − H cos (cid:16) t − t f − ϕ (cid:17) ˙Ω = 3(Ω − H (cid:16) cos (cid:16)(cid:16) f − (cid:17) t + 2 ϕ (cid:17) + 1 (cid:17) + √ √ Ω cos ( t − t f − ϕ ) sin (cid:32) √ √ Ω H sin ( t − t f − ϕ ) f (cid:33) H ˙ ϕ = fH sin (( f − ) t +2 ϕ ) − f + sin ( t − t f − ϕ ) sin (cid:32) √ √ Ω H sin ( t − t f − ϕ ) f (cid:33) √ √ Ω H , , (61)is obtained, where f cos √ √ Ω H sin (cid:16) t − t f − ϕ (cid:17) f + 3(Ω − H = 0 . (62)Expanding in Taylor’s series around H = 0 the following holds: ˙ H = −
3Ω cos (cid:18) t − t f − ϕ (cid:19) H + O (cid:0) H (cid:1) , (63a) ˙Ω = Ω sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f − − Ω)Ω (cid:18) cos (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) + 1 (cid:19) H − (cid:16) Ω cos (cid:16) t − t f − ϕ (cid:17) sin (cid:16) t − t f − ϕ (cid:17)(cid:17) H f + O (cid:0) H (cid:1) , (63b) ˙ ϕ = − cos (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f −
32 sin (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) H − Ω sin (cid:16) t − t f − ϕ (cid:17) H f + O (cid:0) H (cid:1) . (63c)However, the - dimensional system (61) is not in the standard form: ˙ x = Hf ( t, x ) + H g ( t, x, H ) , x (0) = x , t ≥ , (64)but, it is on the form of ˙ x = x ( t, x ) + Hf ( t, x ) + H g ( t, x, H ) , x (0) = x , t ≥ , (65)where x ( t, x ) is T -periodic in t , with T = πf | − f | and with zero average. That is, with Ω and ϕ , viewed asconstants with respect to t , it follows T (cid:90) T Ω sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f dt = 0 , T (cid:90) T (cid:16) t − t f − ϕ (cid:17) − f dt = 0 . (66)The idea is to solve the problem for H = 0 , and use the unperturbed solution to formulate variational equationswhich can be averaged.As H → the unperturbed system follows: P (0) : ˙ H = 0˙Ω = Ω sin (( − f ) t − ϕ ) f ˙ ϕ = − cos (( − f ) t − ϕ ) f , (67)which admits the general solution: Ω ( t ) = c (cid:16) cosh (cid:16) (cid:113) − ff ( t − c ) (cid:17) − f + 1 (cid:17) , < f < c (cid:16) cos (cid:16) (cid:113) f − f ( t − c ) (cid:17) − f + 1 (cid:17) , f > , (68a) ϕ ( t ) = t − t f − tan − (cid:32) tanh (cid:16)(cid:113) − ff ( t − c ) (cid:17)(cid:113) − ff (cid:33) , < f < t − t f − tan − (cid:32) tan (cid:16)(cid:113) f − f ( t − c ) (cid:17)(cid:113) f − f (cid:33) , f > . (68b) eneralized Scalar Field Cosmologies: a Perturbative Analysis f cos √ √ Ω H sin (cid:16) t − t f − ϕ (cid:17) f + 3(Ω − H = 0 . (69)Assuming for a while that f > , is used the variation of constants method to propose: Ω( t ) = c ( t ) (cid:32) cos (cid:32) (cid:115) f − f ( t − c ( t )) (cid:33) − f + 1 (cid:33) , (70a) ϕ ( t ) = t − t f − tan − tan (cid:16)(cid:113) f − f ( t − c ( t )) (cid:17)(cid:113) f − f . (70b)Therefore, the system in the standard form is obtained: c (cid:48) ( t ) = 3 H (cid:16) cos (cid:16) (cid:113) f − f ( t − c ( t )) (cid:17) − f + 1 (cid:17) sin (cid:32) − (cid:32) tan (cid:16)(cid:113) f − f ( t − c ( t )) (cid:17)(cid:113) f − f (cid:33)(cid:33) f − H c ( t ) sin (cid:16)(cid:113) f − f ( t − c ( t )) (cid:17) ( f − f + O ( H ) , (71a) c (cid:48) ( t ) = − Hc ( t ) (2( f − c ( t ) + 1) cos (cid:32)(cid:115) f − f ( t − c ( t )) (cid:33) H c ( t ) sin (cid:16)(cid:113) f − f ( t − c ( t )) (cid:17) cos (cid:16)(cid:113) f − f ( t − c ( t )) (cid:17) √ f − f / + O ( H ) . (71b)The right-hand-side of the previous equations is T -periodic with T = π (cid:113) f − f .Performing the averaging process f (0) ( y ) = 1 T (cid:90) T f ( t, y ) dt, (72)where y as well as H are considered as parameters that are kept constants during integration, is obtained c (cid:48) ( t ) = 3 c ( t ) H f − f , c (cid:48) ( t ) = − c ( t ) H (2 c ( f −
1) + 1) , (73)where H ∼ H , a constant. Hence, c ( t ) ∼ H (cid:0) ln (cid:0) e tH − e c ( f − (cid:1) − tH (cid:1) f − f + c , c ( t ) ∼ e tH − c − f + 2 . (74)Now, taking into account the slow variation of Ω , ϕ, H on t and plugging them back on (70) it is obtained: Ω( t ) = cos (cid:18) (cid:113) f − f (cid:18) − H ( t ) ( ln ( e tH ( t ) − e c ( f − ) − tH ( t ) ) f − f − c + t (cid:19)(cid:19) − f + 1 e tH ( t ) − c − f + 2 , (75) ϕ ( t ) = − tan − tan (cid:18)(cid:113) f − f (cid:18) − H ( t ) ( ln ( e tH ( t ) − e c ( f − ) − tH ( t ) ) f − f − c + t (cid:19)(cid:19)(cid:113) f − f − t f + t. (76)Hence, from the evolution equation of H in (61) it follows: H (cid:48) ( t ) = − e c ( f − H ( t ) cos (cid:18)(cid:113) f − f (cid:18) − H ( t ) ( ln ( e tH ( t ) − e c ( f − ) − tH ( t ) ) f − f − c + t (cid:19)(cid:19) e c ( f − − e tH ( t ) = − H ( t ) (cid:16) e c ( f −
1) cos (cid:16)(cid:113) f − f ( t − c ) (cid:17)(cid:17) e c ( f − − O (cid:0) H ( t ) (cid:1) . (77) eneralized Scalar Field Cosmologies: a Perturbative Analysis H ( t ) ∼ (cid:113) f − f (2 e c ( f − − (cid:113) f − f ( e c ( f −
1) ( − c − c + 3 t ) + c ) + 3 e c ( f −
1) sin (cid:16) (cid:113) f − f ( t − c ) (cid:17) . (78)That satisfies H ( t ) → as t → ∞ .Now, the perturbation theory tools are used.Given the nature of the problem and given that H is not properly a parameter but a function of time, it isproposed an expansion of the type: Ω ≡ Ω( t ) = Ω ( t ) + H ( t )Ω ( t ) + O ( H ) (79a) ϕ ≡ ϕ ( t ) = ϕ ( t ) + H ( t ) ϕ ( t ) + O ( H ) , (79b)where Ω ( t ) and ϕ ( t ) are solutions of the unperturbed system P (0) given by (68). Applying the chain rule andusing the fact that Ω dHdt = O ( H ) according to (63a) is obtained: d Ω dt = d Ω dt + H d Ω dt + Ω dHdt + O ( H ) = d Ω dt + H d Ω dt + O ( H ) , (80a) dϕdt = dϕ dt + H dϕ dt + ϕ dHdt + O ( H ) = dϕ dt + H dϕ dt + O ( H ) . (80b)It follows that: H d Ω dt = d Ω dt − d Ω dt (Ω + H Ω ) sin (cid:16)(cid:16) − f (cid:17) t − ϕ − Hϕ (cid:17) f − Ω sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f − H (1 − Ω − H Ω ) (Ω + H Ω ) (cid:18) cos (cid:18)(cid:18) − f (cid:19) t − ϕ − Hϕ (cid:19) + 1 (cid:19) + O ( H )= H Ω sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f + Ω (3 f (Ω − − ϕ ) cos (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f + 3(Ω − + O (cid:0) H (cid:1) ⇒ d Ω dt = Ω sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f + Ω (3 f (Ω − − ϕ ) cos (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f − − Ω )Ω . (81a) H dϕ dt = dϕdt − dϕ dt = − cos (cid:16)(cid:16) − f (cid:17) t − ϕ − Hϕ (cid:17) f + cos (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f − H sin (cid:18)(cid:18) − f (cid:19) t − ϕ − Hϕ (cid:19) + O ( H )= H (3 f − ϕ ) sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f + O (cid:0) H (cid:1) ⇒ dϕ dt = (3 f − ϕ ) sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f . (81b) Therefore, to determine the analytical expressions of Ω and ϕ the equations: f d Ω dt = Ω sin (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) + Ω (3 f (Ω − − ϕ ) cos (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) − − Ω )Ω f, (82a) f dϕ dt = (3 f − ϕ ) sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) , (82b) have to be solved with the substitution of (68). Additionally, ˙ H = − H Ω cos (cid:18) − t f + t − ϕ (cid:19) + O (cid:0) H (cid:1) . (83) eneralized Scalar Field Cosmologies: a Perturbative Analysis f cos √ H √ Ω + Ω H sin (cid:16) t − t f − Hϕ − ϕ (cid:17) f + 3 H (Ω + Ω H ) + O ( H ) = f + O ( H ) , (84)is deduce.Notice that for f (cid:28) the system has two time scales, the fast-time (or slow time for f (cid:29) ) τ = t/f and theslow time (or fast time for f (cid:29) ) t = f τ .The following systems are obtained:To order O (1) : d Ω dτ = Ω sin(2( f τ − ϕ ) − τ ) , (85a) dϕ dτ = −
12 cos(2( f τ − ϕ ) − τ ) , (85b)with solution ϕ ( τ ) = fτ − τ − tan − (cid:18) tanh ( √ − f √ f ( τ − c ) ) (cid:113) − ff (cid:19) , < f < fτ − τ − tan − (cid:18) tan ( √ f − √ f ( τ − c ) ) (cid:113) f − f (cid:19) , f > , (86a) Ω ( τ ) = (cid:26) c (cid:0) cosh (cid:0) √ − f √ f ( τ − c ) (cid:1) − f + 1 (cid:1) , < f < c (cid:0) cos (cid:0) √ f − √ f ( τ − c ) (cid:1) − f + 1 (cid:1) , f > . (86b)(86c) To order O ( H ) : d Ω dτ = Ω sin (2 f τ − τ − ϕ ) + Ω (3 f (Ω − − ϕ ) cos (2 f τ − τ − ϕ ) − − Ω )Ω f, (87a) dϕ dτ = (3 f − ϕ ) sin (2 f τ − τ − ϕ )2 , (87b)Integrating for ϕ the following is obtained: ϕ ( τ ) = f (4 c ( f − f )cosh ( √ − f √ f ( τ − c ) ) − f +1 + f , < f < f (4 c ( f − f )cos ( √ f − √ f ( τ − c ) ) − f +1 + f , f > . (88) Finally, solving the equation for Ω , after the substitution of Ω , ϕ , ϕ it follows: Ω ( τ ) = c (cid:0) cosh (cid:0) √ − f √ f ( τ − c ) (cid:1) − f + 1 (cid:1) + c f (cid:40) cosh (cid:0) √ − f √ f ( τ − c ) (cid:1) − f + 1 (cid:41) × (cid:40) − c − f − c + 3) ( τ − c ) − c (cid:113) f − f sinh (cid:0) √ − f √ f ( τ − c ) (cid:1) −
24 tan − (cid:18) tanh ( √ − f √ f ( τ − c ) ) (cid:113) − ff (cid:19) − c +3) f − c ) sinh ( √ − f √ f ( τ − c ) ) √ − f √ f ( cosh ( √ − f √ f ( τ − c ) ) − f +1 ) (cid:41) , < f < c (cid:0) cos (cid:0) √ f − √ f ( τ − c ) (cid:1) − f + 1 (cid:1) + c f (cid:40) cos (cid:0) √ f − √ f ( τ − c ) (cid:1) − f + 1 (cid:41) × (cid:40) − c − f − c + 3) ( τ − c ) − c (cid:113) f − f sin (cid:0) √ f − √ f ( τ − c ) (cid:1) −
24 tan − (cid:18) tan ( √ f − √ f ( τ − c ) ) (cid:113) f − f (cid:19) − c +3) f − c ) sin ( √ f − √ f ( τ − c ) ) √ f − √ f ( cos ( √ f − √ f ( τ − c ) ) − f +1 ) (cid:41) , f > . (89) eneralized Scalar Field Cosmologies: a Perturbative Analysis dYdη = HG ( Y, t, H ) , dtdη = H, (90)where G ( Y, t, H ) = − H cos (cid:16) t − t f − ϕ (cid:17) − H (cid:16) cos (cid:16)(cid:16) f − (cid:17) t + 2 ϕ (cid:17) + 1 (cid:17) + √ √ Ω cos (cid:16) t − t f − ϕ (cid:17) sin (cid:32) √ √ Ω H sin ( t − t f − ϕ ) f (cid:33) H fH sin (cid:16)(cid:16) f − (cid:17) t +2 ϕ (cid:17) − f + sin (cid:16) t − t f − ϕ (cid:17) sin (cid:32) √ √ Ω H sin ( t − t f − ϕ ) f (cid:33) √ √ Ω H , (91) where Y denotes the vector of states (Ω , ϕ ) T . Given that the right-hand-side of the system (90) is T -periodic with T = πf | f − | the averaged system (where the terms O ( H ) were neglected) is obtained: (cid:68) P ( H ) (cid:69) : (cid:26) ˙¯Ω = − − ¯Ω) ¯Ω H ˙¯ ϕ = 0 . (92)Defining the time variable τ = ln a , it follows the guiding equation (where the equations for H and ¯ ϕ aredecoupled): ∂ τ ¯Ω = − − ¯Ω) ¯Ω . (93)This system is integrable, and from the local stability analysis of the equilibrium points of the system (93) it followsthat ¯Ω = 0 has eigenvalue − and is a source; and ¯Ω = 1 has eigenvalue and is a source.A general idea of averaging is to express: t = t + Hω ( H, t , Ω , ϕ , η ) , (94a) Ω = Ω + Hω ( H, t , Ω , ϕ , η ) , (94b) ϕ = ϕ + Hω ( H, t , Ω , ϕ , η ) , (94c)and then, to prove that the equations for t , Ω , ϕ have the same asymptotic of the averaged equations for ¯ t, ¯Ω , ¯ ϕ .Then, dtdη = (cid:18) H ∂ω ∂t (cid:19) dt dη + dHdη ω + H (cid:20) ∂ω ∂H dHdη + ∂ω ∂ Ω d Ω dη + ∂ω ∂ϕ dϕ dη + ∂ω ∂η (cid:21) , (95a) d Ω dη = (cid:18) H ∂ω ∂ Ω (cid:19) d Ω dη + dHdη ω + H (cid:20) ∂ω ∂H dHdη + ∂ω ∂t dt dη + ∂ω ∂ϕ dϕ dη + ∂ω ∂η (cid:21) , (95b) dϕdη = (cid:18) H ∂ω ∂ϕ (cid:19) dϕ dη + dHdη ω + H (cid:20) ∂ω ∂H dHdη + ∂ω ∂t dt dη + ∂ω ∂ Ω d Ω dη + ∂ω ∂η (cid:21) . (95c) The left-hand-side of the above equations can be expressed as: dtdη = H, (96a) d Ω dη = 2Ω H sin (cid:16) − t f − ϕ + t (cid:17) cos (cid:16) − t f − ϕ + t (cid:17) f + 3 H (cid:18) Ω cos (cid:18)(cid:18) f − (cid:19) t + 2 ϕ (cid:19) − Ω cos (cid:18)(cid:18) f − (cid:19) t + 2 ϕ (cid:19) + Ω − Ω (cid:19) + O ( H ) , (96b) dϕdη = H (cid:16) (cid:16) − t f − ϕ + t (cid:17) − (cid:17) f + 32 H sin (cid:18)(cid:18) f − (cid:19) t + 2 ϕ (cid:19) + O ( H ) . (96c) Combining the expressions (95) with (96) and (94), it follows: dt dη = H (1 − ω η ) + H f (cid:40) ω ϕ (cid:18) f ω η + cos (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19)(cid:19) − (cid:34) ω (cid:18) Ω sin (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) − f ω η (cid:19) + f ((1 − ω η ) ω t + ω Hη ) (cid:35)(cid:41) + O ( H ) , (97a) eneralized Scalar Field Cosmologies: a Perturbative Analysis d Ω dη = H Ω sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f − ω η + H f (cid:40) f (cid:34) f ω η ω t + 2 f ω η ω − f ω t − f ω H − sin (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) ω + cos (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) ω ϕ + 2 sin (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) ω + 2 f ω ϕ ω η − cos (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) ω + 6 f Ω cos (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) − f Ω cos (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) + 6 f Ω − f Ω (cid:35) + 2(2 f − cos (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) ω (cid:41) + O ( H ) , (97b) dϕ dη = − H (cid:16) f ω η + cos (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17)(cid:17) f + H f (cid:110) f (cid:34) f ω η ω t + 2 f ω η ω − f ω t − f ω Hη − sin (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) ω + ω ϕ (cid:18) f ω η + cos (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19)(cid:19) − (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) ω − f sin (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) (cid:35) + (2 f −
1) sin (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) ω (cid:41) + O ( H ) . (97c)where the sub-indexes H, t , Ω , ϕ , η , of the ω i ’s denote partial derivatives.Now, it is proved that t , Ω , ϕ evolve at first order exactly as for the averaged quantities ¯ t, ¯Ω , ¯ ϕ . Indeed, bychoosing ω η = 0 , ω η = Ω sin (cid:16) t − t f − ϕ (cid:17) f , ω η = − cos (cid:16) t − t f − ϕ (cid:17) f , (98)it follows by integration: ω ( H, t , Ω , ϕ , η ) = c ( H, t , Ω , ϕ ) , (99a) ω ( H, t , Ω , ϕ , η ) = c ( H, t , Ω , ϕ ) + η Ω sin (cid:16) t − t f − ϕ (cid:17) f , (99b) ω ϕ ( H, t , Ω , ϕ , η ) = c ( H, t , Ω , ϕ ) − η cos (cid:16) t − t f − ϕ (cid:17) f , (99c)and the equations for (92) are recovered at the order O ( H ) : dt dη = H − H c t + O (cid:0) H (cid:1) , (100a) eneralized Scalar Field Cosmologies: a Perturbative Analysis d Ω dη = H (cid:40) sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) c f − c t + Ω (cid:34) c (2 f −
1) cos (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f − c cos (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f + (cid:0) − f − f η + η (cid:1) cos (cid:16)(cid:16) f − (cid:17) t + 2 ϕ (cid:17) − f + ηf (cid:35) + 3Ω (cid:18) cos (cid:18)(cid:18) − f (cid:19) t − ϕ (cid:19) + 1 (cid:19) (cid:41) , (100b) dϕ dη = H (cid:40) − c t + c (2 f −
1) sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f − c sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f + sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) (cid:16) − f − f η + η cos (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) + η (cid:17) f (cid:41) + O (cid:0) H (cid:1) . (100c)Now choosing c , c , and c , in such a way that they are only functions of t and ϕ that satisfy: c t = 0 , sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) c f − c t = 0 , c t + c sin (cid:16)(cid:16) − f (cid:17) t − ϕ (cid:17) f = 0 . (101)That is, c ( t , ϕ ) = ˜ c ( ϕ ) , c ( t , ϕ ) = ˜ c ( ϕ ) e − cos (( − f ) t − ϕ ) f − , c ( t , ϕ ) = ˜ c ( ϕ ) e cos (( − f ) t − ϕ ) f − , (102)where, the ˜ c i ’s can be chosen by convenience.Then, it follows: ω ( H, t , Ω , ϕ , η ) = ˜ c ( ϕ ) , (103a) ω ( H, t , Ω , ϕ , η ) = ˜ c ( ϕ ) e − cos (( − f ) t − ϕ ) f − + η Ω sin (cid:16) t − t f − ϕ (cid:17) f , (103b) ω ϕ ( H, t , Ω , ϕ , η ) = ˜ c ( ϕ ) e cos (( − f ) t − ϕ ) f − − η cos (cid:16) t − t f − ϕ (cid:17) f , (103c)from which t = t + H ˜ c ( ϕ ) , (104a) Ω = Ω + H ˜ c ( ϕ ) e − cos (( − f ) t − ϕ ) f − + η Ω sin (cid:16) t − t f − ϕ (cid:17) f , (104b) ϕ = ϕ + H ˜ c ( ϕ ) e cos (( − f ) t − ϕ ) f − − η cos (cid:16) t − t f − ϕ (cid:17) f , (104c) eneralized Scalar Field Cosmologies: a Perturbative Analysis t , Ω , ϕ and H satisfy: dt dη = H + O (cid:0) H (cid:1) , (105a) d Ω dη = Ω H (cid:0) (2 f − c ( ϕ ) cos ( ξ ) + 6 f (Ω −
1) cos ( ξ ) (cid:1) f + η Ω H ((1 − f ) cos ( ξ ) + 1) f − H ˜ c ( ϕ ) e cos (( − f ) t − ϕ ) f − cos ( ξ ) f + O (cid:0) H (cid:1) , (105b) dϕ dη = H sin ( ξ ) (cid:20)
32 + η (1 − f + cos ( ξ ))2 f + (1 − f )2 f ˜ c ( ϕ ) (cid:21) + O (cid:0) H (cid:1) , (105c)where c = c ( ϕ ) and c = c ( ϕ ) will be chosen conveniently, and it is used the notation ξ = (cid:16) f − (cid:17) t +2 ϕ .The averaged system is given by dt dη = H, d ¯Ω dη = − − ¯Ω) ¯Ω H , d ¯ ϕdη = 0 . (106)Since, dHdη = − H Ω cos ( ξ ) + O (cid:0) H (cid:1) , (107)it follows that H is a monotonic decreasing function due to ≤ Ω ≤ . Then, the following sequences are defined: η = 0 H = H ( η ) , η n +12 = η n + H n H n +1 = H ( η n +1 ) . (108)such that lim n →∞ η n = ∞ and lim n →∞ H n = 0 . Defining ∆ ϕ ( η ) = ϕ ( η ) − ¯ ϕ ( η ) , and choosing the same initialconditions at η = η n : ϕ ( η n ) = ¯ ϕ ( η n ) , it follows: | ∆ ϕ ( η ) | = (cid:12)(cid:12)(cid:12) (cid:90) ηη n [ ϕ (cid:48) ( s ) − ¯ ϕ (cid:48) ( s )] ds (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:90) ηη n H sin( ξ ) (cid:20)
32 + s (1 − f + cos ( ξ ))2 f + (1 − f )2 f ˜ c ( ϕ ) (cid:21) + O (cid:0) H (cid:1) ds (cid:12)(cid:12)(cid:12) , ≤ (cid:90) ηη n (cid:12)(cid:12)(cid:12) H sin( ξ ) (cid:124) (cid:123)(cid:122) (cid:125) |·|≤ H n (cid:104) (cid:124)(cid:123)(cid:122)(cid:125) |·| = M +2 s (1 − f + cos( ξ ))4 f (cid:124) (cid:123)(cid:122) (cid:125) |·|≤ M + (1 − f )2 f (cid:124) (cid:123)(cid:122) (cid:125) |·|≤ M ˜ c ( ϕ ) (cid:105) + O (cid:0) H (cid:1) (cid:12)(cid:12)(cid:12) ds ≤ H n M | η − η n | + H n M | η + η n || η − η n | + H n M (cid:90) ηη n | ˜ c ( ϕ ) | ds, η ≥ η n , (109)where M = , M = | − f | f , M = | − f | f , and ξ = (cid:16) f − (cid:17) t + 2 ϕ . Then, choosing ˜ c ( ϕ ) = ϕ ( η ) − ¯ ϕ , itis obtained: | ∆ ϕ ( η ) | ≤ H n ( M + M | η + η n | ) | η − η n | + H n M (cid:90) ηη n | ∆ ϕ ( s ) | ds. (110a)Applying Gronwall’s Lemma Appendix A.1, it follows: | ∆ ϕ ( η ) | ≤ ( M + M | η + η n | ) H n | η − η n | (cid:104) H n M ηe H n M η (cid:105) + O ( H n ) ≤ H n | η − η n | [ M + M | η + η n | ] (cid:2) H n M η (cid:3) + O ( H n ) , = H n | η − η n | [ M + M | η + η n | ] + O ( H n ) . (111)Then, for η ∈ [ η n , η n +1 ] with n large enough such that | η + η n | ≥ , it is verified the inequality: M | η − η n | + M | η − η n | ≤ ( M + M ) | η − η n | ≤ ( M + M ) | η n +1 − η n | ≤ ( M + M ) H − n . eneralized Scalar Field Cosmologies: a Perturbative Analysis | ∆ ϕ ( η ) | ≤ KH n , for a positive constant K ≥ M + M . Finally, taking the limit as n → ∞ , it follows H n → , η n → ∞ . Then itis concluded that lim η →∞ | ∆ ϕ ( η ) | = 0 . This means that ϕ and ¯ ϕ have the same asymptotic behavior as η → ∞ .On the other hand, defining ∆Ω = Ω − ¯Ω and choosing ˜ c ( ϕ ) = ˜ c ( ϕ ) = ∆ ϕ := ϕ ( η ) − ¯ ϕ , it follows: ∆Ω (cid:48) ( η ) = − H ∆Ω(1 − Ω − ¯Ω) + 3(Ω − H (cos( ξ ) + 1)+ η Ω H ( − f cos( ξ ) + cos( ξ ) + 1) f − Ω H ∆ ϕ cos( ξ ) (cid:16) f (cid:16) e cos( ξ )2 f − − (cid:17) + 1 (cid:17) f . (112)Then, for all η ∈ [ η n , η n +1 ] with n large enough such that | η + η n | ≥ , there exists K ≥ M + M such that | ∆ z ( η ) | ≤ KH n , it follows: | ∆Ω (cid:48) ( s ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − H ∆Ω(1 − Ω − ¯Ω) + 3(Ω − H (cos( ξ ) + 1)+ 2 sH Ω ( − f cos( ξ ) + cos( ξ ) + 1)2 f + O (cid:0) H n (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (113)Then, choosing the same initial conditions at η = η n , i.e., Ω ( η n ) = ¯Ω( η n ) , it follows: | ∆Ω( η ) | = (cid:12)(cid:12)(cid:12) (cid:90) ηη n [Ω (cid:48) ( s ) − ¯Ω (cid:48) ( s )] ds (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:90) ηη n − H (1 − Ω − ¯Ω) (cid:124) (cid:123)(cid:122) (cid:125) |·|≤ +3Ω H (1 − Ω )(cos( ξ ) + 1) (cid:124) (cid:123)(cid:122) (cid:125) |·|≤ +2 sH Ω ( − f cos( ξ ) + cos( ξ ) + 1)2 f (cid:124) (cid:123)(cid:122) (cid:125) |·|≤ M + O ( H n ) ds (cid:12)(cid:12)(cid:12) (114) ≤ H n (cid:90) ηη n (cid:12)(cid:12)(cid:12) ∆Ω( s ) (cid:12)(cid:12)(cid:12) ds + 6 H n (cid:90) ηη n (cid:12)(cid:12)(cid:12) Ω ( s ) (cid:12)(cid:12)(cid:12) ds + H n M (cid:90) ηη n s Ω ( s ) ds ≤ H n (cid:90) ηη n (cid:12)(cid:12)(cid:12) ∆Ω( s ) (cid:12)(cid:12)(cid:12) ds + 6 M H n | η − η n | + M M H n | η − η n || η + η n | , η ≥ η n , (115)where the bound M has been chosen as M = max s ∈ [ η n ,η ] (cid:110)(cid:12)(cid:12)(cid:12) Ω ( s ) (cid:12)(cid:12)(cid:12)(cid:111) , which exists due to the continuity of Ω onthe compact set [ η n , η ] and M = f +1 f .Applying Gronwall’s Lemma Appendix A.1, it follows: | ∆Ω( η ) | ≤ M H n [6 + M | η + η n | ] | η − η n | (cid:104) H n ηe H n η (cid:105) + O ( H n )= M H n [6 + M | η + η n | ] | η − η n | (cid:2) H n η (cid:3) + O ( H n )= M H n [6 + M | η + η n | ] | η − η n | + O ( H n ) . (116)Hence, for η ∈ [ η n , η n +1 ] and for n large enough such that | η + η n | ≥ , it follows: M [6 + M | η + η n | ] | η − η n | ≤ M (6 + M ) | η − η n | ≤ M [6 + M ] H − n . Then, the inequality | ∆Ω( η ) | ≤ KH n holds for a positive constant K ≥ M (6 + M ) . Finally, taking the limit as n → ∞ , it follows H n → , η n → ∞ . Therefore, it follows lim η →∞ | ∆Ω( η ) | = 0 . This means that Ω and ¯Ω have the same limit as η → ∞ . eneralized Scalar Field Cosmologies: a Perturbative Analysis (cid:45) (cid:45) V (cid:72) Φ (cid:76) (a) f = 0 . (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) V' (cid:72) Φ (cid:76) (b) f = 0 . (cid:45) (cid:45) V (cid:72) Φ (cid:76) (c) f = 0 . (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) V' (cid:72) Φ (cid:76) (d) f = 0 . (cid:45) (cid:45) V (cid:72) Φ (cid:76) (e) f = 1 (cid:45) (cid:45) (cid:45) (cid:45) V' (cid:72) Φ (cid:76) (f) f = 1 Figure 2. (a) Generalized harmonic potential V ( φ ) = − f cos (cid:16) φf (cid:17) + f + φ and (b) its derivative. V ( φ ) = − f cos (cid:16) φf (cid:17) + f + φ , f > . In this section the perturbation methods are applied for analyzing the dynamic of a scalar field in vacuum withgeneralized harmonic potential: V ( φ ) = − f cos (cid:18) φf (cid:19) + f + φ f + 1) φ f + O (cid:0) φ (cid:1) . (117)This potential belongs to the class of potentials studied by [102]. In the Fig. 2, it is presented the generalizedharmonic potential V ( φ ) = − f cos (cid:16) φf (cid:17) + f + φ and its derivative for f = 0 . , f = 0 . and f = 1 . In firstcase the potential have three local minimums and two local maximums. In other two cases the origin is the uniquestationary point and the global minimum of the potential.An amplitude-phase variables [112]: ˙ φ = r cos( t − ϑ ) , φ = r sin( t − ϑ ) , (118) ϑ = t − tan − (cid:18) φ ˙ φ (cid:19) (119) eneralized Scalar Field Cosmologies: a Perturbative Analysis ˙ r = − cos( t − ϑ ) sin (cid:18) r sin( t − ϑ ) f (cid:19) − rH cos ( t − ϑ ) , (120a) ˙ ϑ = − sin( t − ϑ ) sin (cid:16) r sin( t − ϑ ) f (cid:17) r − H sin( t − ϑ ) cos( t − ϑ ) , (120b)with restriction: f cos (cid:18) r sin( t − ϑ ) f (cid:19) − f − r H = 0 . (120c)Defining the transformation r → Ω = r H , it follows P ( H ) : ˙ H = − H cos ( t − ϑ )˙Ω = 6(Ω − H cos ( t − ϑ ) − √ √ Ω cos( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) H ˙ ϑ = − sin( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) √ √ Ω H − H sin(2( t − ϑ ) , (121)where f cos (cid:32) √ √ Ω H sin( t − ϑ ) f (cid:33) − f − − H = 0 . (122)Using Taylor series in a neighborhood of H = 0 the following holds: ˙ H = − H cos ( t − ϑ ) + O ( H ) , (123a) ˙Ω = − Ω sin(2( t − ϑ )) f + 6(Ω − H cos ( t − ϑ ) + O ( H ) , (123b) ˙ ϑ = − sin ( t − ϑ ) f − H sin(2( t − ϑ )) + O ( H ) . (123c)Then, as H → it follows the unperturbed problem: P (0) : ˙ H = 0˙Ω = − Ω sin(2( t − ϑ )) f ˙ ϑ = − sin ( t − ϑ ) f , (124)whose solution is given by: Ω ( t ) = c cos c + t ) (cid:113) ff +1 + 2 f + 1 , (125a) ϑ ( t ) = t − tan − (cid:115) ff + 1 tan c + t (cid:113) ff +1 . (125b)and f cos (cid:32) √ √ Ω H sin( t − ϑ ) f (cid:33) − f − − H = 0 . (126)Using Taylor series in a neighborhood of H = 0 in Eq. (122) it follows f + O (cid:0) H (cid:1) − f = 0 , (127)which is verified up to order O (cid:0) H (cid:1) .Now, continuing with the applications of the perturbation theory tools is proposed an expansion of kind: Ω ≡ Ω( t ) = Ω ( t ) + H ( t )Ω ( t ) + O ( H ) (128a) ϑ ≡ ϑ ( t ) = ϑ ( t ) + H ( t ) ϑ ( t ) + O ( H ) , (128b) eneralized Scalar Field Cosmologies: a Perturbative Analysis Ω ( t ) , y ϑ ( t ) are the solutions of the unperturbed problem P (0) . Applying the chain rule and using thefact that Ω dHdt = O ( H ) , according to (63a), it follows: d Ω dt = d Ω dt + H d Ω dt + Ω dHdt + O ( H ) = d Ω dt + H d Ω dt + O ( H ) , (129a) dϑdt = dϑ dt + H dϑ dt + ϑ dHdt + O ( H ) = dϑ dt + H dϑ dt + O ( H ) . (129b)Hence, H d Ω dt = d Ω dt − d Ω dt = 6((Ω + H Ω ) − + H Ω ) H cos ( t − ( ϑ + Hϑ )) − (Ω + H Ω ) sin(2( t − ( ϑ + Hϑ ))) f + Ω sin (( t − ϑ )) f + O ( H )= H (cid:0) f (Ω − cos ( t − ϑ ) + 2 ϑ Ω cos(2( t − ϑ )) − Ω sin(2( t − ϑ )) (cid:1) f + O (cid:0) H (cid:1) , (130a) H dϑ dt = dϑdt − dϑ dt = − H sin(2( t − ( ϑ + Hϑ ))) − sin ( t − ( ϑ + Hϑ )) f + sin ( t − ϑ ) f + O ( H )= H (2 ϑ − f ) sin(2( t − ϑ ))2 f + O ( H ) . (130b)Therefore, to find analytically the functions Ω y ϑ , the following equations have to be solved: d Ω dt = (cid:0) f (Ω − cos ( t − ϑ ) + 2 ϑ Ω cos(2( t − ϑ )) − Ω sin(2( t − ϑ )) (cid:1) f , (131a) dϑ dt = (2 ϑ − f ) sin(2( t − ϑ ))2 f , (131b)with the substitution of Ω and ϑ in (131).Integrating for ϑ , it follows: ϑ ( t ) = 12 c − f (2 f + 1)cos (cid:16) (cid:113) f + 1 (2 c + t ) (cid:17) + 2 f + 1 + 3 f . (132a)For Ω the following quadrature is obtained: Ω ( t ) = exp (cid:18) − (cid:90) t sin(2( s − ϑ ( s ))) f ds (cid:19) (cid:90) t g ( s ) exp (cid:16)(cid:82) s s − ϑ ( s ))) f ds (cid:17) f ds + c , (133)where g ( t ) = c cos c + t ) (cid:113) ff +1 + 2 f + 1 × (cid:32) f cos (cid:32) c + t ) (cid:113) ff +1 (cid:33) + 2 c (cid:33) cos (cid:32) − (cid:32)(cid:113) ff +1 tan (cid:32) c + t (cid:113) ff +1 (cid:33)(cid:33)(cid:33) cos (cid:32) c + t ) (cid:113) ff +1 (cid:33) + 2 f + 1+ f ( f + 1) (cid:32) c cos (cid:32) c + t ) (cid:113) ff +1 (cid:33) + 2 c f + c − (cid:33) f tan (cid:32) c + t (cid:113) ff +1 (cid:33) + f + 1 . (134) eneralized Scalar Field Cosmologies: a Perturbative Analysis ˙ H = − H cos ( t − ϑ ) , (135a) ˙Ω = 6(Ω − H cos ( t − ϑ ) − (cid:113) √ Ω cos( t − ϑ ) sin (cid:16) cos − (cid:16) f +3Ω H − H f (cid:17)(cid:17) H , (135b) ˙ ϑ = − H sin(2( t − ϑ )) − sin( t − ϑ ) sin (cid:16) cos − (cid:16) f +3Ω H − H f (cid:17)(cid:17) √ √ Ω H . (135c)The previous system can be expressed as: dYdη = HG ( Y, t, H ) , dtdη = H, (136)where G ( Y, t, H ) = − H cos ( t − ϑ )6(Ω − H cos ( t − ϑ ) − √ √ Ω cos( t − ϑ ) sin (cid:16) cos − (cid:16) f +3Ω H − H f (cid:17)(cid:17) H − H sin(2( t − ϑ )) − sin( t − ϑ ) sin (cid:16) cos − (cid:16) f +3Ω H − H f (cid:17)(cid:17) √ √ Ω H , (137)where Y denotes the phase vector (Ω , ϑ ) T .For the problem (135) the following averaged system is deduced: (cid:68) P ( H ) (cid:69) : ˙ H = − H , ˙¯Ω = − − ¯Ω) ¯Ω H, ˙¯ ϑ = 0 . (138)Introducing the new variable τ = ln a , the following guiding equation is obtained: ∂ τ ¯Ω = − − ¯Ω) ¯Ω , (139)for which ¯Ω = 0 is a sink and ¯Ω = 1 is a source.Starting with the averaged equations (138), is proved that Ω , ϑ evolve at first order according to the averagedequations for ¯Ω , ¯ ϑ . It is easy to see that the system (123) can be conveniently written as: dtdη = H, (140a) d Ω dη = − H Ω sin(2( t − ϑ )) f + 6(Ω − H cos ( t − ϑ ) + O ( H ) , (140b) dϑdη = − H sin ( t − ϑ ) f − H sin(2( t − ϑ )) + O ( H ) . (140c)and the averaged problem is: dtdη = H, d ¯Ω dη = − − ¯Ω) ¯Ω H , d ¯ ϑdη = 0 . (141)Now, the following expansion is proposed: t = t + Hα ( t , ϑ ) , (142a) Ω = Ω + H (cid:20) α ( t , ϑ ) − ηf sin(2( t − ϑ ))Ω (cid:21) , (142b) ϑ = ϑ + H (cid:20) α ( t , ϑ ) − ηf sin ( t − ϑ ) (cid:21) . (142c)Next, it is proved that the equations for t , Ω , ϑ have the same asymptotic than the averaged equations for ¯ t, ¯Ω , ¯ ϑ .After some algebraic manipulations and recalling that dHdη = − H cos ( t − ϑ ) = O ( H ) , (143) eneralized Scalar Field Cosmologies: a Perturbative Analysis dt dη = H − H α t + O (cid:0) H (cid:1) , (144a) d Ω dη = H (cid:110) − α t − sin(2( t − ϑ )) α + Ω (cid:2) t − ϑ ))( α − α ) + 2( η −
3) cos ( t − ϑ ) (cid:3) + 6Ω cos ( t − ϑ ) (cid:111) + O (cid:0) H (cid:1) , (144b) dϑ dη = H (cid:110) −
12 sin(2( t − ϑ ))(2 α − α + 3) − α t + 2 η sin( t − ϑ ) cos ( t − ϑ ) (cid:111) + O (cid:0) H (cid:1) . (144c) Imposing the conditions α t = 0 , α t = − sin(2( t − ϑ )) α , α t = −
12 sin(2( t − ϑ ))(2 α − α + 3) , (145)it follows: α ( t , ϑ ) = c ( ϑ ) , (146a) α ( t , ϑ ) = c ( ϑ ) + c ( ϑ ) e − cos(2( t − ϑ )) + 32 , (146b) α ( t , ϑ ) = c ( ϑ ) e cos(2( t − ϑ )) . (146c)Hence, the following equations are deduced: dt dη = H + O (cid:0) H (cid:1) , (147a) d Ω dη = H (cid:110) − (1 − Ω ) + 2Ω (cid:104) c ( ϑ ) e − cos(2( t − ϑ )) cos(2( t − ϑ )) + η cos ( t − ϑ ) (cid:105) − c ( ϑ ) sin( t − ϑ ) e cos(2( ϑ − t )) cos(2( ϑ − t )) + 3Ω (2 cos ( t − ϑ ) − (cid:111) + O (cid:0) H (cid:1) , (147b) dϑ dη = 2 ηH sin( t − ϑ ) cos ( t − ϑ ) + O (cid:0) H (cid:1) . (147c)From the equation ˙ H = − H cos ( t − ϑ ) + O ( H ) , (148)or its averaged version, it follows H is a monotonic decreasing function of t due to ≤ Ω , ¯Ω ≤ . This allows todefine recursively the sequences: η = 0 H = H ( η ) , η n +12 = η n + H n H n +1 = H ( η n +1 ) , (149)such that lim n →∞ H n = 0 and lim n →∞ η n = ∞ .Defining ∆ ϑ ( η ) = ϑ ( η ) − ¯ ϑ ( η ) and taking the same initial conditions at η = η n , ϑ ( η n ) = ¯ ϑ ( η n ) , it follows: | ∆ ϑ ( η ) | = (cid:12)(cid:12)(cid:12) (cid:90) ηη n [ ϑ (cid:48) ( s ) − ¯ ϑ (cid:48) ( s )] ds (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:90) ηη n s (cid:104) H sin( t − ϑ ) cos ( t − ϑ ) (cid:124) (cid:123)(cid:122) (cid:125) |·|≤ H n + O (cid:0) H (cid:1) (cid:105) ds (cid:12)(cid:12)(cid:12) , ≤ H n (cid:12)(cid:12)(cid:12) (cid:90) ηη n sds (cid:12)(cid:12)(cid:12) + O (cid:0) H (cid:1) ≤ H n | η + η n || η − η n | + O (cid:0) H (cid:1) , (150)for all η ≥ η n . Then, for η ∈ [ η n , η n +1 ] , it follows the inequality: | ∆ ϑ ( η ) | ≤ H n . eneralized Scalar Field Cosmologies: a Perturbative Analysis n → ∞ , it follows H n → , η n → ∞ , then, it follows lim η →∞ | ∆ ϑ ( η ) | = 0 . Thismeans that ϕ and ¯ ϕ have the same limit as η → ∞ . Without losing generality, are chosen c ( ϑ ) ≡ and c ( ϑ ) ≡ . Therefore, it follows: d Ω dη = H (cid:110) − (1 − Ω ) + 2Ω (cid:2) η cos ( t − ϑ ) (cid:3) + 3Ω (2 cos ( t − ϑ ) − (cid:111) + O (cid:0) H (cid:1) . (151)Defining ∆Ω = Ω − ¯Ω , it follows ∆Ω (cid:48) ( s ) = − H (1 − ¯Ω − Ω ) + 2 H Ω (cid:2) η cos ( t − ϑ ) (cid:3) + 3 H Ω (2 cos ( t − ϑ ) − . (152)Choosing the same initial conditions at η = η n , Ω ( η n ) = ¯Ω( η n ) , it follows: | ∆Ω( η ) | = (cid:12)(cid:12)(cid:12) (cid:90) ηη n [Ω (cid:48) ( s ) − ¯Ω (cid:48) ( s )] ds (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:90) ηη n − H (1 − ¯Ω − Ω ) (cid:124) (cid:123)(cid:122) (cid:125) |·|≤ +2 H s Ω cos ( t − ϑ ) (cid:124) (cid:123)(cid:122) (cid:125) |·|≤ (153) +3 H Ω (2 cos ( t − ϑ ) − (cid:124) (cid:123)(cid:122) (cid:125) |·|≤ + O (cid:0) H (cid:1) ds (cid:12)(cid:12)(cid:12) , ≤ H n (cid:90) ηη n (cid:12)(cid:12)(cid:12) ∆Ω( s ) (cid:12)(cid:12)(cid:12) ds + M H n | η + η n || η − η n | + 9 M H n | η − η n | , η ≥ η n , (154)where the bound M have been chosen as M = max s ∈ [ η n ,η ] (cid:110)(cid:12)(cid:12)(cid:12) Ω ( s ) (cid:12)(cid:12)(cid:12)(cid:111) , which exists due to the continuity of Ω on the compact set [ η n , η ] .Applying Gronwall’s Lemma Appendix A.1, it follows | ∆Ω( η ) | ≤ (cid:2) M H n | η + η n || η − η n | + 9 M H n | η − η n | (cid:3) (cid:104) H n ηe H n η (cid:105) + O ( H n )= (cid:2) M H n | η + η n || η − η n | + 9 M H n | η − η n | (cid:3) (cid:2) H n η (cid:3) + O ( H n )= M H n | η + η n || η − η n | + 9 M H n | η − η n | + O ( H n ) . (155)Then, for η ∈ [ η n , η n +1 ] and for n large enough such that | η + η n | ≥ , it follows: M | η + η n || η − η n | + 9 M | η − η n | ≤ (cid:0) M + 9 M (cid:1) | η − η n |≤ (cid:0) M + 9 M (cid:1) | η n +1 − η n | = (cid:0) M + 9 M (cid:1) H − n . Therefore, it follows the inequality | ∆Ω( η ) | ≤ KH n , for a positive constant K ≥ M + 9 M . Finally, taking the limit as n → ∞ , it follows H n → , η n → ∞ . Then,it follows lim η →∞ | ∆Ω( η ) | = 0 . This means that Ω and ¯Ω have the same limit as η → ∞ . V ( φ ) = φ + f cos (cid:16) φf (cid:17) , f > in presence of matter. In this section the averaging methods are applied to a scalar field cosmology with generalized harmonic potentialof the type V ( φ ) = φ + f cos( φf ) with f > in the presence of matter for FLRW metrics and for Bianchi Imetrics. eneralized Scalar Field Cosmologies: a Perturbative Analysis The field equations are: ¨ φ + 3 H ˙ φ + φ − sin (cid:18) φf (cid:19) = 0 , (156a) ˙ ρ m + 3 γHρ m = 0 , (156b) ˙ a = aH, (156c) ˙ H = − (cid:16) γρ m + ˙ φ (cid:17) + ka , (156d) H = 12 ˙ φ + φ f cos (cid:18) φf (cid:19) + ρ m − ka , k ∈ {− , } . (156e)Defining the amplitude-phase variables [112]: ˙ φ = r cos( t − ϑ ) , φ = r sin( t − ϑ ) , (157) ϑ = t − tan − (cid:18) φ ˙ φ (cid:19) , (158)and defining Ω = r H , Ω m = ρ m H , Ω k = − ka H , (159)satisfying f cos (cid:32) √ √ Ω H sin( t − ϑ ) f (cid:33) + 3 H (Ω + Ω k + Ω m −
1) = 0 , (160)it follows the dynamical system: P ( H ) : ˙ H = − H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) ˙Ω = Ω H (3 γ Ω m + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω + 2Ω k − √ √ Ω cos( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) H ˙Ω m = Ω m H (cid:0) γ (Ω m −
1) + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) ˙Ω k = Ω k H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k − (cid:1) ˙ ϑ = − H sin(2( t − ϑ )) + sin( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) √ √ Ω H . (161)Substituting √ √ Ω H sin( t − ϑ ) f = cos − (cid:18) H (1 − Ω − Ω k − Ω m ) f (cid:19) , (162)it follows: ˙ H = − H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) , (163a) ˙Ω = Ω H (3 γ Ω m + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω + 2Ω k − (cid:113) √ Ω cos( t − ϑ ) sin (cid:16) cos − (cid:16) H (1 − Ω − Ω k − Ω m ) f (cid:17)(cid:17) H , (163b) ˙Ω m = Ω m H (cid:0) γ (Ω m −
1) + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) , (163c) ˙Ω k = Ω k H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k − (cid:1) , (163d) ˙ ϑ = − H sin(2( t − ϑ )) + sin( t − ϑ ) sin (cid:16) cos − (cid:16) H (1 − Ω − Ω k − Ω m ) f (cid:17)(cid:17) √ √ Ω H . (163e)Considering only the leading terms of the system (163) when H → , the resulting system is: dtdη = H, d Ω dη = (cid:114) √ Ω cos( t − ϑ ) , dϑdη = sin( t − ϑ ) √ √ Ω . (164) eneralized Scalar Field Cosmologies: a Perturbative Analysis H is approximately a constant H (cid:28) , it follows the closed system: d Ω dη = (cid:114) √ Ω cos( ηH − ϑ ) , dϑdη = sin( ηH − ϑ ) √ √ Ω . (165)Integrating the first equation, it follows: Ω( η ) = 14 (cid:32) (cid:90) η (cid:114)
23 cos( H s − ϑ ( s )) ds (cid:33) . (166)For ϑ it follows the integral-differential equation: dϑdη = (cid:113) sin( ηH − ϑ ( η ))2Ω(0) (cid:82) η (cid:113) cos( H s − ϑ ( s )) ds , (167)that can be rewritten as the higher order differential equation: ϑ (cid:48)(cid:48) ( η ) = ϑ (cid:48) ( η ) ( H − ϑ (cid:48) ( η )) cot( ηH − ϑ ( η )) . (168)Now, the time variables η, ω = H η are used and treated them as independent variables. Using series expansionof the form ϑ ∼ ϑ ( η, ω ) + H ϑ ( η, ω ) + O ( H ) + . . . , (169)it follows − ϑ ηη + 2 ϑ η cot( ϑ )+ H (cid:40) − ϑ ηω − ϑ ηη + 2 ϑ η csc ( ϑ ) (cid:34) sin(2 ϑ ) ( ϑ ω + ϑ η ) − ϑ η ϑ (cid:35)(cid:41) + O ( H ) = 0 . (170)At order O (1) it follows the zeroth order equation: ϑ ηη = 2 ϑ η cot( ϑ ) , (171)with solution ϑ ( η, ω ) = − cot − ( c ( ω ) ( η + c ( ω ))) . (172)Substituting back at the equation at order O ( H ) it is obtained: − ϑ ηω − ϑ ηη + 2 ϑ η csc ( ϑ ) [sin(2 ϑ ) ( ϑ ω + ϑ η ) − ϑ η ϑ ] = 0 , (173)it follows c ( ω ) (( c ( ω ) + η ) (( c ( ω ) + η ) ϑ ηη + 4 ϑ η ) + 2 ϑ ) + ϑ ηη + 2 c (cid:48) ( ω ) = 0 . (174)The last equation is integrable, with general solution: ϑ ( η, ω ) = ηc ( ω ) + c ( ω ) c ( ω ) ( c ( ω ) + η ) + 1 − c (cid:48) ( ω ) c ( ω ) . (175)Hence, it follows: ϑ ( η ) = − cot − ( c ( ηH ) ( c ( ηH ) + η )) + H (cid:18) c ( ηH ) + ηc ( ηH ) c ( ηH ) ( c ( ηH ) + η ) + 1 − c (cid:48) ( ηH ) c ( ηH ) (cid:19) . (176)Finally, Ω( η ) is given by the quadrature (166).For the problem (163) it follows the averaged system: (cid:68) P ( H ) (cid:69) : ˙ H = − H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k )˙¯Ω = ¯Ω H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − m = ¯Ω m H (3 γ ( ¯Ω m −
1) + 3 ¯Ω + 2 ¯Ω k )˙¯Ω k = ¯Ω k H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − ϑ = 0 . (177) eneralized Scalar Field Cosmologies: a Perturbative Analysis ( ¯Ω , ¯Ω m , ¯Ω k ) Eigenvalues Stability P (0 , , { γ, γ − , γ − } nonhyperbolic for γ = 1 , , / Saddle for < γ < or < γ < / Sink for γ > / P (0 , , { , − , − γ } Saddle for γ < / or γ > / nonhyperbolic for γ = 2 / P (0 , , {− , − , − γ } Sink for γ > nonhyperbolic for γ = 0 P (1 , , { , , − γ − } Source for γ < Saddle for γ > nonhyperbolic for γ = 1 Table 1.
Stability criteria for the equilibrium point of (178).
Introducing the time variable τ = ln a , it follows the guiding system: ∂ τ ¯Ω = ¯Ω(3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − , (178a) ∂ τ ¯Ω m = ¯Ω m (3 γ ( ¯Ω m −
1) + 3 ¯Ω + 2 ¯Ω k ) , (178b) ∂ τ ¯Ω k = ¯Ω k (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − . (178c)The equilibrium points of the system (178) are P = (0 , , , P (0 , , , P = (0 , , and P = (1 , , . Theirstability criteria are summarized in Table 1. Figure 3.
Phase portrait of system (178) for γ = 1 . Furthermore, in Fig. 3 is shown that the origin is a sink as indicated in Table 1.
In this metric the field equations are given by: ¨ φ + 3 H ˙ φ + φ − sin (cid:18) φf (cid:19) = 0 , (179a) ˙ ρ m + 3 γHρ m = 0 , (179b) ˙ a = aH, (179c) ˙ H = − (cid:16) γρ m + ˙ φ (cid:17) − σ a , (179d) H = 12 ˙ φ + φ f cos (cid:18) φf (cid:19) + ρ m + σ a . (179e) eneralized Scalar Field Cosmologies: a Perturbative Analysis ˙ φ = r cos( t − ϑ ) , φ = r sin( t − ϑ ) , (180) ϑ = t − tan − (cid:18) φ ˙ φ (cid:19) , (181)and Ω = r H , Ω m = ρ m H , Σ = σ a H . (182)which satisfy f cos (cid:32) √ √ Ω H sin( t − ϑ ) f (cid:33) + H (cid:0) Σ + 3(Ω + Ω m − (cid:1) = 0 , (183)it follows the dynamical system: P ( H ) : ˙ H = − H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) ˙Ω = Ω H (cid:0) γ Ω m + Ω −
1) + 2Σ + 3(Ω −
1) cos(2( t − ϑ )) (cid:1) + √ √ Ω cos( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) H ˙Ω m = Ω m H (cid:0) γ (Ω m −
1) + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) ˙Σ = Σ H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) − (cid:1) ˙ ϑ = − H sin(2( t − ϑ )) + sin( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) √ √ Ω H . (184)Using the relation √ √ Ω H sin( t − ϑ ) f = cos − (cid:32) − H (cid:0) Σ + 3Ω + 3Ω m − (cid:1) f (cid:33) (185)it follows ˙ H = − H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) , (186a) ˙Ω = Ω H (cid:0) γ Ω m + Ω −
1) + 2Σ + 3(Ω −
1) cos(2( t − ϑ )) (cid:1)(cid:113) √ Ω cos( t − ϑ ) sin (cid:18) cos − (cid:18) H ( − Ω − Ω m ) − Σ ) f (cid:19)(cid:19) H , (186b) ˙Ω m = Ω m H (cid:0) γ (Ω m −
1) + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) , (186c) ˙Σ = 12 Σ H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) − (cid:1) , (186d) ˙ ϑ = − H sin(2( t − ϑ )) + sin( t − ϑ ) sin (cid:18) cos − (cid:18) H ( − Ω − Ω m ) − Σ ) f (cid:19)(cid:19) √ √ Ω H . (186e)For the problem (186), it follows the averaged system: (cid:68) P ( H ) (cid:69) : ˙ H = − H (cid:0) γ ¯Ω m + Ω) + 2 ¯Σ (cid:1) ˙¯Ω = ¯Ω H (cid:0) γ ¯Ω m + Ω −
1) + 2 ¯Σ (cid:1) ˙¯Ω m = ¯Ω m H (cid:0) γ ( ¯Ω m −
1) + 2 ¯Σ + 3 ¯Ω (cid:1) ˙¯Σ = ¯Σ H (cid:0) γ ¯Ω m + ¯Ω −
2) + 2 ¯Σ (cid:1) ˙¯ ϑ = 0 . (187)Introducing the time variable τ = ln a , it follows the guiding system: ∂ τ ¯Ω = ¯Ω (cid:0) γ ¯Ω m + Ω −
1) + 2 ¯Σ (cid:1) , (188a) ∂ τ ¯Ω m = ¯Ω m (cid:0) γ ( ¯Ω m −
1) + 2 ¯Σ + 3 ¯Ω (cid:1) , (188b) ∂ τ ¯Σ = 12 ¯Σ (cid:0) γ ¯Ω m + ¯Ω −
2) + 2 ¯Σ (cid:1) . (188c) eneralized Scalar Field Cosmologies: a Perturbative Analysis ( ¯Ω , ¯Ω m , Σ) Eigenvalues Stability P (0 , , { γ − , γ − , γ } saddle for < γ < or < γ < nonhyperbolic for γ = 0 , , P (1 , , { , − , − γ − } saddle for γ > or γ < nonhyperbolic for γ = 1 P (0 , , −√ { , , − γ − } source for γ < nonhyperbolic for γ = 2 P (0 , , √ { , , − γ − } saddle for γ < nonhyperbolic for γ = 2 P (0 , , {− , − , − γ } sink for γ > nonhyperbolic for γ = 0 Table 2.
Stability criteria for the equilibrium points of the system (188).
Observe that the system (188) is invariant under the change of coordinates Σ → − Σ , therefore it can be investigateonly one part of the phase portrait.The equilibrium points of the system (188) are P = (0 , , , P = (1 , , , P = (0 , , −√ , P = (0 , , √ and P = (0 , , . The stability criteria of them are summarized in Table 2. In figure 4, it can be corroborated thatthe origin is a sink as it was indicated in table 2. Figure 4.
Phase portrait of the system (188) for γ = 1 . V ( φ ) = φ + f (cid:104) − cos (cid:16) φf (cid:17)(cid:105) , f > in presence of matter. In this section, a scalar field cosmology is investigated in the presence of matter for FLRW metrics and forBianchi I metrics. The averaging methods are applied for a generalized harmonic potential of the type V ( φ ) = φ + f (cid:16) − cos (cid:16) φf (cid:17)(cid:17) with f > . In each case the stability criteria of their equilibrium points are obtained. For the FLRW metrics, the field equations are given by: ¨ φ + 3 H ˙ φ + φ + sin (cid:18) φf (cid:19) = 0 , (189a) ˙ ρ m + 3 γHρ m = 0 , (189b) ˙ a = aH, (189c) ˙ H = − (cid:16) γρ m + ˙ φ (cid:17) + ka , (189d) H = 12 ˙ φ + φ f (cid:20) − cos (cid:18) φf (cid:19)(cid:21) + ρ m − ka , k ∈ {− , } (189e)Introducing the amplitude-phase variables: ˙ φ = r cos( t − ϑ ) , φ = r sin( t − ϑ ) , (190) eneralized Scalar Field Cosmologies: a Perturbative Analysis ϑ = t − tan − (cid:18) φ ˙ φ (cid:19) , (191)and defining Ω = r H , Ω m = ρ m H , Ω k = − ka H , (192)such that f cos (cid:32) √ √ Ω H sin( t − ϑ ) f (cid:33) = f − H (1 − Ω − Ω k − Ω m ) , (193)it follows the dynamical system: P ( H ) : ˙ H = − H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) ˙Ω = Ω H (3 γ Ω m + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω + 2Ω k − − √ √ Ω cos( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) H ˙Ω m = Ω m H (cid:0) γ (Ω m −
1) + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) ˙Ω k = Ω k H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k − (cid:1) ˙ ϑ = − H sin(2( t − ϑ )) − sin( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) √ √ Ω H . (194)Using the relation √ √ Ω H sin( t − ϑ ) f = cos − (cid:18) f + 3 H (Ω + Ω k + Ω m − f (cid:19) , (195)it follows: ˙ H = − H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) , (196a) ˙Ω = Ω H (3 γ Ω m + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω + 2Ω k − − (cid:113) √ Ω cos( t − ϑ ) sin (cid:16) cos − (cid:16) f +3 H (Ω+Ω k +Ω m − f (cid:17)(cid:17) H , (196b) ˙Ω m = Ω m H (cid:0) γ (Ω m −
1) + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) , (196c) ˙Ω k = Ω k H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k − (cid:1) , (196d) ˙ ϑ = − H sin(2( t − ϑ )) − sin( t − ϑ ) sin (cid:16) cos − (cid:16) f +3 H (Ω+Ω k +Ω m − f (cid:17)(cid:17) √ √ Ω H . (196e)For the problem (196) it follows the averaged system: (cid:68) P ( H ) (cid:69) : ˙ H = − H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k )˙¯Ω = ¯Ω H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − m = ¯Ω m H (3 γ ( ¯Ω m −
1) + 3 ¯Ω + 2 ¯Ω k )˙¯Ω k = ¯Ω k H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − ϑ = 0 . (197)Introducing the time variable τ = ln a , it is obtained the guiding system: ∂ τ ¯Ω = ¯Ω(3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − , (198a) ∂ τ ¯Ω m = ¯Ω m (3 γ ( ¯Ω m −
1) + 3 ¯Ω + 2 ¯Ω k ) , (198b) ∂ τ ¯Ω k = ¯Ω k (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − . (198c)Observing that the unperturbed system is the same system as (178), therefore, it follows the same equilibriumpoints P = (0 , , , P (0 , , , P = (0 , , and P = (1 , , . Their stability conditions are summarized inTable 1. eneralized Scalar Field Cosmologies: a Perturbative Analysis For the Bianchi I metric the field equations become: ¨ φ + 3 H ˙ φ + φ + sin (cid:18) φf (cid:19) = 0 , (199a) ˙ ρ m + 3 γHρ m = 0 , (199b) ˙ a = aH, (199c) ˙ H = − (cid:16) γρ m + ˙ φ (cid:17) + σ a , (199d) H = 12 ˙ φ + φ f (cid:20) − cos (cid:18) φf (cid:19)(cid:21) + ρ m + σ a , (199e)Using the amplitude-phase transformations ˙ φ = r cos( t − ϑ ) , φ = r sin( t − ϑ ) , (200) ϑ = t − tan − (cid:18) φ ˙ φ (cid:19) , (201)and defining Ω = r H , Ω m = ρ m H , Σ = σ a H , (202)such that f cos (cid:32) √ √ Ω H sin( t − ϑ ) f (cid:33) = f − H (cid:0) − Ω − Ω m ) − Σ (cid:1) , (203)it is derived the dynamical system: P ( H ) : ˙ H = − H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) ˙Ω = Ω H (cid:0) γ Ω m + Ω −
1) + 2Σ + 3(Ω −
1) cos(2( t − ϑ )) (cid:1) − √ √ Ω cos( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) H ˙Ω m = Ω m H (cid:0) γ (Ω m −
1) + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) ˙Σ = Σ H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) − (cid:1) ˙ ϑ = − H sin(2( t − ϑ )) − sin( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) √ √ Ω H . (204)Substituting: √ √ Ω H sin( t − ϑ ) f = cos − (cid:32) f + H (cid:0) Σ + 3(Ω + Ω m − (cid:1) f (cid:33) , (205)it follows: ˙ H = − H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) , (206a) ˙Ω = Ω H (cid:0) γ Ω m + Ω −
1) + 2Σ + 3(Ω −
1) cos(2( t − ϑ )) (cid:1) , − (cid:113) √ Ω cos( t − ϑ ) sin (cid:18) cos − (cid:18) f + H ( Σ +3(Ω+Ω m − ) f (cid:19)(cid:19) H , (206b) ˙Ω m = Ω m H (cid:0) γ (Ω m −
1) + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) , (206c) ˙Σ = 12 Σ H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) − (cid:1) , (206d) ˙ ϑ = − H sin(2( t − ϑ )) − sin( t − ϑ ) sin (cid:18) cos − (cid:18) f + H ( Σ +3(Ω+Ω m − ) f (cid:19)(cid:19) √ √ Ω H . (206e) eneralized Scalar Field Cosmologies: a Perturbative Analysis (cid:68) P ( H ) (cid:69) : ˙ H = − H (cid:0) γ ¯Ω m + ¯Ω) + 2 ¯Σ (cid:1) ˙¯Ω = ¯Ω H (cid:0) γ ¯Ω m + ¯Ω −
1) + 2 ¯Σ (cid:1) ˙Ω m = ¯Ω m H (cid:0) γ ( ¯Ω m −
1) + 2 ¯Σ + 3 ¯Ω (cid:1) ˙¯Σ = ¯Σ H (cid:0) γ ¯Ω m + ¯Ω −
2) + 2 ¯Σ (cid:1) ˙¯ ϑ = 0 . (207)Introducing the time variable τ = ln a , it is obtained the guiding system: ∂ τ ¯Ω = ¯Ω (cid:0) γ ¯Ω m + ¯Ω −
1) + 2 ¯Σ (cid:1) (208a) ∂ τ Ω m = ¯Ω m (cid:0) γ ( ¯Ω m −
1) + 2 ¯Σ + 3 ¯Ω (cid:1) (208b) ∂ τ ¯Σ = 12 ¯Σ (cid:0) γ ¯Ω m + ¯Ω −
2) + 2 ¯Σ (cid:1) . (208c)The previous system is exactly the system (188). Therefore, the equilibrium points are the same: P = (0 , , , P = (1 , , , P = (0 , , −√ , P = (0 , , √ and P = (0 , , . The stability criteria of the equilibriumpoints of the system (208) are summarized in table 2. V ( φ ) = φ + f cos (cid:16) φf (cid:17) , f > nonminimally coupled tomatter with coupling χ ( φ ) = χ e λφ − γ . In this section the averaging methods are applied for FLRW metrics and Bianchi I metrics for the generalizedharmonic potential V ( φ ) = φ + f cos (cid:16) φf (cid:17) , f > , coupled to matter with coupling function χ = χ e λφ − γ ,where λ is a constant and γ (cid:54) = . In the following sections it will be studied the FLRW and Bianchi I modelsseparately. In this case the system (55) becomes: ¨ φ + 3 H ˙ φ + φ − sin (cid:18) φf (cid:19) = λ ρ m , (209a) ˙ ρ m + 3 γHρ m = − λ ρ m ˙ φ, (209b) ˙ a = aH, (209c) ˙ H = − (cid:16) γρ m + ˙ φ (cid:17) + ka , (209d) H = ρ m + 12 ˙ φ + φ f cos (cid:18) φf (cid:19) − ka , (209e)Using the amplitude-phase variables [112]: ˙ φ = r cos( t − ϑ ) , φ = r sin( t − ϑ ) , (210) ϑ = t − tan − (cid:18) φ ˙ φ (cid:19) , (211)it follows: ˙ r = ˙ φr (cid:104) ¨ φ + φ (cid:105) = ˙ φr (cid:20) − H ˙ φ + sin (cid:18) φf (cid:19) + λ ρ m (cid:21) = − Hr cos ( t − ϑ ) + cos( t − ϑ ) sin (cid:18) r sin( t − ϑ ) f (cid:19) + λ ρ m cos( t − ϑ ) , (212)and ˙ ϑ = φr (cid:104) ¨ φ + φ (cid:105) = φr (cid:20) − H ˙ φ + sin (cid:18) φf (cid:19) + λ ρ m (cid:21) + sin( t − ϑ ) sin (cid:16) r sin( t − ϑ ) f (cid:17) r − H sin( t − ϑ ) cos( t − ϑ ) + λ ρ m sin( t − ϑ ) r . (213) eneralized Scalar Field Cosmologies: a Perturbative Analysis ˙ r = − Hr cos ( t − ϑ ) + cos( t − ϑ ) sin (cid:18) r sin( t − ϑ ) f (cid:19) + λ ρ m cos( t − ϑ ) , (214a) ˙ ϑ = sin( t − ϑ ) sin (cid:16) r sin( t − ϑ ) f (cid:17) r − H sin( t − ϑ ) cos( t − ϑ ) + λ ρ m sin( t − ϑ ) r , (214b) ˙ ρ m = − γHρ m , (214c) ˙ a = aH, (214d) − ρ m − r − f cos (cid:18) r sin( t − ϑ ) f (cid:19) + 3 ka + 3 H = 0 . (214e)Defining Ω = r H , Ω m = ρ m H , Ω k = − ka H , (215)which satisfy f cos (cid:32) √ √ Ω H sin( t − ϑ ) f (cid:33) + 3 H (Ω + Ω k + Ω m −
1) = 0 , (216)it is deduced the dynamical system: P ( H ) : ˙ H = − H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) ˙Ω = √ √ Ω cos( t − ϑ ) sin (cid:18) √ √ Ω H sin( t − ϑ ) f (cid:19) H + H (cid:16) γ Ω m + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω + 2Ω k −
3) + √ λ √ ΩΩ m cos( t − ϑ ) (cid:17) ˙Ω m = Ω m H (cid:16) γ (Ω m − − √ λ √ Ω cos( t − ϑ ) + 6Ω cos(2( t − ϑ )) + 6Ω + 4Ω k (cid:17) ˙Ω k = Ω k H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k − (cid:1) )˙ ϑ = sin( t − ϑ ) sin (cid:18) √ √ Ω H sin( t − ϑ ) f (cid:19) √ √ Ω H + H (cid:16) √ λ Ω m sin( t − ϑ ) √ Ω − t − ϑ )) (cid:17) . (217) Substituting √ √ Ω H sin( t − ϑ ) f = cos − (cid:18) H (1 − Ω − Ω k − Ω m ) f (cid:19) , (218)it follows: ˙ H = − H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) , (219a) ˙Ω = (cid:113) √ Ω cos( t − ϑ ) sin (cid:16) cos − (cid:16) H (1 − Ω − Ω k − Ω m ) f (cid:17)(cid:17) H + 12 H (cid:16) γ Ω m + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω + 2Ω k −
3) + √ λ √ ΩΩ m cos( t − ϑ ) (cid:17) , (219b) ˙Ω m = 12 Ω m H (cid:16) γ (Ω m − − √ λ √ Ω cos( t − ϑ ) + 6Ω cos(2( t − ϑ )) + 6Ω + 4Ω k (cid:17) , (219c) ˙Ω k = Ω k H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k − (cid:1) , (219d) ˙ ϑ = sin( t − ϑ ) sin (cid:16) cos − (cid:16) H (1 − Ω − Ω k − Ω m ) f (cid:17)(cid:17) √ √ Ω H + 14 H (cid:18) √ λ Ω m sin( t − ϑ ) √ Ω − t − ϑ )) (cid:19) . (219e) Considering only the dominant terms of the system (219) as H → , it follows dtdη = H, d Ω dη = (cid:114) √ Ω cos( t − ϑ ) , dϑdη = sin( t − ϑ ) √ √ Ω . (220)Assuming H as approximately constant, H (cid:28) , it follows the system: d Ω dη = (cid:114) √ Ω cos( ηH − ϑ ) , dϑdη = sin( ηH − ϑ ) √ √ Ω . (221) eneralized Scalar Field Cosmologies: a Perturbative Analysis Ω( η ) = 14 (cid:32) (cid:90) η (cid:114)
23 cos( H s − ϑ ( s )) ds (cid:33) . (222)Furthermore, for ϑ it is deduced the higher order differential equation: ϑ (cid:48)(cid:48) ( η ) = ϑ (cid:48) ( η ) ( H − ϑ (cid:48) ( η )) cot( ηH − ϑ ( η )) . (223)As before, assuming H (cid:28) , introducing the time variables η, ω = H η treated as independent variables, andusing series expansion of the form ϑ ∼ ϑ ( η, ω ) + H ϑ ( η, ω ) + O ( H ) + . . . , (224)it follows the equation at order O (1) : ϑ ηη = 2 ϑ η cot( ϑ ) , (225)with solution ϑ ( η, ω ) = − cot − ( c ( ω ) ( η + c ( ω ))) . (226)Substituting back at the equation at order O ( H ) , − ϑ ηω − ϑ ηη + 2 ϑ η csc ( ϑ ) [sin(2 ϑ ) ( ϑ ω + ϑ η ) − ϑ η ϑ ] = 0 , (227)it follows: c ( ω ) (( c ( ω ) + η ) (( c ( ω ) + η ) ϑ ηη + 4 ϑ η ) + 2 ϑ ) + ϑ ηη + 2 c (cid:48) ( ω ) = 0 . (228)The last equation is integrable, with general solution ϑ ( η, ω ) = ηc ( ω ) + c ( ω ) c ( ω ) ( c ( ω ) + η ) + 1 − c (cid:48) ( ω ) c ( ω ) . (229)Hence, it follows ϑ ( η ) = − cot − ( c ( ηH ) ( c ( ηH ) + η )) + H (cid:18) c ( ηH ) + ηc ( ηH ) c ( ηH ) ( c ( ηH ) + η ) + 1 − c (cid:48) ( ηH ) c ( ηH ) (cid:19) . (230)Finally, Ω( η ) is given by the quadrature (222). These results are the same as discussed in section 3.3.1.Now it is implemented the averaging methods for a scalar field cosmology with generalized harmonic potentialof the type V ( φ ) = φ + f cos( φf ) , with f > minimally coupled to matter with coupling function χ = χ e λφ − γ ,where λ is a constant and γ (cid:54) = for the FLRW metrics.For the problem (219) it follows the averaged system: (cid:68) P ( H ) (cid:69) : ˙ H = − H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k )˙¯Ω = ¯Ω H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − m = ¯Ω m H (3 γ ( ¯Ω m −
1) + 3 ¯Ω + 2 ¯Ω k )˙¯Ω k = ¯Ω k H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − ϑ = 0 . (231)By using the new temporary variable τ = ln a , the guiding system is obtained: ∂ τ ¯Ω = ¯Ω(3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − , (232a) ∂ τ ¯Ω m = ¯Ω m (3 γ ( ¯Ω m −
1) + 3 ¯Ω + 2 ¯Ω k ) , (232b) ∂ τ ¯Ω k = ¯Ω k (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − . (232c)Analyzing the equilibrium points of the system (232) it follows the points P = (0 , , , P (0 , , , P = (0 , , and P = (1 , , . By evaluating the linearization of the system (232) and calculating the eigenvalues, it followsthe results summarized in the table 1. Let us note that the system (232) is exactly (178), which means that onaverage the qualitative behavior is the same, and the results are independent of the coupling function. eneralized Scalar Field Cosmologies: a Perturbative Analysis In this example the system (55) becomes: ¨ φ + 3 H ˙ φ + φ − sin (cid:18) φf (cid:19) = λ ρ m , (233a) ˙ ρ m + 3 γHρ m = − λ ρ m ˙ φ, (233b) ˙ a = aH, (233c) ˙ H = − (cid:16) γρ m + ˙ φ (cid:17) − σ a , (233d) H = 12 ˙ φ + φ f cos (cid:18) φf (cid:19) + ρ m + σ a (233e)Using the amplitude- phase transformation [112]: ˙ φ = r cos( t − ϑ ) , φ = r sin( t − ϑ ) , (234) ϑ = t − tan − (cid:18) φ ˙ φ (cid:19) , (235)and defining Ω = r H , Ω m = ρ m H , Σ = σ a H , (236)which satisfy f cos (cid:32) √ √ Ω H sin( t − ϑ ) f (cid:33) + H (cid:0) Σ + 3(Ω + Ω m − (cid:1) = 0 , (237)it follows the dynamical system: P ( H ) : ˙ H = − H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) ˙Ω = √ √ Ω cos( t − ϑ ) sin (cid:18) √ √ Ω H sin( t − ϑ ) f (cid:19) H + H (cid:16) (cid:0) γ Ω m + 2Σ + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω − (cid:1) + √ λ √ ΩΩ m cos( t − ϑ ) (cid:17) ˙Ω m = Ω m H (cid:16) γ (Ω m −
1) + 4Σ − √ λ √ Ω cos( t − ϑ ) + 6Ω cos(2( t − ϑ )) + 6Ω (cid:17) ˙Σ = Σ H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) − (cid:1) ˙ ϑ = sin( t − ϑ ) sin (cid:18) √ √ Ω H sin( t − ϑ ) f (cid:19) √ √ Ω H + H (cid:16) √ λ Ω m sin( t − ϑ ) √ Ω − t − ϑ )) (cid:17) . (238) Using the relation √ √ Ω H sin( t − ϑ ) f = cos − (cid:32) H (cid:0) − − m − Σ (cid:1) f (cid:33) (239)it follows ˙ H = − H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) , (240a) ˙Ω = (cid:113) √ Ω cos( t − ϑ ) sin (cid:18) cos − (cid:18) H ( − − m − Σ ) f (cid:19)(cid:19) H + 12 H (cid:16) (cid:0) γ Ω m + 2Σ + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω − (cid:1) + √ λ √ ΩΩ m cos( t − ϑ ) (cid:17) , (240b) ˙Ω m = 12 Ω m H (cid:16) γ (Ω m −
1) + 4Σ − √ λ √ Ω cos( t − ϑ ) + 6Ω cos(2( t − ϑ )) + 6Ω (cid:17) , (240c) ˙Σ = 12 Σ H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) − (cid:1) , (240d) ˙ ϑ = sin( t − ϑ ) sin (cid:18) cos − (cid:18) H ( − − m − Σ ) f (cid:19)(cid:19) √ √ Ω H + 14 H (cid:18) √ λ Ω m sin( t − ϑ ) √ Ω − t − ϑ )) (cid:19) . (240e) eneralized Scalar Field Cosmologies: a Perturbative Analysis (cid:68) P ( H ) (cid:69) : ˙ H = − H (cid:0) γ ¯Ω m + Ω) + 2 ¯Σ (cid:1) ˙¯Ω = ¯Ω H (cid:0) γ ¯Ω m + Ω −
1) + 2 ¯Σ (cid:1) ˙¯Ω m = ¯Ω m H (cid:0) γ ( ¯Ω m −
1) + 2 ¯Σ + 3 ¯Ω (cid:1) ˙¯Σ = ¯Σ H (cid:0) γ ¯Ω m + ¯Ω −
2) + 2 ¯Σ (cid:1) ˙¯ ϑ = 0 . (241)Introducing the new variable τ = ln a , it follows the guiding system: ∂ τ ¯Ω = ¯Ω (cid:0) γ ¯Ω m + Ω −
1) + 2 ¯Σ (cid:1) , (242a) ∂ τ ¯Ω m = ¯Ω m (cid:0) γ ( ¯Ω m −
1) + 2 ¯Σ + 3 ¯Ω (cid:1) , (242b) ∂ τ ¯Σ = 12 ¯Σ (cid:0) γ ¯Ω m + ¯Ω −
2) + 2 ¯Σ (cid:1) . (242c)Analyzing the equilibrium points of the system (242), which is the same as (188), it follows points of theform P = (0 , , , P = (1 , , , P = (0 , , −√ , P = (0 , , √ and P = (0 , , . By evaluating thepoints in the linearization of (242) and calculating their respective eigenvalues, it follows the results shown in thetable 2. This means that the dynamics of the models (242) and (188) is the same on average. This indicates thatthe asymptotic behavior of the model on average is independent of the coupling function and of the geometry ofmodel (FLRW, Bianchi I). Although, obviously, non-averaged systems have quite different dynamics. V ( φ ) = φ + f (cid:104) − cos (cid:16) φf (cid:17)(cid:105) , f > nonminimallycoupled to matter with coupling χ ( φ ) = χ e λφ − γ . In this section the averaging methods are applied for FLRW metrics and Bianchi I metrics for the generalizedharmonic potential V ( φ ) = φ + f (cid:104) − cos (cid:16) φf (cid:17)(cid:105) , f > coupled to matter with coupling function χ = χ e λφ − γ where λ is a constant and γ (cid:54) = . In the following sections the FLRW and Bianchi I models will be studiedseparately. In this case the field equations are: ¨ φ + 3 H ˙ φ + φ + sin (cid:18) φf (cid:19) = λ ρ m , (243a) ˙ ρ m + 3 γHρ m = − λ ρ m ˙ φ, (243b) ˙ a = aH, (243c) ˙ H = − (cid:16) γρ m + ˙ φ (cid:17) + ka , (243d) H = ρ m + 12 ˙ φ + φ f (cid:20) − cos (cid:18) φf (cid:19)(cid:21) − ka . (243e)Using the amplitude-phase variables [112]: ˙ φ = r cos( t − ϑ ) , φ = r sin( t − ϑ ) , (244) ϑ = t − tan − (cid:18) φ ˙ φ (cid:19) , (245)and defining Ω = r H , Ω m = ρ m H , Ω k = − ka H , (246)such that f cos (cid:32) √ √ Ω H sin( t − ϑ ) f (cid:33) = f − H (1 − Ω − Ω k − Ω m ) , (247) eneralized Scalar Field Cosmologies: a Perturbative Analysis P ( H ) : ˙ H = − H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) ˙Ω = H (cid:16) γ Ω m + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω + 2Ω k − √ λ √ ΩΩ m cos( t − ϑ ) (cid:17) − √ √ Ω cos( t − ϑ ) sin (cid:18) √ √ Ω H sin( t − ϑ ) f (cid:19) H ˙Ω m = Ω m H (cid:16) γ (Ω m − − √ λ √ Ω cos( t − ϑ ) + 6Ω cos(2( t − ϑ )) + 6Ω + 4Ω k (cid:17) ˙Ω k = Ω k H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k − (cid:1) ˙ ϑ = H (cid:16) √ λ Ω m sin( t − ϑ ) √ Ω − t − ϑ )) (cid:17) − sin( t − ϑ ) sin (cid:18) √ √ Ω H sin( t − ϑ ) f (cid:19) √ √ Ω H . (248) Using the restriction √ √ Ω H sin( t − ϑ ) f = cos − (cid:18) f + 3 H (Ω + Ω k + Ω m − f (cid:19) , (249)it follows ˙ H = − H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k (cid:1) , (250a) ˙Ω = 12 H (cid:16) γ Ω m + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω + 2Ω k −
3) + √ λ √ ΩΩ m cos( t − ϑ ) (cid:17) − (cid:113) √ Ω cos( t − ϑ ) sin (cid:16) cos − (cid:16) f +3 H (Ω+Ω k +Ω m − f (cid:17)(cid:17) H , (250b) ˙Ω m = 12 Ω m H (cid:16) γ (Ω m − − √ λ √ Ω cos( t − ϑ ) + 6Ω cos(2( t − ϑ )) + 6Ω + 4Ω k (cid:17) , (250c) ˙Ω k = Ω k H (cid:0) γ Ω m + 6Ω cos ( t − ϑ ) + 2Ω k − (cid:1) , (250d) ˙ ϑ = 14 H (cid:18) √ λ Ω m sin( t − ϑ ) √ Ω − t − ϑ )) (cid:19) − sin( t − ϑ ) sin (cid:16) cos − (cid:16) f +3 H (Ω+Ω k +Ω m − f (cid:17)(cid:17) √ √ Ω H . (250e)
For the problem (250) it follows the averaged system: (cid:68) P ( H ) (cid:69) : ˙ H = − H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k )˙¯Ω = ¯Ω H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − m = ¯Ω m H (3 γ ( ¯Ω m −
1) + 3 ¯Ω + 2 ¯Ω k )˙¯Ω k = ¯Ω k H (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − ϑ = 0 . (251)Introducing the time variable τ = ln a , it follows the guiding system: ∂ τ ¯Ω = ¯Ω(3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − , (252a) ∂ τ ¯Ω m = ¯Ω m (3 γ ( ¯Ω m −
1) + 3 ¯Ω + 2 ¯Ω k ) , (252b) ∂ τ ¯Ω k = ¯Ω k (3 γ ¯Ω m + 3 ¯Ω + 2 ¯Ω k − . (252c)Analyzing the equilibrium points of the system (252) it follows the points of the form P = (0 , , , P = (0 , , , P = (0 , , and P = (1 , , . By evaluating the linearization of the system (252) and calculating theeigenvalues, it follows the results that are summarized in the table 1. Let us observe that the system (252) isexactly (178) which means that on average the qualitative behavior is the same and the results are independent ofthe coupling function. eneralized Scalar Field Cosmologies: a Perturbative Analysis In this case the field equations are: ¨ φ + 3 H ˙ φ + φ + sin (cid:18) φf (cid:19) = λ ρ m , (253a) ˙ ρ m + 3 γHρ m = − λ ρ m ˙ φ, (253b) ˙ a = aH, (253c) ˙ H = − (cid:16) γρ m + ˙ φ (cid:17) − σ a , (253d) H = ρ m + 12 ˙ φ + φ f (cid:20) − cos (cid:18) φf (cid:19)(cid:21) + σ a . (253e)Using the amplitude-phase variables [112]: ˙ φ = r cos( t − ϑ ) , φ = r sin( t − ϑ ) , (254) ϑ = t − tan − (cid:18) φ ˙ φ (cid:19) , (255)and defining Ω = r H , Ω m = ρ m H , Σ = σ a H . (256)such that f cos (cid:32) √ √ Ω H sin( t − ϑ ) f (cid:33) = f − H (cid:0) − Ω − Ω m ) − Σ (cid:1) , (257)it follows the dynamical system P ( H ) : ˙ H = − H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) ˙Ω = H (cid:16) (cid:0) γ Ω m + 2Σ + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω − (cid:1) + √ λ √ ΩΩ m cos( t − ϑ ) (cid:17) − √ √ Ω cos( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) H ˙Ω m = Ω m H (cid:16) γ (Ω m −
1) + 4Σ − √ λ √ Ω cos( t − ϑ ) + 6Ω cos(2( t − ϑ )) + 6Ω (cid:17) ˙Σ = Σ H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) − (cid:1) H (cid:16) √ λ Ω m sin( t − ϑ ) √ Ω − t − ϑ )) (cid:17) − sin( t − ϑ ) sin (cid:16) √ √ Ω H sin( t − ϑ ) f (cid:17) √ √ Ω H . (258)Using the substitution √ √ Ω H sin( t − ϑ ) f = cos − (cid:32) f − H (cid:0) − Ω − Ω m ) − Σ (cid:1) f (cid:33) (259)it follows ˙ H = − H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) (cid:1) , (260a) ˙Ω = 12 H (cid:16) (cid:0) γ Ω m + 2Σ + 3(Ω −
1) cos(2( t − ϑ )) + 3Ω − (cid:1) + √ λ √ ΩΩ m cos( t − ϑ ) (cid:17) − (cid:113) √ Ω cos( t − ϑ ) sin (cid:18) cos − (cid:18) f − H ( − Ω − Ω m ) − Σ ) f (cid:19)(cid:19) H , (260b) ˙Ω m = 12 Ω m H (cid:16) γ (Ω m −
1) + 4Σ − √ λ √ Ω cos( t − ϑ ) + 6Ω cos(2( t − ϑ )) + 6Ω (cid:17) , (260c) ˙Σ = 12 Σ H (cid:0) γ Ω m + 2Σ + 6Ω cos ( t − ϑ ) − (cid:1) , (260d) ˙ ϑ = 14 H (cid:18) √ λ Ω m sin( t − ϑ ) √ Ω − t − ϑ )) (cid:19) − sin( t − ϑ ) sin (cid:18) cos − (cid:18) f − H ( − Ω − Ω m ) − Σ ) f (cid:19)(cid:19) √ √ Ω H (260e) eneralized Scalar Field Cosmologies: a Perturbative Analysis (cid:68) P ( H ) (cid:69) : ˙ H = − H (cid:0) γ ¯Ω m + Ω) + 2 ¯Σ (cid:1) ˙¯Ω = ¯Ω H (cid:0) γ ¯Ω m + Ω −
1) + 2 ¯Σ (cid:1) ˙¯Ω m = ¯Ω m H (cid:0) γ ( ¯Ω m −
1) + 2 ¯Σ + 3 ¯Ω (cid:1) ˙¯Σ = ¯Σ H (cid:0) γ ¯Ω m + ¯Ω −
2) + 2 ¯Σ (cid:1) ˙¯ ϑ = 0 . (261)Introducing the time variable τ = ln a , it follows the guiding system: ∂ τ ¯Ω = ¯Ω (cid:0) γ ¯Ω m + Ω −
1) + 2 ¯Σ (cid:1) , (262a) ∂ τ ¯Ω m = ¯Ω m (cid:0) γ ( ¯Ω m −
1) + 2 ¯Σ + 3 ¯Ω (cid:1) , (262b) ∂ τ ¯Σ = 12 ¯Σ (cid:0) γ ¯Ω m + ¯Ω −
2) + 2 ¯Σ (cid:1) . (262c)Analyzing the equilibrium points of the system (262), which is the same as (188), it follows points of the form P = (0 , , , P = (1 , , , P = (0 , , −√ , P = (0 , , √ and P = (0 , , . By evaluating the points inthe linearization of (262) and calculating their respective eigenvalues, it follows the results shown in the table (2).This means that on average the dynamics of the models (262), (242) and (188) is the same which indicates thatthe asymptotic behavior of the model on average is independent of the coupling function, of the model geometry(FLRW, Bianchi I). Although, obviously, non-averaged systems have different dynamics.
4. Results and Conclusions
This paper was devoted to the study of perturbation problems in scalar field cosmologies in vacuum and withmatter, with minimal and non-minimal couplings. There were investigated scalar field cosmologies in Friedmann-Lemaitre Robertson-Walker metrics and Bianchi I metric. There were used qualitative techniques, asymptoticmethods and averaging theory to obtain relevant information about the solution’s space of the aforementionedcosmologies. There were discussed some basic examples of applications of perturbation techniques. There werecommented on the concepts of fast and slow systems with respect to an small parameter. Combining results ofthese two limiting problems, information of the dynamics for small values of ε is obtained. This technique isused to construct uniformly valid approximations to the solutions of perturbation problems in which the solutionsdepend simultaneously on quite different scales by using seed solutions which satisfy some version of the originalequations in the limit of ε → .There were illustrated the expansion in ε technique to solve approximately the initial value problem with t > : d ydt = − εy ) , y (0) = 0 , y (cid:48) (0) = 0 . (263)The approximated solution is given by y ( t ) ∼ t (cid:0) − t (cid:1) + εt (cid:0) − t (cid:1) , and it is uniformly valid as ε → .Next, it was illustrated the two-timing technique to a classical textbook example [110], given by the ordinarydifferential equation y (cid:48)(cid:48) + εy (cid:48) + y = 0 , t > , y (0) = 0 , y (cid:48) (0) = 1 . (264)In this method one incorporates the two scales in the model by introducing the time variables t, τ = εt , and treatthem as independent variables. After that, one uses Taylor expansions in the parameter ε . Then, one solves theequations order by order eliminating secular terms. In this classical example a good approximated solution of theproblem is y ∼ e − τ sin( t ) , (265)valid up to the first order of ε .As per we are interested the cosmological set-up, the two-timing procedure was illustrated with the so-calledinduced gravity model with the action [113, 114]: S IG = (cid:90) √− g (cid:18) σ ω R − g µν ∂ µ σ∂ ν σ − γ U σ − γ (cid:19) , (266) eneralized Scalar Field Cosmologies: a Perturbative Analysis ω > , γ ≥ . A massless scalar field was added to the action in [115]: S IGφ = S IG + (cid:90) √− g (cid:18) − g µν ∂ µ φ∂ ν φ (cid:19) . (267)To define the order parameter ε it was started by the equation of motion for a massless scalar field that is given by ¨ φ + 3 ˙ aa ˙ φ = 0 , (268)and it admits the solution ˙ φ = εa − , where ε is an integration constant, that it was set ε → for a slow-varyingscalar field.Using the parametrization [113]: a = σ − exp( u + v ) , (269a) σ = exp( A ( u − v )) , (269b)with A = (cid:113) γ, and applying the two-timing method it was obtained the first order approximations: u ( t ; ε ) = c − (cid:40) √ γ − , γ < √ γ − , γ ≥ . + ε √ c tanh(∆) √ U (6 − γ ) , γ < √ c tan(∆) √ U ( γ − , γ ≥ O ( ε ) , (270a) v ( t ; ε ) = c + (cid:40) √ γ +6 , γ < √ γ +6 , γ ≥ ε − ( √ γ − √ ) c coth(∆) ( √ γ +6 ) √ U (6 − γ ) , γ < ( √ γ − √ ) c cot(∆) ( √ γ +6 ) √ U ( γ − , γ ≥ O ( ε ) , (270b)where ∆ := ∆( t ) = (24 c + t ) √ U √ | γ − | √ . These solutions are valid on the time scale εt = O (1) .Regarding the cosmological applications of these techniques (which is the core of the present research) therewere obtained the following results.In sections 3.1 and 3.2 were analyzed the periodic solutions for potentials V ( φ ) = f cos (cid:16) φf (cid:17) + φ , and V ( φ ) = − f cos (cid:16) φf (cid:17) + f + φ , respectively, with minimal coupling to the matter field using the PerturbationTheory Methods.In sections 3.5 and 3.6 there were considered scalar fields with potentials V ( φ ) and V ( φ ) as defined above,respectively. It was considered an interaction with the background matter with strength Q = λρ m arising from thecoupling function χ ( φ ) = χ e λφ − γ . Using averaging methods for periodic functions of a given period T , it canbe concluded that regardless of whether the scalar field is minimally or not minimally coupled to the matter field(at least for interactions of the type Q = λρ m ) there is not difference in dynamics when performing the averagingprocess. Obviously, non averaged systems have quite different dynamics. This indicates that the asymptotic resultswhen H → are independent of the coupling function. In addition, when comparing the averaged systemsfor the FLRW metrics and the Bianchi I metric, the stability results are qualitatively the same as for the averagedsystems, indicating that the asymptotic behavior when H → of the averaged model is independent of the couplingfunction, and in addition of the geometry (FLRW, Bianchi I). Acknowledgments
Genly Leon and Felipe Orlando Franz Silva have the financial support of Agencia Nacional de Investigación yDesarrollo - ANID through the program FONDECYT Iniciación grant no. 11180126. Additionally, this researchis funded by Vicerrectoría de Investigación y Desarrollo Tecnológico at Universidad Católica del Norte. AlfredoDavid Millano Mejías is acknowledged by helpful comments and encouraging discussions. Ellen de los M.Fernández Flores is acknowledged for proofreading this manuscript and improving the English. eneralized Scalar Field Cosmologies: a Perturbative Analysis Appendix A. Gronwall’s LemmaLemma Appendix A.1.
Gronwall’s Lemma (i)
Differential form .i) Let be η ( t ) a nonnegative function, absolutely continuous over [0 , T ] , which satisfies almost everywherethe differential inequality η (cid:48) ( t ) ≤ φ ( t ) η ( t ) + ψ ( t ) , (A.1) where φ ( t ) and ψ ( t ) are nonnegative summable functions over [0 , T ] . Then, η ( t ) ≤ e (cid:82) t φ ( s ) ds (cid:20) η (0) + (cid:90) t ψ ( s ) ds (cid:21) , (A.2) is satisfied for all ≤ t ≤ T .ii) In particular, if η (cid:48) ( t ) ≤ φ ( t ) η ( t ) , t ∈ [0 , T ] , η (0) = 0 , (A.3) then η ≡ , t ∈ [0 , T ] . (A.4) (ii) Integral form i) Let be ξ ( t ) a nonnegative function, summable over [0 , T ] which satisfies almost everywhere the integralinequality ξ ( t ) ≤ C (cid:90) t ξ ( s ) ds + C , C , C ≥ . (A.5) Then ξ ( t ) ≤ C (1 + C te C t ) , (A.6) almost everywhere for t in ≤ t ≤ T .ii) In particular, if ξ ( t ) ≤ C (cid:90) t ξ ( s ) ds, C ≥ , (A.7) almost everywhere for t in ≤ t ≤ T , then η ≡ , (A.8) almost everywhere for t in ≤ t ≤ T . [1] C. Brans and R. H. 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