Averaging generalized scalar field cosmologies III: Kantowski--Sachs and closed Friedmann--Lemaître--Robertson--Walker models
Genly Leon, Esteban González, Samuel Lepe, Claudio Michea, Alfredo D. Millano
AAveraging Generalized Scalar Field Cosmologies III: Kantowski-Sachs and ClosedFriedmann-Lemaˆıtre-Robertson-Walker Models
Genly Leon, ∗ Esteban Gonz´alez,
2, †
Samuel Lepe,
3, ‡
Claudio Michea,
1, § and Alfredo D. Millano
1, ¶ Departamento de Matem´aticas, Universidad Cat´olica del Norte,Avda. Angamos 0610, Casilla 1280 Antofagasta, Chile Universidad de Santiago de Chile (USACH), Facultad de Ciencia, Departamento de F´ısica, Chile. Instituto de F´ısica, Facultad de Ciencias, Pontificia Universidad Cat´olica de Valpara´ıso, Av. Brasil 2950, Valpara´ıso, Chile (Dated: February 16, 2021)Scalar field cosmologies with a generalized harmonic potential and a background matter given by abarotropic Equation of State (EoS) are investigated for Kantowski-Sachs metric and closed Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metrics. Using methods from the Theory of Averaging of Nonlinear Dy-namical Systems and numerical simulations it is proved that the full equations of the time dependent system andtheir corresponding time-averaged versions have the same late-time dynamics. Therefore, the simplest time-averaged system determines the future asymptotic of the full system. In particular, for Kantowski-Sachs metricthe late-time attractors of the full and time-averaged systems are: two anisotropic contracting solutions for0 ≤ γ <
2: a non-Flat LRS Kasner Bianchi I and a Taub (flat LRS Kasner) and the flat matter dominated FLRWuniverse if 0 ≤ γ ≤ (mimicking de Sitter, quintessence or zero acceleration solutions). For closed FLRWmetric the late-time attractors are: the flat Friedman Einstein-de-Sitter solution for 0 < γ <
1, the flat matterdominated FLRW universe for 0 ≤ γ ≤ (mimicking de Sitter, quintessence or zero acceleration solutions), anda matter dominated contracting isotropic solution for 1 < γ <
2. With this approach the oscillations enteringthe system through Klein-Gordon equation can be controlled and smoothed out as a relevant time-dependentperturbative parameter goes monotonically to zero for a finite time t < t ∗ . The region where the perturbation pa-rameter changes its monotony to monotonic increasing is analyzed by a discrete symmetry and by using propervariables that brings infinity to a finite interval. PACS numbers: 98.80.-k, 95.35.+d, 95.36.+xKeywords: Generalized scalar field cosmologies, Anisotropic models, Early universe, Equilibrium-points, Harmonic oscillator
Contents
1. Introduction
2. Spatial homogeneous and anisotropic scalarfield cosmologies
3. Averaging scalar field cosmologies
4. Qualitative analysis of time-averaged systems
5. Analysis as D → ∞ for Kantowski-Sachs and ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] closed FLRW models
6. Conclusions Acknowledgements A. Main equations B. Proof of Theorem 3
C. Numerical simulation
References
1. INTRODUCTION
Scalar fields with arbitrary potentials and arbitrarycouplings have played important roles in the physicaldescription of the Universe [1–16]. These models canbe examined by means of qualitative techniques of dy-namical systems [18–34], which permit the stabilityanalysis of the solutions. Complementary, Asymptotic a r X i v : . [ g r- q c ] F e b methods and Averaging Theory [35–41] are proved tobe helpful to obtain relevant information about the so-lution’s space of scalar field cosmologies: (i) in thevacuum, (ii) in the presence of matter [42, 43]. In thisprocess, one idea is to construct a time-averaged ver-sion of the original system. Solving it the oscillationsof the original system are smoothed out [46]. This canbe achieved for Bianchi I, flat FLRW, Bianchi III andnegatively curved FLRW metrics, where the Hubbleparameter H plays the role of a time dependent per-turbation parameter which controls the magnitude ofthe error between the solutions of the full and the time-averaged problems as H → γ minimally coupled to a scalarfield with generalized harmonic potential V ( φ ) = µ (cid:104) φ µ + b f (cid:16) − cos (cid:16) φ f (cid:17)(cid:17)(cid:105) , b > , inspired in thepotential of [51, 52]. This research program has threesteps according to the three cases of study: (I) BianchiIII and Open FLRW model [42], (II) Bianchi I andFlat FLRW model [43] and (III) Kantowski-Sachs andClosed FLRW.In Paper I [42] was proved that late-time attractorsof the original and time-averaged systems for LRSBianchi III are: a matter dominated flat FLRW uni-verse if 0 ≤ γ ≤ (mimicking de Sitter, quintessenceor zero acceleration solutions), a matter-curvature scal-ing solution if < γ < ≤ γ ≤
2. For FLRW metric with k = − ≤ γ ≤ (mimicking de Sitter,quintessence or zero acceleration solutions) and theMilne solution if < γ <
2. In all metrics, the matterdominated flat FLRW universe represents quintessencefluid if 0 < γ < .Following this research program in paper II [43] wasstudied the case (II) Bianchi I and flat FLRW model. Itwas proved that the late-time attractors of the originaland time-averaged systems for both metrics are: theflat matter dominated FLRW universe for 0 ≤ γ < < γ ≤ t and c = c can be associated withthe flat Friedman Einstein-de-Sitter solution.Paper III is devoted to the case (III) Kantowski-Sachs metrics and positively curved FLRW models.We will prove that the quantity D = (cid:113) H + K , where K is the Gauss spatial curvature, plays the role of atime dependent perturbation parameter which controlsthe magnitude of the error between the solutions ofthe full and the time-averaged problems. The anal-ysis of the system is therefore reduced to study thecorresponding time-averaged equations. With this ap-proach the oscillations of the scalar field through Klein-Gordon equation can be controlled and smoothed outas the time-dependent perturbative parameter D goesmonotonically to zero for a finite time t < t ∗ . The re-gion where the perturbation parameter D changes itsmonotony from monotonic decreasing to monotonic in-creasing is analyzed by a discrete symmetry and by us-ing proper variables that brings infinity to a finite inter-val is examined the limit D → + ∞ .The paper is organized as follows. In section 2 weintroduce the model under study. In section 3 we ap-ply Averaging Methods to analyze the periodic solu-tions of a scalar field with self-interacting potentialswithin the class of generalized harmonic potentials[50]. In particular, section 3.1 is devoted to the maintheorem proved in [42] for LRS Bianchi III and neg-atively curved FLRW and in section 3.2 is presentedthe main theorem proved in [43] for LRS Bianchi I andflat FLRW. In section 3.3 is studied Kantowski-Sachsmodel by using D -normalization, rather than Hubble-normalization, because the Hubble factor is not mono-tonic for closed universes. FLRW models with k = + k = + D → ∞ for Kantowski-Sachs andclosed FLRW models. Finally, in section 6 are dis-cussed our main results. Main equations are presentedin appendix A. In appendix B is given the Proof of maintheorem. In appendix C we present numerical evidencesupporting the results of section 3.
2. SPATIAL HOMOGENEOUS AND ANISOTROPICSCALAR FIELD COSMOLOGIES
The spatial homogeneous but anisotropic spacetimesare known as either Bianchi or Kantowski-Sachs (KS)cosmologies. In Bianchi models, the spacetime mani-fold is foliated along with the time axis with three di-mensional homogeneous hypersurfaces. On the otherhand, the isometry group of KS spacetime is R × SO ( ) and does not act simply transitively on spacetime, nordoes it possess a subgroup with simple transitive ac-tion. Hence, this model is spatially homogeneousbut does not belong to the Bianchi classification. KSmodel, when isotropizes, approaches a closed FLRWmodel [53–56]. In General Relativity (GR) the Hub-ble parameter H is always monotonic for Bianchi I andBianchi III and for Bianchi I the anisotropy decays ontime for H >
0. Therefore, isotropization occurs [57].For Kantowski-Sachs as well as for closed FLRW, theHubble parameter is not in general monotonic andanisotropies would increase rather than vanish as H change the sign. We refer the reader to [58–83], andreferences therein, for applications of KS models, spa-tially homogeneous and LRS metrics. The typical be-havior of KS metric for perfect fluids, Vlasov matter,etc., is that the generic solutions are past and futureasymptotic to the non-flat LRS Kasner vacuum solu-tion, which have a big-bang (or big-crunch). There arenon-generic solutions which are past (future) asymp-totic to a non-isotropic Bianchi I matter solution. (c)There are non-generic solutions which are past (fu-ture) asymptotic to the flat Friedman matter solution.The qualitative properties of positive-curvature mod-els and the KS models with a barotropic fluid and anon-interacting scalar field with exponential potential V ( φ ) = Λ e k φ were examined, e.g., in [84]. The mainresults are the following. For positively curved FLRWmodels, and for k >
2, all the solutions start from andrecollapse to a singularity. In general these solutionsare not inflationary. For k < k >
2, all the solutions start from and recollapse toa singularity; these solutions are not inflationary nei-ther isotropizes. For k < k > k < S = (cid:90) d x (cid:112) | g | (cid:20) R + L φ + L m (cid:21) , (1)expressed in a system of units in which 8 π G = c = (cid:125) =
1. In equation (1), R is the curvature scalar and φ is thescalar field with Lagrangian density L φ = − g µν ∇ µ φ ∇ ν φ − V ( φ ) . (2) ∇ α is the covariant derivative and the potential is V ( φ ) = µ (cid:20) b f (cid:18) − cos (cid:18) φ f (cid:19)(cid:19) + φ µ (cid:21) , b > . (3)The metric element in hyper-spherical curvature-normalized coordinates [ t , r , ϑ , ζ ] is: ds = − dt + (cid:2) e ( t ) (cid:3) − dr + (cid:2) e ( t ) (cid:3) − (cid:2) d ϑ + sin ( ϑ ) d ζ (cid:3) , (4)where e , e and e = e / sin ( ϑ ) are functions of t which are the components of the frame vectors [82].This family of metrics contains the Kantowski-Sachsmetric ( e ( t ) (cid:54) = e ( t ) ) or the Closed (positive curva-ture) universe ( k = +
1) when e ( t ) = e ( t ) = a ( t ) − (where a ( t ) is the scale factor) [59]. In the anisotropiccases are defined the average scale factor a ( t ) and theanisotropic parameter σ + through˙ aa = H : = − ddt ln (cid:2) e ( t )( e ( t )) (cid:3) , (5) σ + = ddt ln (cid:2) e ( t )( e ( t )) − (cid:3) . (6)Taking variation of (1), for the 1-parameter family ofmetrics (4) leads to [59]:3 H + K = σ + + ρ m +
12 ˙ φ + V ( φ ) , (7) − ( σ + + H ) − σ + − H − K = ( γ − ) ρ m +
12 ˙ φ − V ( φ ) , (8) − σ + + σ + H − H + ˙ σ + − H = ( γ − ) ρ m +
12 ˙ φ − V ( φ ) , (9)where it is used for the matter component a barotropicEoS p m = ( γ − ) ρ m , with p m the pressure of the mattercomponent, ρ m the energy density and γ is known asbarotropic index with 0 ≤ γ ≤ R = H + σ + + H + K , (10)while the Gauss curvature and 3-curvature scalar are K = ( e ( t )) , ( ) R = K . (11)Furthermore, the evolution of Gauss curvature is˙ K = − ( σ + + H ) K , (12)while the evolution for e is given by [82]:˙ e = − ( H − σ + ) e . (13)Combining (8) and (9) is obtained the shear evolutionequation ˙ σ + = − H σ + − K . (14)Furthermore, relations (7), (8), (9) and (14) give theRaychaudhuri equation˙ H = − H − σ + − ( γ − ) ρ m −
13 ˙ φ + V ( φ ) . (15)Finally, the evolution equation for matter is˙ ρ m = − γ H ρ m , (16)and the Klein-Gordon equation is¨ φ = − H ˙ φ − dV ( φ ) d φ . (17)The potential (3) can be expressed as V ( φ ) = µ φ + f (cid:0) ω − µ (cid:1) (cid:18) − cos (cid:18) φ f (cid:19)(cid:19) . (18) To obtain (18) we have imposed the condition b µ + f µ − f ω = ω ∈ R and satisfies ω − µ > φ =
0, we have V ( φ ) ∼ ω φ + O (cid:0) φ (cid:1) , as φ →
0, where ω can be relatedto the mass of the scalar field near its global minimum.As φ → ± ∞ the cosine- correction is bounded, then, V ( φ ) ∼ µ φ + O ( ) as φ → ± ∞ .
3. AVERAGING SCALAR FIELD COSMOLOGIES
The term averaging here is related to the approxima-tion of initial value problems in ordinary differentialequations involving perturbations (chapter 11, [41]).
In the reference [42] we have deduced the time-averaged equations:˙¯ Ω = H ¯ Ω (cid:0) − γ (cid:0) ¯ Σ + ¯ Ω + ¯ Ω k − (cid:1) + Σ + Ω + Ω k − (cid:1) , (19)˙¯ Σ = H (cid:0) ¯ Σ (cid:0) − γ (cid:0) ¯ Σ + ¯ Ω + ¯ Ω k − (cid:1) + Σ + Ω + Ω k − (cid:1) + Ω k (cid:1) , (20)˙¯ Ω k = − H ¯ Ω k (cid:0) γ (cid:0) ¯ Σ + ¯ Ω + ¯ Ω k − (cid:1) − Σ + Σ − Ω − Ω k + (cid:1) , (21)˙¯ Φ = , (22)˙ H = − H (cid:16) γ (cid:0) − ¯ Σ − ¯ Ω − ¯ Ω k (cid:1) + Σ + Ω + Ω k (cid:17) . (23) Theorem 1
Let be defined the functions ¯ Ω , ¯ Σ , ¯ Ω k , ¯ Φ ,and H, which satisfy the time-averaged equations (19) , (20) , (21) , (22) and (23) . Then, there exists functionsg , g , g and g , such that if Ω = Ω + Hg ( H , Ω , Σ , Ω k , Φ , t ) , Σ = Σ + Hg ( H , Ω , Σ , Ω k , Φ , t ) , Ω k = Ω k + Hg ( H , Ω , Σ , Ω k , Φ , t ) , Φ = Φ + Hg ( H , Ω , Σ , Ω k , Φ , t ) , (24) the functions Ω , Σ , Ω k , Φ and ¯ Ω , ¯ Σ , ¯ Ω k , ¯ Φ have thesame limit as t → ∞ . Setting Σ = Σ = , are derivedanalogous results for negatively curved FLRW model. Theorem 1 implies that Ω , Σ , Ω k , Φ evolve according tothe time-averaged equations (19), (20), (21) and (22) as H →
0. The proof is given in [42].
In the reference [42] we have deduced the time-averaged equations:˙¯ Ω = H ¯ Ω (cid:0) γ (cid:0) − ¯ Σ − ¯ Ω (cid:1) + Σ + ¯ Ω − (cid:1) , (25)˙¯ Σ = H ¯ Σ (cid:0) γ (cid:0) − ¯ Σ − ¯ Ω (cid:1) + Σ + ¯ Ω − (cid:1) , (26)˙¯ Φ = , (27)˙ H = − H (cid:0) γ (cid:0) − ¯ Σ − ¯ Ω (cid:1) + Σ + ¯ Ω (cid:1) . (28) Theorem 2
Let be defined the functions ¯ Ω , ¯ Σ , ¯ Φ , andH, which satisfy the time-averaged equations (25) , (26) , (27) and (28) . Then, there exists three functionsg , g and g such that if Ω = Ω + Hg ( H , Ω , Σ , Φ , t ) , Σ = Σ + Hg ( H , Ω , Σ , Φ , t ) , Φ = Φ + Hg ( H , Ω , Σ , Φ , t ) , (29) the functions Ω , Σ , Φ and ¯ Ω , ¯ Σ , ¯ Φ have the samelimit as t → ∞ . Setting Σ = Σ = , are derived analo-gous results for flat FLRW model. Theorem 2 implies that Ω , Σ , and Φ evolve according tothe time-averaged equations(25), (26), (27) as H → Now we investigate models where the Hubble scalaris not monotonic, but the systems admit a function thatplays the role of the order time depending parameterwhich is monotonic in a finite time-interval t < t ∗ in thecase of closed universes: Kantowski-Sachs and posi-tively curved FLRW. These models are analyzed sepa-rately due to in this case H is not a monotonic function. The equations for Kantowski-Sachs metric are:¨ φ = − H ˙ φ − V (cid:48) ( φ ) , (30)˙ ρ m = − γ H ρ m , (31)˙ K = − ( σ + + H ) K , (32)˙ H = − H − σ + − ( γ − ) ρ m −
13 ˙ φ + V ( φ ) , (33)˙ σ + = − H σ + − K , (34)3 H + K = σ + + ρ m +
12 ˙ φ + V ( φ ) . (35)Defining D = (cid:114) H + K , Q = HD , Ω = (cid:115) ω φ + ˙ φ D , Σ = σ + D , Φ = t ω − tan − (cid:18) ωφ ˙ φ (cid:19) , (36)we obtain the equations (A2).Assuming ω > µ and setting f = b µ ω − µ >
0, weobtain:˙ x = f ( t , x ) D + O ( D ) , x = ( Ω , Σ , Q , Φ ) T , ˙ D = − D (cid:0) Σ ( − Q + Q Σ ) + Q Ω cos ( Φ − t ω ) + γ Q ( − Σ − Ω ) (cid:1) + O ( D ) , (37)where f ( t , x ) = Ω (cid:16) Q (cid:0) Ω − (cid:1) ( − γ + ( Φ − t ω )) − Σ (cid:0) Q + ( γ − ) Q Σ − (cid:1) (cid:17) (cid:16) (cid:0) Σ − (cid:1) (cid:0) − Q − ( γ − ) Q Σ + (cid:1) + Q ΣΩ ( − γ + ( Φ − t ω )) (cid:17) (cid:0) Q − (cid:1) (cid:16) − ( γ − ) Σ + γ − Q Σ + Ω ( − γ + ( Φ − t ω )) − (cid:17) − Q sin ( ( t ω − Φ )) . (38)Replacing ˙ x = f ( t , x ) D with x = ( Ω , Σ , Q , Φ ) T and f ( t , x ) as in (38) by ˙ y = H ¯ f ( y ) with y = (cid:0) ¯ Ω , ¯ Q , ¯ Σ , ¯ Φ (cid:1) T with the time averaging¯ f ( · ) : = L (cid:90) L f ( · , t ) dt , L = πω , (39) we obtain the time-averaged system:˙¯ Ω = D ¯ Ω (cid:0) ¯ Q (cid:0) − γ (cid:0) ¯ Σ + ¯ Ω − (cid:1) − Q ¯ Σ + Σ + Ω − (cid:1) + Σ (cid:1) , (40)˙¯ Σ = D (cid:0)(cid:0) ¯ Σ − (cid:1) (cid:0) − Q − ( γ − ) ¯ Q ¯ Σ + (cid:1) − ( γ − ) ¯ Q ¯ Σ ¯ Ω (cid:1) , (41)˙¯ Q = − D (cid:0) ¯ Q − (cid:1) (cid:0) γ (cid:0) ¯ Σ + ¯ Ω − (cid:1) + Σ ( Q − Σ ) − Ω + (cid:1) , (42)˙ Φ = , (43)˙ D = − D (cid:16) Σ ( − ¯ Q + Q ¯ Σ ) + Q ¯ Ω + γ ¯ Q ( − ¯ Σ − ¯ Ω ) (cid:17) . (44) Theorem 3
Let be defined the functions ¯ Ω , ¯ Σ , ¯ Ω k , ¯ Φ ,and D, satisfying the time-averaged equations (40) , (41) , (42) , (43) and (44) . Then, there exists functionsg , g , g and g , such that if Ω = Ω + Dg ( D , Ω , Σ , Q , Φ , t ) , Σ = Σ + Dg ( D , Ω , Σ , Q , Φ , t ) , Q = Q + Dg ( D , Ω , Σ , Q , Φ , t ) , Φ = Φ + Dg ( D , Ω , Σ , Q , Φ , t ) , (45) the functions Ω , Σ , Q , Φ and ¯ Ω , ¯ Σ , ¯ Q , ¯ Φ have thesame limit on a time scale tD = O ( ) . Setting Σ = Σ = are derived analogous results for positively curvedFLRW model. Proof.
The proof is based on the following steps:1. Proving that D is a monotonic decreasing func-tion of t whenever 0 < Q < , Σ + Ω < , − Q + Q Σ >
0. The last restriction holds, in par-ticular, by choosing Q > , Σ >
0. In the lastcase, we define the bootstrapping sequences t = t , η = , D = D ( t ) , t n + = t n + D n D n + = D ( t n + ) , η n + = η n + (cid:82) t n + t n D ( s ) dsD n + = D ( t n + ) . (46)These sequences, however, are finite in the sensethat t n < t ∗ , with t ∗ satisfying lim t → t ∗ D ( t ) = ∞ .A lower bound for t ∗ is estimated assup { t : 1 − Q ( t ) + Q ( t ) Σ ( t ) > } . The last restriction holds, in particular, if Q > , Σ > ( t , η , Σ , Q , Φ ) (cid:55)→ ( − t , − η , − Σ , − Q , − Φ ) leaves invariant the sys-tem. Therefore, the solution can be completedby using the above symmetry. Hence, if Q ( t n ) < Σ ( t n ) < n we stop integration because D changes the monotony to become amonotonically increasing function. In this case,the solution is completed using the above sym-metry.3. Finding g i such that can be defined Ω , Σ , Q , Φ through equation (45) andproving that g i , i = , . . . , t ∈ [ t n , t n + ] . By continuity of¯ Ω , Ω , ¯ Σ , Σ , ¯ Q , Q , Φ , can be defined con-stants L , M , M , M , M which are finite andindependent of D .4. Using the expansion (45) and defining ∆Ω = Ω − ¯ Ω , ∆Σ = Σ − ¯ Σ , ∆ Q = Q − ¯ Q , ∆Φ = Φ − ¯ Φ and taking the same initial conditions at t = t n : Ω ( t n ) = ¯ Ω ( t n ) = Ω n , Σ ( t n ) = ¯ Σ ( t n ) = Σ n , Q ( t n ) = ¯ Q ( t n ) = Q n , Φ ( t n ) = ¯ Φ ( t n ) = Φ n , < Q n < , Σ n + Ω n < , − Q n + Q n Σ n > . Using Gronwall’s Lemma 6 for t ∈ [ t n , t n + ] : (cid:12)(cid:12)(cid:12) ∆Ω ( t ) (cid:12)(cid:12)(cid:12) ≤ M D n ( t − t n ) (cid:104) + LD n ( t − t n ) e LD n ( t − t n ) (cid:105) (cid:46) M ( t − t n ) D n + O ( D n ) ≤ M D n + O ( D n ) , (cid:12)(cid:12)(cid:12) ∆Σ ( t ) (cid:12)(cid:12)(cid:12) ≤ M D n ( t − t n ) (cid:104) + LD n ( t − t n ) e LD n ( t − t n ) (cid:105) (cid:46) M ( t − t n ) D n + O ( H n ) ≤ M D n + O ( D n ) , (cid:12)(cid:12)(cid:12) ∆Ω k ( t ) (cid:12)(cid:12)(cid:12) ≤ M D n ( t − t n ) (cid:104) + LH n ( t − t n ) e LD n ( t − t n ) (cid:105) (cid:46) M ( t − t n ) D n + O ( H n ) ≤ M H n + O ( D n ) , | ∆Φ ( t ) | ≤ M D n , due to t − t n ≤ t n + − t n = D n .
5. Finally, given the initial region0 < Q < , Σ + Ω < , − Q + Q Σ > . It is not invariant. Thus, the functions Ω , Σ , Q , Φ and ¯ Ω , ¯ Σ , ¯ Q , ¯ Φ have the same limiton a finite time scale tD = O ( ) and not for all t .The complete proof is given in appendix B.Theorem 3 implies that Ω , Σ , Ω k , and Φ evolve atfirst order in D according to the time-averaged equa-tions (40), (41), (42), (43) and (44) on a time scale tD = O ( ) . Remark 1
The initial region < Q < , Σ + Ω < , − Q + Q Σ > , is not invariant for the full sys-tem (A2) and for the averaged equations (40) , (41) , (42) , (43) and (44) . Hence, although for t (cid:28) t ∗ , D ( t ) remains close to zero. Once the orbit crosses the ini-tial region, D changes its monotony and will strictlyincrease without bound. Hence, Theorem 3 is valid ona time scale tD = O ( ) . Remark 2
Theorems 1 and 2 however are based on lim t → ∞ H ( t ) = due to the invariance of the initialregion for large t. That is, Theorem 1 implies that Ω , Σ , Ω k , and Φ evolve according to the time-averagedequations (25) , (26) , (27) and (28) as H → . Theo-rem 2 implies that Ω , Σ , Φ evolve according to the time-averaged equations (25) , (26) , (27) as H → . For FLRW metric with positive curvature the fieldequations are:¨ φ = − H ˙ φ − V (cid:48) ( φ ) , (47a)˙ ρ m = − γ H ρ m , (47b)˙ a = aH , (47c)˙ H = − (cid:0) γρ m + ˙ φ (cid:1) + a , (47d)3 H + a = ρ m +
12 ˙ φ + V ( φ ) . (47e)Defining D = (cid:114) H + a , Q = HD , Ω = (cid:115) ω φ + ˙ φ D , Φ = t ω − tan − (cid:18) ωφ ˙ φ (cid:19) , (48)we obtain equations (A6). Setting f = b µ ω − µ >
0, we obtain the series expan-sion near D = x = f ( t , x ) D + O (cid:0) D (cid:1) , x = ( Ω , Q , Φ ) T , ˙ D = − Q (cid:0) γ ( − Ω ) + ( t ω − Φ ) Ω (cid:1) D + O (cid:0) D (cid:1) , (49)where the vector function is defined f ( t , x ) = − Q Ω (cid:0) − Ω (cid:1) (cid:0) ( t ω − Φ ) − γ (cid:1) (cid:0) − Q (cid:1) (cid:16) − γ − Ω (cid:0) ( t ω − Φ ) − γ (cid:1) (cid:17) − Q sin ( t ω − Φ ) . (50)The simultaneous change ( t , Q , Φ ) (cid:55)→ ( − t , − Q , − Φ ) left invariant the systems (A6) and (49).Replacing ˙ x = f ( t , x ) D with x = ( Ω , Q , Φ ) T where f ( t , x ) is given by (49) by ˙ y = D ¯ f ( y ) with y = (cid:0) ¯ Ω , ¯ Q , ¯ Φ (cid:1) T with the time averaging (39), we obtain for γ (cid:54) = Ω = ( γ − ) D ¯ Q ¯ Ω (cid:0) − ¯ Ω (cid:1) , (51)˙¯ Q = − D (cid:0) − ¯ Q (cid:1) (cid:0) γ (cid:0) − ¯ Ω (cid:1) + Ω − (cid:1) , (52)˙¯ Φ = . (53)Theorem 3 applies for Kantowski-Sachs and the invari-ant set Σ =
4. QUALITATIVE ANALYSIS OF TIME-AVERAGEDSYSTEMS
According to Theorem 3 for Kantowski-Sachsmetrics and positively curved FLRW models D = (cid:113) H + K plays the role of time dependent perturba-tion parameter controlling the magnitude of error be-tween the solutions of the full and time-averaged prob-lems. The oscillations are thus viewed as perturbations.In the time-averaged system the Raychaudhuri equa-tion decouples. The analysis of the system is there-fore reduced to study the corresponding time-averagedequations. In this case the time-averaged system is (40), (41),(42) and (43). Introducing the new time variable η through d fd η = D d fdt , we obtain the guiding system: d ¯ Ω d η =
12 ¯ Ω (cid:0) ¯ Q (cid:0) − γ (cid:0) ¯ Σ + ¯ Ω − (cid:1) − Q ¯ Σ + Σ + Ω − (cid:1) + Σ (cid:1) , (54) d ¯ Σ d η = (cid:0)(cid:0) ¯ Σ − (cid:1) (cid:0) − Q − ( γ − ) ¯ Q ¯ Σ + (cid:1) − ( γ − ) ¯ Q ¯ Σ ¯ Ω (cid:1) , (55) d ¯ Qd η = − (cid:0) ¯ Q − (cid:1) (cid:0) γ (cid:0) ¯ Σ + ¯ Ω − (cid:1) + Σ ( Q − Σ ) − Ω + (cid:1) . (56)We have¯ Q ¯ Ω m : = − ¯ Σ − ¯ Ω , ¯ Q Ω k : = − Q (57)where ¯ Ω m , Ω k are the time-averaged values of Ω m : = ρ m H , Ω k : = a H . (58)Then, the phase space is (cid:8) ( ¯ Ω , ¯ Σ , ¯ Q ) ∈ R : − ≤ Q ≤ , ¯ Σ + ¯ Ω ≤ (cid:9) . (59)Furthermore, we have the auxiliary equations˙ e = − D ( Q − Σ ) e , ˙ K = − D ( Q + Σ ) K . (60)The equilibrium points of the guiding system(54),(55),(56) are:1. P : ( ¯ Ω , ¯ Σ , ¯ Q ) = ( , − , − ) with eigenvalues (cid:8) − , − , ( γ − ) (cid:9) . i) It is a sink for 0 ≤ γ < . ii) It is nonhyperbolic for γ = . For P we obtain:˙ D = D , ˙ H = − D , ˙ σ + = − D , ˙ e = − De , ˙ K = DK . Imposing the initial conditions (where t = , τ ( ) = K ( ) = / c , e ( ) = c , H ( ) = H , D ( ) = D , σ ( ) = σ we obtain by integration: D ( t ) = D − D t , H ( t ) = D D t − + D + H , σ + ( t ) = D D t − + D + σ , e ( t ) − = c ( D t − ) / , K ( t ) − = c ( D t − ) / , where D = H + c . This point represents anon-Flat LRS Kasner ( p = − , p = p = )Bianchi I contracting solution ([83] Sect. 6.2.2and Sect. 9.1.1 (2)). This solution is singular atthe finite time t = D , and it is valid for t > D .2. P : ( ¯ Ω , ¯ Σ , ¯ Q ) = ( , , ) with eigenvalues (cid:8) , , − γ (cid:9) . i) It is a source for 0 ≤ γ < . ii) It is nonhyperbolic for γ = . Evaluating at P we obtain:˙ D = − D , ˙ H = − D , ˙ σ + = − D , ˙ e = De , ˙ K = DK . Imposing the initial conditions K ( ) = / c , e ( ) = c , H ( ) = H , D ( ) = D , σ ( ) = σ we obtain by integration: D ( t ) = D D t + , H ( t ) = D (cid:18) D t + − (cid:19) + H , σ + ( t ) = σ − D t D t + , e ( t ) − = c ( D t + ) / , K ( t ) − = c ( D t + ) / , where D = H + c . This point represents anon-Flat LRS Kasner ( p = − , p = p = )Bianchi I expanding solution ([83] Sect. 6.2.2and Sect. 9.1.1 (2)). It is valid for all t .3. P : ( ¯ Ω , ¯ Σ , ¯ Q ) = ( , , − ) with eigenvalues (cid:8) − , − , ( γ − ) (cid:9) . i) It is a sink for 0 ≤ γ < . ii) It is nonhyperbolic for γ = . Evaluating at P we obtain:˙ D = D , ˙ H = − D , ˙ σ + = D , ˙ e = De , ˙ K = . Imposing the initial conditions K ( ) = / c , e ( ) = c , H ( ) = H , D ( ) = D , σ ( ) = σ we obtain by integration: D ( t ) = D − D t , H ( t ) = D D t − + D + H , σ + ( t ) = D t − D t + σ , e ( t ) − = ( − D t ) c , K ( t ) − = c , where D = H + c . This point representsa Taub (flat LRS Kasner) contracting solution( p = , p = , p =
0) [83] (Sect 6.2.2 and p193, Eq. (9.6)).4. P : ( ¯ Ω , ¯ Σ , ¯ Q ) = ( , − , ) with eigenvalues (cid:8) , , − γ (cid:9) . i) It is a source for 0 ≤ γ < . ii) It is nonhyperbolic for γ = . Evaluating at P we obtain:˙ D = − D , ˙ H = − D , ˙ σ + = D , ˙ e = − De , ˙ K = . Imposing the initial conditions K ( ) = / c , e ( ) = c , H ( ) = H , D ( ) = D , σ ( ) = σ we obtain by integration: D ( t ) = D D t + , H ( t ) = D (cid:18) D t + − (cid:19) + H , σ + ( t ) = D t D t + + σ , e ( t ) − = ( D t + ) c , K ( t ) − = c , where D = H + c . This point representsa Taub (flat LRS Kasner) expanding solution( p = , p = , p =
0) [83] (Sect 6.2.2 and p193, Eq. (9.6)).5. P : ( ¯ Ω , ¯ Σ , ¯ Q ) = ( , , − ) with eigenvalues (cid:110) − γ , − ( γ − ) , − γ (cid:111) . i) It is a source for 0 ≤ γ < . ii) It is a saddle for < γ < < γ < . iii) It is nonhyperbolic for γ = or γ = γ = . Evaluating at P we obtain:˙ D = γ D , ˙ H = − γ D , ˙ σ + = , ˙ e = De , ˙ K = DK . Imposing the initial conditions K ( ) = / c , e ( ) = c , H ( ) = H , D ( ) = D , σ ( ) = D ( t ) = D − γ D t , H ( t ) = D γ D t − + D + H , σ + ( t ) = , e ( t ) − = (cid:16) − γ D t (cid:17) γ c , K ( t ) − = c (cid:18) − γ D t (cid:19) γ , D = H + c . The corresponding solu-tion is a flat matter dominated FLRW contractingsolution with ¯ Ω m = P : ( ¯ Ω , ¯ Σ , ¯ Q ) = ( , , ) with eigenvalues (cid:110) ( γ − ) , ( γ − ) , γ − (cid:111) . i) It is a sink for 0 ≤ γ < . ii) It is a saddle for < γ < < γ < . iii) It is nonhyperbolic for γ = or γ = γ = . Evaluating at P we obtain:˙ D = − γ D , ˙ H = − γ D , ˙ σ + = , ˙ e = − De , ˙ K = − DK . Imposing the initial conditions K ( ) = / c , e ( ) = c , H ( ) = H , D ( ) = D , σ ( ) = D ( t ) = D γ D t + , H ( t ) = D (cid:18) γ D t + − (cid:19) + H , σ + ( t ) = , e ( t ) − = (cid:16) γ D t + (cid:17) γ c , K ( t ) − = c (cid:18) γ D t + (cid:19) γ , where D = H + c . The corresponding so-lution is a flat matter dominated FLRW uni-verse with ¯ Ω m = F represents a quintessencefluid for 0 < γ < or a zero-acceleration modelfor γ = . In the limit γ = a ( t ) ∝ (cid:16) + γ D t (cid:17) γ → a e H t , i.e., a de Sitter solution.7. P : ( ¯ Ω , ¯ Σ , ¯ Q ) = ( , , − ) with eigenvalues (cid:8) − , , − γ (cid:9) . This point is always a saddlebecause it has a negative and a positive eigen-value. For γ = P we obtain:˙ D = − b γ µ ω − µ sin (cid:113) D (cid:0) ω − µ (cid:1) sin ( t ω − Φ ) b µ ω − γ D (cid:18) µ sin ( t ω − Φ ) ω − (cid:19) − ( γ − ) D cos ( t ω − Φ )= ( t ω − Φ ) D + O (cid:0) D (cid:1) , ˙ H = b γ µ ω − µ sin (cid:113) D (cid:0) ω − µ (cid:1) sin ( t ω − Φ ) b µ ω + D (cid:18) γ µ sin ( t ω − Φ ) ω − γ (cid:19) + ( γ − ) D cos ( t ω − Φ )= − ( t ω − Φ ) D + O (cid:0) D (cid:1) , ˙ σ + = , ˙ e = De , ˙ K = DK . Therefore,˙ D ∼ D cos ( t ω − Φ ) ˙ H ∼ − D cos ( t ω − Φ ) , for large t . In average, Φ is constant, setting for1simplicity and integrating we obtain: D ( t ) ≈ − ω αω + t ω + ( t ω ) , H ( t ) ≈ ω αω + t ω + ( t ω ) . Then D ( t ) ∼ − t , H ( t ) ∼ t (61)as t → ∞ . Finally, e ( t ) = c t / , K ( t ) = c t / , σ = . (62)The line element (4) becomes ds = − dt + c − t / dr + c − t / (cid:2) d ϑ + ϑ d ζ (cid:3) . (63) Hence for large t and c = c the equilibriumpoint can be associated with the flat FriedmanEinstein-de-Sitter solution ([83], Sec 9.1.1 (1))with γ =
1. It is a contracting solution.8. P : ( ¯ Ω , ¯ Σ , ¯ Q ) = ( , , ) with eigenvalues (cid:8) , − , ( γ − ) (cid:9) . This point is always a saddlebecause it has a negative and a positive eigen-value. For γ = P we obtain:˙ D = b γ µ ω − µ sin (cid:113) D (cid:0) ω − µ (cid:1) sin ( t ω − Φ ) b µ ω + ( γ − ) D cos ( t ω − Φ ) + γ D (cid:18) µ sin ( t ω − Φ ) ω − (cid:19) = − ( t ω − Φ ) D + O (cid:0) D (cid:1) , ˙ H = b γ µ ω − µ sin (cid:113) D (cid:0) ω − µ (cid:1) sin ( t ω − Φ ) b µ ω + D (cid:18) γ µ sin ( t ω − Φ ) ω − γ (cid:19) + ( γ − ) D cos ( t ω − Φ )= − ( t ω − Φ ) D + O (cid:0) D (cid:1) , ˙ σ + = , ˙ e = − De , ˙ K = − DK . Therefore,˙ D ∼ − D cos ( t ω − Φ ) ˙ H ∼ − D cos ( t ω − Φ ) , for large t . In average, Φ is constant, setting forsimplicity and integrating we obtain: D ( t ) ≈ ω − αω + t ω + ( t ω ) , H ( t ) ≈ ω − αω + t ω + ( t ω ) . Then D ( t ) ∼ t , H ( t ) ∼ t (64) as t → ∞ . Finally, e ( t ) = c t / , K ( t ) = c t / , σ ( t ) = . (65)The line element (4) becomes ds = − dt + c − t / dr + c − t / (cid:2) d ϑ + ϑ d ζ (cid:3) . (66)Hence for large t and c = c the equilibriumpoint can be associated with the flat FriedmanEinstein-de-Sitter solution ([83], Sec 9.1.1 (1))with γ =
1. It is a expanding solution.9. P : ( ¯ Ω , ¯ Σ , ¯ Q ) = (cid:18) , − γ √ ( − γ ) , − √ ( − γ ) (cid:19) witheigenvalues2 (cid:110) γ − + , (cid:16) γ −√ − γ √ γ ( γ − )+ − (cid:17) γ − , (cid:16) γ + √ − γ √ γ ( γ − )+ − (cid:17) γ − (cid:111) . It exists for 0 ≤ γ ≤ or γ = ≤ γ < . ii) It is nonhyperbolic for γ = or γ = . Evaluating at P we obtain for 0 ≤ γ < the fol-lowing:˙ D = − γ D γ − , ˙ H = − γ D ( γ − ) , ˙ σ + = − γ ( γ − ) D ( γ − ) , ˙ e = ( γ − ) De γ − , ˙ K = − γ DK γ − . Imposing the initial conditions K ( ) = / c , e ( ) = c , H ( ) = H , D ( ) = D , σ ( ) = D ( t ) = ( γ − ) D γ ( D t + ) − , H ( t ) = D (cid:18) − γ + γ + γ D t − (cid:19) + H , σ + ( t ) = ( − γ ) γ D t ( γ − )( γ + γ D t − ) + σ , e ( t ) − = ( γ − ) − γ ( γ ( D t + ) − ) γ − c , K ( t ) − = c ( − γ ( D t + )) ( − γ ) , where D = H + c . The line element (4) be-comes ds = − dt + ( γ − ) − γ ( γ ( D t + ) − ) γ − c dr + c ( − γ ( D t + )) ( − γ ) (cid:2) d ϑ + sin ( ϑ ) d ζ (cid:3) . (67)For γ = D = D , ˙ H = − D , ˙ σ + = − D , ˙ e = − De , ˙ K = DK . Imposing the initial conditions K ( ) = / c , e ( ) = c , H ( ) = H , D ( ) = D , σ ( ) = D ( t ) = D − D t , H ( t ) = D D t − + D + H , σ + ( t ) = D t D t − + σ , e ( t ) − = ( c − c D t ) , K ( t ) − = c ( − D t ) , where D = H + c .10. P : ( ¯ Ω , ¯ Σ , ¯ Q ) = (cid:18) , γ − √ ( − γ ) , √ ( − γ ) (cid:19) witheigenvalues (cid:110) ( γ − ) − γ , (cid:16) γ + √ − γ √ γ ( γ − )+ − (cid:17) − γ , − γ + √ − γ √ γ ( γ − )+ + γ − (cid:111) . It exists for 0 ≤ γ ≤ or γ = ≤ γ < . ii) It is nonhyperbolic for γ = or γ = . Evaluating at P we obtain for 0 ≤ γ < thefollowing:˙ D = γ D γ − , ˙ H = − γ D ( γ − ) , ˙ σ + = − γ ( γ − ) D ( γ − ) , ˙ e = − ( γ − ) De γ − , ˙ K = γ DK γ − . Imposing the initial conditions K ( ) = / c , e ( ) = c , H ( ) = H , D ( ) = D , σ ( ) = D ( t ) = ( − γ ) D γ ( D t − ) + , H ( t ) = D (cid:18) γ − + − γ + γ D t + (cid:19) + H , σ + ( t ) = γ ( γ − ) D t ( γ − )( − γ + γ D t + ) + σ , e ( t ) − = (cid:16) − γ + γ D t + − γ (cid:17) γ − c , K ( t ) − = c ( γ ( D t − ) + ) ( − γ ) , where D = H + c . The line element (4) be-3comes ds = − dt + (cid:16) − γ + γ D t + − γ (cid:17) γ − c dr + c ( γ ( D t − ) + ) ( − γ ) (cid:2) d ϑ + sin ( ϑ ) d ζ (cid:3) . (68)For γ = D = − D , ˙ H = − D , ˙ σ + = − D , ˙ e = De , ˙ K = − DK Imposing the initial conditions K ( ) = / c , e ( ) = c , H ( ) = H , D ( ) = D , σ ( ) = D ( t ) = D D t + , H ( t ) = D (cid:18) D t + − (cid:19) + H , σ + ( t ) = σ − D t D t + , e ( t ) − = ( c D t + c ) , K ( t ) − = c ( D t + ) , where D = H + c .In figure 1(a) are presented some orbits in the phasespace of the guiding system (54), (55), (56) for γ = P , P and P ; the late-time at-tractors are P , P and P . The saddles are P , P , P and P .In figure 1(b) are presented some orbits of the phasespace of the guiding system (56) for γ = . In figure2(a) are presented some orbits in the phase space of theguiding system (54), (55), (56) for γ = γ = corresponding to radiation. In three cases theearly-time attractors are P and P ; the late-time attrac-tors are P and P . The saddles are P , P , P , and P .Points P and P do not exist.In figure 3 are presented some orbits in the phasespace of guiding system (54), (55), (56) for γ = P , P and P is invariant and it is the early-timeattractor and the line connecting the point P , P and P is invariant and it is the late-time attractor. P , and P are saddle. Points P and P do not exist. The results from the linear stability analysis com-bined with Theorem 3 lead to:
Theorem 4
The late-time attractors of full system (A2) and time-averaged system (C2) for Kantowski-Sachsline element are(i) The anisotropic solution P with ¯ Ω m = andds = − dt + c ( D t − ) / dr c ( D t − ) / (cid:2) d ϑ + ϑ d ζ (cid:3) (69) if ≤ γ < . This point represents a non-FlatLRS Kasner (p = − , p = p = ) Bianchi Icontracting solution with H < ([83] Sect. 6.2.2and Sect. 9.1.1 (2)). This solution is singular atthe finite time t = D , and it is valid for t > D .(ii) The anisotropic solution P with ¯ Ω m = andds = − dt + ( − D t ) c dr c (cid:2) d ϑ + ϑ d ζ (cid:3) (70) if ≤ γ < , representing a contracting solutionwith H < . This point represents a Taub (flatLRS Kasner) contracting solution (p = , p = , p = ) [83] (Sect 6.2.2 and p 193, Eq. (9.6)).(iii) The flat matter dominated FLRW universe P withds = − dt + (cid:16) γ D t + (cid:17) γ c dr c (cid:18) γ D t + (cid:19) γ (cid:2) d ϑ + ϑ d ζ (cid:3) (71) if < γ < . P represents a quintessence fluid ora zero-acceleration model for γ = . In the limit γ = we have a ( t ) ∝ (cid:16) γ D t + (cid:17) γ → e D t , i.e.,the de Sitter solution. In this case the time-averaged system is (51), (52)and (53). Introducing the new time variable η through4 Q P1 P2P3P4P5 P6P7 P8P9P10 (a) γ = Q P1 P2P3P4P5 P6P7 P8 (b) γ = . Figure 1: Phase space of the guiding system (54), (55), (56) for γ = , . Q P1 P2P3P4P5 P6P7 P8 (a) γ = Q P1 P2P3P4P5 P6P7 P8 (b) γ = . Figure 2: Phase space of the guiding system (54), (55), (56) for γ = , . Q P1 P2P3P4P5 P6P7 P8
Figure 3: Phase space of the guiding system (54), (55), (56) for γ = d fd η = D d fdt we obtain the guiding system: d ¯ Ω d η = ( γ − ) ¯ Q ¯ Ω (cid:0) − ¯ Ω (cid:1) , (72) d ¯ Qd η = − (cid:0) − ¯ Q (cid:1) (cid:0) γ (cid:0) − ¯ Ω (cid:1) + Ω − (cid:1) . (73)We have ¯ Q ¯ Ω m : = − ¯ Ω , ¯ Q Ω k : = − Q (74)where ¯ Ω m , Ω k are the time-averaged values of Ω m : = ρ m H , Ω k : = a H . (75)Then, the phase space is (cid:8) ( ¯ Ω , ¯ Q ) ∈ R : − ≤ Q ≤ , ≤ ¯ Ω ≤ (cid:9) . (76)In some special cases we relax the condition ¯ Ω ≤ Ω >
1. The equilibrium points of (72) and(73) are:1. P : ( ¯ Ω , ¯ Q ) = ( , − ) with eigenvalues (cid:8) − γ , − ( γ − ) (cid:9) .i) It is a source for 0 < γ < .ii) It is a saddle for < γ <
1. iii) It is nonhyperbolic for γ = and γ = < γ < P of Kantowski-Sachs with D ( t ) = D − γ D t , H ( t ) = D γ D t − + D + H , σ + ( t ) = , e ( t ) − = (cid:16) − γ D t (cid:17) γ c , K ( t ) − = c (cid:18) − γ D t (cid:19) γ . The corresponding solution is a flat matter dom-inated FLRW contracting solution with ¯ Ω m = P : ( ¯ Ω , ¯ Q ) = ( , ) with eigenvalues (cid:8) ( γ − ) , γ − (cid:9) .i) It is a sink for 0 < γ < .ii) It is a saddle for < γ < γ = and γ = < γ < P P P P E - - Q Ω γ = (a) P P P P E - - Q Ω γ = / (b) P P P P - - Q Ω γ = (c) P P P P E - - Q Ω γ = / (d) Figure 4: Phase plane for system (72), (73) for different choices of γ . This equilibrium point is related to the isotropicpoint P of Kantowski-Sach with D ( t ) = D γ D t + , H ( t ) = D (cid:18) γ D t + − (cid:19) + H , σ + ( t ) = , e ( t ) − = (cid:16) γ D t + (cid:17) γ c , K ( t ) − = c (cid:18) γ D t + (cid:19) γ . P represents a quintessence fluid or a zero-acceleration model for γ = . In the limit γ = a ( t ) ∝ (cid:16) γ D t + (cid:17) γ → e D t , i.e., the deSitter solution.3. P : ( ¯ Ω , ¯ Q ) = ( , − ) with eigenvalues {− , ( γ − ) } .i) It is a sink for 0 < γ < γ = < γ < P of Kantowski-Sachs with D ( t ) ∼ − t , H ( t ) ∼ t (77)and e ( t ) = c t / , K ( t ) = c t / , σ ( t ) = . (78)as t → ∞ . The line element (4) becomes ds = − dt + c − t / dr + c − t / (cid:2) d ϑ + ϑ d ζ (cid:3) . (79)Hence for large t and c = c the equilibriumpoint can be associated with the flat FriedmanEinstein-de-Sitter solution ([83], Sec 9.1.1 (1))with γ =
1. It is a contracting solution.4. P : ( ¯ Ω , ¯ Q ) = ( , ) with eigenvalues { − γ , } .i) Is is a source for 0 < γ < γ = < γ < P of Kantowski-Sachs with D ( t ) ∼ t , H ( t ) ∼ t (80) and e ( t ) = c t / , K ( t ) = c t / , σ ( t ) = . (81)as t → ∞ . The line element (4) becomes ds = − dt + c − t / dr + c − t / (cid:2) d ϑ + ϑ d ζ (cid:3) . (82)Hence for large t and c = c the equilibriumpoint can be associated with the flat FriedmanEinstein-de-Sitter solution ([83], Sec 9.1.1 (1))with γ =
1. It is a expanding solution.5. E : ( ¯ Ω , ¯ Q ) = (cid:16)(cid:113) γ − γ − , (cid:17) with eigenvalues (cid:110) − √ − γ √ , √ − γ √ (cid:111) . This point exists for 0 ≤ γ ≤ or 1 < γ ≤ < γ < .ii) nonhyperbolic for γ = or 1 < γ < D -expansionwhen γ =
1. Using Taylor expansion up to the fourthorder in D the following time-averaged system is ob-tained d ¯ Ω d η = D ¯ Q ¯ Ω (cid:0) ω − µ (cid:1) b µ ω (83a) d ¯ Qd η = − (cid:0) − ¯ Q (cid:1) (cid:32) D ¯ Ω (cid:0) ω − µ (cid:1) b µ ω + (cid:33) , (83b) d ¯ Φ d η = D ¯ Ω (cid:0) ω − µ (cid:1) b µ ω − D ¯ Ω (cid:0) ω − µ (cid:1) b µ ω , (83c) dDd η = − D ¯ Q − D ¯ Q ¯ Ω (cid:0) ω − µ (cid:1) b µ ω . (83d)The system (83) admits the first integral D ( η ) = e − (cid:82) ¯ Q ( η ) d η (cid:112) − ¯ Q ( η ) , ¯ Ω ( η ) = √ b µ / ω e (cid:82) ¯ Q ( η ) d η √ ( ω − µ ) / (cid:113) ¯ Q ( η ) − Q (cid:48) ( η ) − , Φ ( η ) = (cid:0) ω − µ (cid:1) / b µ ω (cid:90) (cid:112) ¯ Q ( η ) − Q (cid:48) ( η ) − (cid:112) − ¯ Q ( η ) d η − (cid:0) ω − µ (cid:1) b µ ω (cid:90) D ( η ) (cid:0) ¯ Q ( η ) − Q (cid:48) ( η ) − (cid:1)(cid:0) − ¯ Q ( η ) (cid:1) d η , E - - - - x y + y Figure 5: Compacted phase plane of system (84). where ¯ Q satisfies the differential equation ¯ Q (cid:48)(cid:48) = ¯ Q (cid:18) ( ¯ Q − ) + ¯ Q (cid:48) (cid:18) Q (cid:48) ¯ Q − − (cid:19)(cid:19) , where the comma means derivative with respect to η .Introducing the variables x = ¯ Q ( η ) , y = ¯ Q (cid:48) ( η ) we ob-tain: x (cid:48) ( η ) = y ( η ) , (84a) y (cid:48) ( η ) = x ( η ) (cid:18) ( x ( η ) − ) + y ( η ) (cid:18) y ( η ) x ( η ) − − (cid:19)(cid:19) . (84b) The origin is an equilibrium point with eigenval-ues (cid:110) i √ , − i √ (cid:111) . The dynamics of system (84) inthe coordinates ( x , y / (cid:112) + y ) is presented in Figure5, where the origin is a stable center and the points ( x , y ) = ( − , ) and ( x , y ) = ( , ) are saddles.The results from the linear stability analysis com-bined with Theorem 3 for Σ = Theorem 5
The late-time attractors of full system (A6) and time-averaged system (C3) for closed FLRW met-ric with positive curvature line element are (i) The isotropic solution P withds = − dt + (cid:16) − γ D t (cid:17) γ c dr c (cid:18) − γ D t (cid:19) γ (cid:2) d ϑ + ϑ d ζ (cid:3) (85) if < γ < . The corresponding solution is aflat matter dominated FLRW contracting solu-tion with ¯ Ω m = .(ii) The flat matter dominated FLRW universe P withds = − dt + (cid:16) γ D t + (cid:17) γ c dr c (cid:18) γ D t + (cid:19) γ (cid:2) d ϑ + ϑ d ζ (cid:3) (86) if < γ < . P represents a quintessence fluid ora zero-acceleration model for γ = . In the limit γ = we have a ( t ) ∝ (cid:16) γ D t + (cid:17) γ → e D t , i.e.,the de Sitter solution.(iii) The equilibrium point P withds = − dt + c − t / dr + c − t / (cid:2) d ϑ + ϑ d ζ (cid:3) (87) for < γ < . For large t and c = c the equilib-rium point can be associated with the flat Fried-man Einstein-de-Sitter solution.
5. ANALYSIS AS D → ∞ FOR KANTOWSKI-SACHSAND CLOSED FLRW MODELS
According to Remark 1, Theorem 3 is valid on a timescale t < t ∗ where D remains close to zero but at a crit-ical time t ∗ : lim t → t ∗ D ( t ) = ∞ . A lower bound for t ∗ is estimated as¯ t = sup { t > − Q ( t ) + Q ( t ) Σ ( t ) > } . (88)0According with the results derived in appendix 1 wehave the asymptotic inner-outer expansion x ( t ) − ¯ x ( t ) = t ∗ ( − r ) C + C t ∗ D ( t )( − r )+ N ∑ j = C j t ∗ j − ( − r ) j − + O (cid:32) t ∗ N ( − r ) N (cid:33) . (89)For r : = tt ∗ (cid:28) t ∗ C + ∑ Nj = C j t ∗ j − = x ( t ) − ¯ x ( t ) = C t ∗ D ( t ) verifying the result of Theorem 3.These results are supported by the numerical evidence in appendices 1 and 2.In this section we analyze qualitatively the system as D → ∞ by introducing z = D . (90) The time-averaged system (C2) becomes ∂ η ¯ Ω = ¯ Ω (cid:32) ¯ Q (cid:16) − γ (cid:0) ¯ Σ + ¯ Ω − (cid:1) − Q ¯ Σ + Σ + Ω − (cid:17) + Σ (cid:33) , ∂ η ¯ Q = − (cid:0) ¯ Q − (cid:1) (cid:16) γ (cid:0) ¯ Σ + ¯ Ω − (cid:1) + Σ ( Q − Σ ) − Ω + (cid:17) , ∂ η ¯ Σ = (cid:16) (cid:0) ¯ Σ − (cid:1) (cid:0) − Q − ( γ − ) ¯ Q ¯ Σ + (cid:1) − ( γ − ) ¯ Q ¯ Σ ¯ Ω (cid:17) , ∂ η z = z (cid:16) Σ ( − ¯ Q + Q ¯ Σ ) + Q ¯ Ω + γ ¯ Q ( − ¯ Σ − ¯ Ω ) (cid:17) . (91)We are interested in late-time or early-time attractorsand in discussing the relevant saddle equilibrium pointsof (91). So we do not present all the stability regions inthe parameter space. In this regard, have the followingresults:1. P : ( Ω , Σ , Q , z ) = ( , − , − , ) with eigenvalues (cid:8) − , − , − , − ( − γ ) (cid:9) . It is an attractor for0 ≤ γ < P : ( Ω , Σ , Q , z ) = ( , , , ) with eigenvalues (cid:8) , , , − γ (cid:9) . It is a source for 0 ≤ γ < P : ( Ω , Σ , Q , z ) = ( , , − , ) with eigenvalues (cid:8) − , − , − , − ( − γ ) (cid:9) . It is an attractor for0 ≤ γ < P : ( Ω , Σ , Q , z ) = ( , − , , ) with eigenvalues (cid:8) , , , − γ (cid:9) . It is a source for 0 ≤ γ < P : ( Ω , Σ , Q , z ) = ( , , − , ) with eigenvalues (cid:110) − γ , − ( γ − ) , − γ , − γ (cid:111) . It is a saddlein the extended phase space.6. P : ( Ω , Σ , Q , z ) = ( , , , ) with eigenvalues (cid:110) ( γ − ) , γ , ( γ − ) , γ − (cid:111) . It is a saddle in theextended phase space.7. P : ( Ω , Σ , Q , z ) = ( , , − , ) with eigenvalues (cid:8) − , − , , ( γ − ) (cid:9) . It is a saddle. 8. P : ( Ω , Σ , Q , z ) = ( , , , ) with eigenvalues (cid:8) , , − , − γ (cid:9) . It is a saddle.9. P : ( Ω , Σ , Q , z ) = (cid:16) , − γ | − γ | , − | − γ | , (cid:17) witheigenvalues (cid:40) − (cid:16) γ + √ ( − γ )( γ ( γ − )+ ) − (cid:17) | − γ | , (cid:16) − γ + √ ( − γ )( γ ( γ − )+ )+ (cid:17) | − γ | , − γ | − γ | , − γ | − γ | (cid:41) .Exists for 0 ≤ γ ≤ or γ = P : ( Ω , Σ , Q , z ) = (cid:16) , γ − | − γ | , | − γ | , (cid:17) witheigenvalues (cid:40) − (cid:16) − γ + √ ( − γ )( γ ( γ − )+ )+ (cid:17) | − γ | , (cid:16) γ + √ ( − γ )( γ ( γ − )+ ) − (cid:17) | − γ | , ( γ − ) | − γ | , γ | − γ | (cid:41) .Exists for 0 ≤ γ ≤ or γ = Using the variable (90) the time-averaged system for γ (cid:54) = ∂ η ¯ Ω = ( γ − ) ¯ Q ¯ Ω (cid:0) − ¯ Ω (cid:1) , ∂ η ¯ Q = − (cid:0) − ¯ Q (cid:1) (cid:0) γ (cid:0) − ¯ Ω (cid:1) + Ω − (cid:1) , ∂ η z = ¯ Q (cid:0) γ ( − ¯ Ω ) + ¯ Ω (cid:1) z . (92)We are interested in late-time or early-time attractorsand in discussing the relevant saddle equilibrium pointsof (92). So we do not present all the stability regions inthe parameter space. In this regard, have the followingresults:1. P : ( ¯ Ω , ¯ Q , z ) = ( , − , ) with eigenvalues (cid:110) − γ , − ( γ − ) , − γ (cid:111) is an attractor for γ > P : ( ¯ Ω , ¯ Q , z ) = ( , , ) with eigenvalues (cid:110) ( γ − ) , γ , γ − (cid:111) is a source for γ > P : ( ¯ Ω , ¯ Q , z ) = ( , − , ) with eigenvalues (cid:8) − , − , ( γ − ) (cid:9) is an attractor for γ < P : ( ¯ Ω , ¯ Q , z ) = ( , , ) with eigenvalues (cid:8) , − γ , (cid:9) is a source for γ < E c : ( ¯ Ω , ¯ Q , z ) = (cid:16) √ γ − √ γ − , , z c (cid:17) with eigenvalues (cid:26) , − √ ( − γ ) γ − √ √ γ − , √ ( − γ ) γ − √ √ γ − (cid:27) . This line ofequilibrium points exists for 0 ≤ γ ≤ or 1 < γ ≤ ≤ γ < . For 1 < γ ≤
6. CONCLUSIONS
This paper is the last of the research program “Av-eraging Generalized Scalar Field Cosmologies”. Wehave used Asymptotic Methods and Averaging Theoryto explore the solution’s space of scalar field cosmolo-gies with generalized harmonic potential V ( φ ) = µ (cid:20) b f (cid:18) − cos (cid:18) φ f (cid:19)(cid:19) + φ µ (cid:21) , b > t < t ∗ . For t > t ∗ the monotony of D changes and this parameterincreases without bound.We have proved Theorem 3 which states that thelate-time attractors of full and time-averaged systemsare the same for some homogeneous metrics. Morespecific, according to Theorem 3 for Kantowski-Sachsmetrics and positively curved FLRW models the quan-tity D = (cid:113) H + K is a time dependent perturbationparameter controlling the magnitude of error betweenfull time dependent and time-averaged solutions. Theanalysis of the system is therefore reduced to study thecorresponding time-averaged equations. However, forKantowski-Sachs metric the initial region0 < Q < , Σ + Ω < , − Q + Q Σ > , (and for closed FLRW the initial region Q >
0, respec-tively) is not invariant for the full system (A2) and forthe time-averaged equations (40), (41), (42), (43) and(44). Hence, although for t (cid:28) t ∗ , D ( t ) remains close tozero. Once the orbit crosses the initial region, D changemonotony and strictly increases without bound. Hence,Theorem 3 is valid on a time scale tD = O ( ) .We have formulated theorems 4 and 5 concerningthe late-time behavior of our model, whose proofs arebased on the previous theorems and linear stabilityanalysis. Hence, we can establish the stability of a pe-riodic solution as it matches exactly the stability of thestationary solution of the time-averaged equation.In particular, for Kantowski-Sachs metric the late-time attractor of full system (A2) and time-averagedsystem (C2) are:(i) The anisotropic solution P with ¯ Ω m = ds = − dt + c ( D t − ) / dr c ( D t − ) / (cid:2) d ϑ + ϑ d ζ (cid:3) (93)if 0 ≤ γ <
2. This point represents a non-FlatLRS Kasner ( p = − , p = p = ) Bianchi Icontracting solution with H < t = D , and it is valid for t > D .(ii) The anisotropic solution P with ¯ Ω m = ds = − dt + ( − D t ) c dr c (cid:2) d ϑ + ϑ d ζ (cid:3) (94)if 0 ≤ γ <
2. This point represents a Taub (flatLRS Kasner) contracting solution ( p = , p = , p =
0) [83] (Sect 6.2.2 and p 193, Eq. (9.6)).(iii) The flat matter dominated FLRW universe P ds = − dt + (cid:16) γ D t + (cid:17) γ c dr c (cid:18) γ D t + (cid:19) γ (cid:2) d ϑ + ϑ d ζ (cid:3) (95)if 0 < γ < . P represents a quintessence fluid ora zero-acceleration model for γ = . In the limit γ = a ( t ) ∝ (cid:16) γ D t + (cid:17) γ → e D t , i.e.,the de Sitter solution.Finally, the late-time attractors of full system (A6)and time-averaged system (C3) for closed FLRW met-ric with positive curvature are:(i) The isotropic solution P with ds = − dt + (cid:16) − γ D t (cid:17) γ c dr c (cid:18) − γ D t (cid:19) γ (cid:2) d ϑ + ϑ d ζ (cid:3) (96)if 1 < γ <
2. The corresponding solution is aflat matter dominated FLRW contracting solu-tion with ¯ Ω m = P with ds = − dt + (cid:16) γ D t + (cid:17) γ c dr c (cid:18) γ D t + (cid:19) γ (cid:2) d ϑ + ϑ d ζ (cid:3) (97)if 0 < γ < . P represents a quintessence fluid ora zero-acceleration model for γ = . In the limit γ = a ( t ) ∝ (cid:16) γ D t + (cid:17) γ → e D t , i.e.,the de Sitter solution.(iii) The equilibrium point P with ds = − dt + c − t / dr + c − t / (cid:2) d ϑ + ϑ d ζ (cid:3) (98)for 0 < γ <
1. For large t and c = c the equilib-rium point can be associated with the flat Fried-man Einstein-de-Sitter solution.According to Remark 1, Theorem 3 is valid on a timescale t < t ∗ , where D remains close to zero, but at a critical time t ∗ : lim t → t ∗ D ( t ) = ∞ . According to the derived results in appendix 1 we havethe asymptotic inner-outer expansion x ( t ) − ¯ x ( t ) = t ∗ ( − r ) C + C t ∗ D ( t )( − r )+ N ∑ j = C j t ∗ j − ( − r ) j − + O (cid:32) t ∗ N ( − r ) N (cid:33) . (99)For r : = tt ∗ (cid:28) t ∗ C + ∑ Nj = C j t ∗ j − = x ( t ) − ¯ x ( t ) = C t ∗ D ( t ) verifying the result of Theorem3. Hence, to analyze qualitatively the system as D → ∞ we have used a transformation of coordinates thatbrings infinity to the origin. In the analysis of the dy-namics of Kantowski-Sachs and closed FLRW universein limit D → ∞ we found that the point P becomes asaddle in the extended phase space. This is because for P the parameter D ( t ) = D γ D t + is finite for all positivetimes (it is a sink for 0 ≤ γ ≤ if we neglect behavioralong the axis z = / D ).Summarizing, the results of the Research Program“Averaging Generalized Scalar Field Cosmologies” arethe following.In paper I [42] was proved for LRS Bianchi IIIthe late-time attractors are: a matter dominated flatFLRW universe if 0 ≤ γ ≤ (mimicking de Sitter,quintessence or zero acceleration solutions), a matter-curvature scaling solution if < γ < ≤ γ ≤
2. For FLRW metric with k = − ≤ γ ≤ (mimicking de Sit-ter, quintessence or zero acceleration solutions) and theMilne solution if < γ <
2. In all metrics, the matterdominated flat FLRW universe represents quintessencefluid if 0 < γ < .In paper II [43] was obtained for flat FLRW andLRS Bianchi I metrics that the late-time attractors ofthe full and time-averaged systems are: the flat mat-ter dominated FLRW solutoin (mimicking de Sitter,quintessence or zero acceleration solutions) and the flatFriedman Einstein-de-Sitter solution.It is interesting to note that in the FLRW with nega-tive or zero curvature and for Bianchi I metric when thebackground fluid corresponds to a cosmological con-stant, H tends asymptotically to constant values de-pending on the initial conditions which is consistentwith de Sitter expansion. In addition, for FLRW mod-els with negative curvature, for any γ < and Ω k > Ω k → γ > and Ω k > Ω k → τ → + ∞ ¯ Ω ( τ ) = const., and lim τ → + ∞ H ( τ ) =
0. It wasalso proved numerically that as H → Ω gives an upper bound for the values Ω of the originalsystem. Therefore, by controlling the error of the time-averaged higher order system one can also controls theerror of the original system.Finally, in the present research we have proved thatin Kantowski-Sachs metric the late-time attractors ofthe full and time-averaged systems are: two anisotropiccontracting solutions for 0 ≤ γ <
2: a non-Flat LRSKasner Bianchi I and a Taub (flat LRS Kasner) andthe flat matter dominated FLRW universe and the flatmatter dominated FLRW universe if 0 ≤ γ ≤ (mim-icking de Sitter, quintessence or zero acceleration so-lutions). For FLRW metric with k = + < γ < c = c and large t , the flat matter dominatedFLRW universe for 0 < γ < (mimicking de Sitter,quintessence or zero acceleration solutions), and a mat-ter dominated contracting isotropic solution for 1 < γ <
2. In all metrics, the matter dominated flat FLRW uni-verse represents quintessence fluid if 0 ≤ γ ≤ . There-fore, this analysis completes the characterization of thefull class of homogeneous but anisotropic solutions andtheir isotropic limits.Our analytical results were strongly supported bynumerics in appendix C. We have shown that Asymp-totic Methods and Averaging Theory are powerful tools to investigate scalar field cosmologies with generalizedharmonic potential. Acknowledgements
This research was funded by Agencia Nacional deInvestigaci´on y Desarrollo- ANID through the pro-gram FONDECYT Iniciaci´on grant no. 11180126 andby Vicerrector´ıa de Investigaci´on y Desarrollo Tec-nol´ogico at Universidad Cat´olica del Norte. Ellen delos M. Fern´andez Flores is acknowledged for proof-reading this manuscript and for improving the English.
Appendix A: Main equations
Kantowski-Sachs metric . Defining D = (cid:114) H + K , Q = HD , Ω = (cid:115) ω φ + ˙ φ D , Σ = σ + D , Φ = t ω − tan − (cid:18) ωφ ˙ φ (cid:19) , (A1)we obtain˙ Ω = − b µ √ D cos ( t ω − Φ ) sin (cid:32) √ D sin ( t ω − Φ ) Ω f ω (cid:33) − b f γ Q Ω µ D sin (cid:113) D sin ( t ω − Φ ) Ω f ω + (cid:0) ω − µ (cid:1) Ω ω sin ( t ω − Φ ) + DQ Ω cos ( t ω − Φ ) (cid:18)(cid:18) γ (cid:18) µ ω − (cid:19) + (cid:19) Ω − (cid:19) + D Ω (cid:16) − Σ Q + Σ + (cid:18) − ( γ − ) Σ − γ µ Ω ω + γ (cid:19) Q (cid:17) , (A2a)˙ Σ = − b f γ Q Σ µ D sin (cid:113) D sin ( t ω − Φ ) Ω f ω − D ( γ − ) Q ΣΩ cos ( t ω − Φ )+ D (cid:16) − γ µ Q ΣΩ ω sin ( t ω − Φ ) − (cid:0) Q + ( γ − ) Σ Q − (cid:1) (cid:0) Σ − (cid:1) (cid:17) , (A2b)˙ Q = − b f γ (cid:0) Q − (cid:1) µ D sin (cid:113) D sin ( t ω − Φ ) Ω f ω − D ( γ − ) (cid:0) Q − (cid:1) Ω cos ( t ω − Φ )+ D (cid:0) − Q (cid:1) (cid:16) ( γ − ) Σ + Q Σ − γ + + γ µ Ω ω sin ( t ω − Φ ) (cid:17) , (A2c)˙ Φ = − b µ √ D Ω sin ( t ω − Φ ) sin (cid:32) √ D sin ( t ω − Φ ) Ω f ω (cid:33) + (cid:0) ω − µ (cid:1) ω sin ( t ω − Φ ) , (A2d)4˙ D = b f µ γ Q sin (cid:113) D sin ( t ω − Φ ) Ω f ω + D ( γ − ) Q Ω cos ( t ω − Φ )+ D (cid:16) Σ Q − Σ + ( γ − ) Σ Q + γ (cid:18) µ sin ( t ω − Φ ) Ω ω − (cid:19) Q (cid:17) , (A2e)and the deceleration parameter is q = − b f γ µ D Q sin (cid:113) D sin ( t ω − Φ ) Ω f ω − ( γ − ) cos ( t ω − Φ ) Ω Q + ( − γ ) Σ − γµ sin ( t ω − Φ ) Ω ω + γ − Q . (A3)The simultaneous change ( t , Σ , Q , Φ ) (cid:55)→ ( − t , − Σ , − Q , − Φ ) left invariant the system (A2). Setting the constant b µ + f µ − f ω = = ⇒ f = b µ ω − µ , the fractional energy density of matter Ω m : = ρ m H = ρ m Q D is parame-terized by the equation Q Ω m = − Σ − Ω + Ω (cid:18) − µ ω (cid:19) sin ( Φ − t ω ) + b µ D (cid:0) µ − ω (cid:1) sin (cid:113) D Ω (cid:0) µ − ω (cid:1) sin ( Φ − t ω ) b µ ω = − Σ − Ω + O ( H ) . (A4) FLRW metric with positive curvature . Defining D = (cid:114) H + a , Q = HD , Ω = (cid:115) ω φ + ˙ φ D , Φ = t ω − tan − (cid:18) ωφ ˙ φ (cid:19) , (A5)˙ Ω = − b γ f µ Q Ω D sin (cid:113) D Ω sin ( t ω − Φ ) f ω − b µ √ D cos ( t ω − Φ ) sin (cid:32) √ D Ω sin ( t ω − Φ ) f ω (cid:33) + DQ Ω cos ( t ω − Φ ) (cid:18) Ω (cid:18) γ (cid:18) µ ω − (cid:19) + (cid:19) − (cid:19) + γ DQ Ω (cid:18) − µ Ω ω (cid:19) + (cid:0) ω − µ (cid:1) Ω sin ( t ω − Φ ) ω , (A6a)˙ Q = b γ f µ (cid:0) − Q (cid:1) D sin (cid:113) D Ω sin ( t ω − Φ ) f ω − D (cid:0) − Q (cid:1) (cid:18) γ − γ µ Ω sin ( t ω − Φ ) ω − (cid:19) + ( γ − ) D (cid:0) − Q (cid:1) Ω cos ( t ω − Φ ) , (A6b)˙ Φ = − b µ sin ( t ω − Φ ) √ D Ω sin (cid:32) √ D Ω sin ( t ω − Φ ) f ω (cid:33) − DQ sin ( t ω − Φ ) cos ( t ω − Φ ) + (cid:0) ω − µ (cid:1) sin ( t ω − Φ ) ω , (A6c)5˙ D = b γ f µ Q sin (cid:113) D Ω sin ( t ω − Φ ) f ω + γ D Q (cid:18) µ Ω sin ( t ω − Φ ) ω − (cid:19) + ( γ − ) D Q Ω cos ( t ω − Φ ) . (A6d)and the deceleration parameter is q = γ − Q − b γ f µ D Q sin (cid:113) D Ω sin ( t ω − Φ ) f ω − Ω (cid:18) γ µ sin ( t ω − Φ ) ω Q + ( γ − ) cos ( t ω − Φ ) Q (cid:19) . (A7)The fractional energy density of matter Ω m : = ρ m H = ρ m Q D is parameterized by the equation Q Ω m = − b f µ D sin (cid:32) √ D Ω sin ( t ω − Φ ) f ω (cid:33) − µ Ω ω sin ( t ω − Φ ) − Ω cos ( t ω − Φ ) . (A8)We have1 = Ω + ρ m D + D ( ω − µ ) U ( φ ) , (A9)where the function defined by U ( φ ) = f (cid:16) − cos (cid:16) φ f (cid:17)(cid:17) − φ , has a local degeneratedmaximum of order 2 at φ = ω > µ , Ω and Ω m : = ρ m H can be greaterthan 1. Appendix B: Proof of Theorem 3
Lemma 6 (Gronwall’s Lemma) Differentialform .i) Let be η ( t ) a nonnegative function, absolutelycontinuous over [ , T ] which satisfies almost every-where the differential inequality: η (cid:48) ( t ) ≤ φ ( t ) η ( t ) + ψ ( t ) , where φ ( t ) and ψ ( t ) are nonnegative summable func-tions over [ , T ] . Then, η ( t ) ≤ e (cid:82) t φ ( s ) ds (cid:20) η ( ) + (cid:90) t ψ ( s ) ds (cid:21) is satisfied for all ≤ t ≤ T .ii) In particular, if η (cid:48) ( t ) ≤ φ ( t ) η ( t ) , t ∈ [ , T ] , η ( ) = , then η ≡ , t ∈ [ , T ] .2. Integral form i) Let be ξ ( t ) a nonnegative function, summable over [ , T ] which satisfies almost everywhere the integral in-equality ξ ( t ) ≤ C (cid:90) t ξ ( s ) ds + C , C , C ≥ . Then, ξ ( t ) ≤ C ( + C te C t ) , almost everywhere for t in ≤ t ≤ T .ii) In particular, if ξ ( t ) ≤ C (cid:90) t ξ ( s ) ds , C ≥ almost everywhere for t in ≤ t ≤ T . Then, ξ ≡ almost everywhere for t in ≤ t ≤ T .
Theorem 3 states:
Let be defined the functions ¯ Ω , ¯ Σ , ¯ Ω k , ¯ Φ , and D, satisfying the time-averaged equa-tions (40) , (41) , (42) , (43) and (44) . Then, there existsthree functions g , g , g and g , such that if Ω = Ω + Dg ( D , Ω , Σ , Q , Φ , t ) , Σ = Σ + Dg ( D , Ω , Σ , Q , Φ , t ) , Q = Q + Dg ( D , Ω , Σ , Q , Φ , t ) , Φ = Φ + Dg ( D , Ω , Σ , Q , Φ , t ) , the functions Ω , Σ , Q , Φ and ¯ Ω , ¯ Σ , ¯ Q , ¯ Φ have thesame limit on a time scale tD = O ( ) . Setting Σ = Σ = , are derived analogous results for positively curvedFLRW model. Proof.
From the equation (37) it follows that D is amonotonic decreasing function of t whenever 0 < Q < , Σ + Ω < , − Q + Q Σ >
0. The last restrictionholds, in particular, by choosing Q > , Σ >
0. In thelast case, we define the bootstrapping sequences t = t , η = , D = D ( t ) , t n + = t n + D n D n + = D ( t n + ) . (B1)6This sequence, however, is finite; that is t n < t ∗ with t ∗ satisfying lim t → t ∗ D ( t ) = ∞ . t ∗ is estimated as: t ∗ ≥ sup { t : 1 − Q ( t ) + Q ( t ) Σ ( t ) > } . If Q ( t n ) < Σ ( t n ) < n we stop integration because D changes the monotony to become a mono-tonic increasing function. However, the change ( t , η , Σ , Q , Φ ) (cid:55)→ ( − t , − η , − Σ , − Q , − Φ ) left invariant the system.Therefore, the solution is completed by using the above symmetry. Given the expansions (45), we obtain:˙ Ω = DQ Ω (cid:32) γ (cid:0) − Σ − Ω (cid:1) − Q Σ + Σ + Ω − (cid:33) + DQ (cid:0) Ω − (cid:1) Ω cos ( ( Φ − t ω )) + D Σ Ω − D ∂ g ∂ t + O ( D ) , (B2)˙ Σ = D (cid:32) (cid:0) Σ − (cid:1) (cid:0) − Q − ( γ − ) Q Σ + (cid:1) − ( γ − ) Q Σ Ω (cid:33) + DQ Σ Ω cos ( ( Φ − t ω )) − D ∂ g ∂ t + O ( D ) , (B3)˙ Q = D (cid:0) − Q (cid:1) (cid:32) γ (cid:0) Σ + Ω − (cid:1) + Σ ( Q − Σ ) − Ω + (cid:33) + D (cid:0) Q − (cid:1) Ω cos ( ( Φ − t ω )) − D ∂ g ∂ t + O ( D ) , (B4)˙ Φ = (cid:18) Q sin ( ( Φ − t ω )) − ∂ g ∂ t (cid:19) D + O (cid:0) D (cid:1) . (B5)Let be defined ∆Ω = Ω − ¯ Ω , ∆Σ = Σ − ¯ Σ , ∆ Q = Q − ¯ Q , ∆Φ = Φ − ¯ Φ and taking the same initial conditionsat t = t n : Ω ( t n ) = ¯ Ω ( t n ) = Ω n , Σ ( t n ) = ¯ Σ ( t n ) = Σ n , Q ( t n ) = ¯ Q ( t n ) = Q n , Φ ( t n ) = ¯ Φ ( t n ) = Φ n , such that 0 < Q n < , Σ n + Ω n < , − Q n + Q n Σ n > . Setting ∂ g ∂ t = Ω (cid:32) Q (cid:16) γ (cid:0) − Σ − Ω (cid:1) − Q Σ + Σ + Ω − (cid:17) + Σ − (cid:33) + Q Ω (cid:0) Ω − (cid:1) cos ( ( Φ − t ω ))+ ¯ Ω (cid:32) ¯ Σ (cid:18) ( γ − ) ¯ Q − (cid:19) − ( γ − ) ¯ Q + (cid:0) ¯ Q − (cid:1) ¯ Σ + (cid:33) + ¯ Ω (cid:18) ( γ − ) ¯ Q − (cid:19) + Ω ¯ Σ + Ω ¯ Ω , (B6a) ∂ g ∂ t = (cid:32) Q (cid:0) − Σ (cid:1) − ( γ − ) Q Σ (cid:0) Σ − (cid:1) − ( γ − ) Q Σ Ω + Σ ( Σ − ) (cid:33) + Q Σ Ω cos ( ( Φ − t ω )) + ¯ Ω (cid:32) ¯ Σ (cid:18) ( γ − ) ¯ Q − (cid:19) + Σ (cid:33) + ¯ Σ (cid:18) ( γ − ) ¯ Q − (cid:19) + ¯ Σ (cid:18)(cid:18) − γ (cid:19) ¯ Q + (cid:19) + ¯ Σ (cid:18) ¯ Q + Σ − (cid:19) − ¯ Q , (B6b)7 ∂ g ∂ t = (cid:32) ( γ − ) Σ + ( γ − ) Ω − Q Σ + Q (cid:0) − γ (cid:0) Σ + Ω − (cid:1) + Σ + Ω − (cid:1) + Q ( Σ − ) (cid:33) + (cid:0) Q − (cid:1) Ω cos ( ( Φ − t ω )) + ¯ Σ (cid:32) ( γ − ) ¯ Q − Q + ( − γ + Q + ) (cid:33) + ¯ Ω (cid:32) ( γ − ) ¯ Q − Q + ( − γ + Q + ) (cid:33) + (cid:18) − γ (cid:19) ¯ Q + (cid:0) ¯ Q − ¯ Q (cid:1) ¯ Σ + Q , (B6c) ∂ g ∂ t = Q sin ( ( Φ − t ω )) , (B6d)where we have substituted the explicit dependence on t of the time-averaged functions ¯ Ω , ¯ Σ and ¯ Q .The explicit expressions for the g i obtained by integration of (B6) as follows: g ( D , Ω , Σ , Q , Φ , t ) = Ω (cid:32) Q (cid:16) γ (cid:0) − Σ − Ω (cid:1) − Q Σ + Σ + Ω − (cid:17) + Σ − (cid:33) t + ω Q Ω (cid:0) − Ω (cid:1) sin ( ( Φ − t ω ))+ (cid:90) (cid:34) ¯ Ω (cid:32) ¯ Σ ( t ) (cid:18) ( γ − ) ¯ Q ( t ) − (cid:19) − ( γ − ) ¯ Q ( t ) + (cid:0) ¯ Q ( t ) − (cid:1) ¯ Σ + (cid:33) + ¯ Ω ( t ) (cid:18) ( γ − ) ¯ Q ( t ) − (cid:19) + Ω ¯ Σ ( t ) + Ω ¯ Ω ( t ) (cid:35) dt + C ( Σ , Ω , Q , Φ ) , (B7) g ( D , Ω , Σ , Q , Φ , t ) = (cid:32) Q (cid:0) − Σ (cid:1) − ( γ − ) Q Σ (cid:0) Σ − (cid:1) − ( γ − ) Q Σ Ω + Σ ( Σ − ) (cid:33) t − ω Q Σ Ω sin ( ( Φ − t ω ))+ (cid:90) (cid:34) ¯ Ω ( t ) (cid:32) ¯ Σ ( t ) (cid:18) ( γ − ) ¯ Q ( t ) − (cid:19) + Σ (cid:33) + ¯ Σ ( t ) (cid:18) ( γ − ) ¯ Q ( t ) − (cid:19) + ¯ Σ ( t ) (cid:18)(cid:18) − γ (cid:19) ¯ Q ( t ) + (cid:19) + ¯ Σ ( t ) (cid:18) ¯ Q ( t ) + Σ − (cid:19) − ¯ Q ( t ) (cid:35) dt + C ( Σ , Ω , Q , Φ ) , (B8) g ( D , Ω , Σ , Q , Φ , t )= (cid:32) ( γ − ) Σ + ( γ − ) Ω − Q Σ + Q (cid:0) − γ (cid:0) Σ + Ω − (cid:1) + Σ + Ω − (cid:1) + Q ( Σ − ) (cid:33) t + ω (cid:0) − Q (cid:1) Ω sin ( ( Φ − t ω ))+ (cid:90) (cid:34) ¯ Σ ( t ) (cid:32) ( γ − ) ¯ Q ( t ) − Q ( t ) + ( − γ + Q + ) (cid:33) + ¯ Ω ( t ) (cid:32) ( γ − ) ¯ Q ( t ) − Q ( t ) + ( − γ + Q + ) (cid:33) + (cid:18) − γ (cid:19) ¯ Q ( t ) + (cid:0) ¯ Q ( t ) − ¯ Q ( t ) (cid:1) ¯ Σ ( t ) + Q ( t ) (cid:35) dt + C ( H , Σ , Ω , Q , Φ ) , (B9)8 g ( D , Ω , Σ , Q , Φ , t ) = ( Φ − t ω ) ω + C ( Ω , Σ , Q , Φ ) , (B10)where C i , i = , , , D because theterms ∂ g i ∂ D ˙ D in the procedure contribute with O ( D ) -factors. The terms unbarred inside the integral are taken asconstants during the integration. We set C ≡ ∆Ω = − D ∆Ω ( − ¯ Ω − ¯ Σ ) + O ( D ) , ˙ ∆Σ = − D ∆Σ ( − ¯ Ω − ¯ Σ ) + O (cid:0) D (cid:1) , ˙ ∆ Q = − D ∆ Q ( − ¯ Ω − ¯ Σ ) + O (cid:0) D (cid:1) , ˙ ∆Φ = D (cid:34) (cid:0) µ − ω (cid:1) ¯ Ω ( t ) b µ ω + g sin ( ( Φ − t ω ))+ Q (cid:0) Ω + (cid:1) cos ( Φ − t ω ) cos ( ( Φ − t ω )) ω + Q cos ( Φ − t ω ) ω (cid:32) Q (cid:16) − γ (cid:0) Σ + Ω − (cid:1) − Q Σ + Σ + Ω (cid:17) + Σ (cid:33)(cid:35) + O (cid:0) D (cid:1) . Continuing the proof, let t ∈ [ t n , t n + ] such that t − t n ≤ t n + − t n = D n we have the following: | ∆Ω ( t ) | = | Ω ( t ) − ¯ Ω ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) tt n (cid:16) ˙ Ω ( s ) − ˙¯ Ω ( s ) (cid:17) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) tt n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) − ¯ Σ − ¯ Ω (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:124) (cid:123)(cid:122) (cid:125) ≤ L D (cid:124)(cid:123)(cid:122)(cid:125) ≤ D n | ∆Ω ( s ) | ds + M D n ( t − t n ) + O ( D n ) ≤ LD n (cid:90) tt n | ∆Ω ( s ) | ds + M D n ( t − t n ) + O ( D n ) , | ∆Σ | = | Σ − ¯ Σ | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) tt n (cid:16) ˙ Σ ( s ) − ˙¯ Σ ( s ) (cid:17) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) tt n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) − ¯ Σ − ¯ Ω (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:124) (cid:123)(cid:122) (cid:125) ≤ L D (cid:124)(cid:123)(cid:122)(cid:125) ≤ D n | ∆Σ ( s ) | ds + M D n ( t − t n ) + O ( D n ) ≤ LD n (cid:90) tt n | ∆Σ ( s ) | ds + M D n ( t − t n ) + O ( D n ) , | ∆ Q | = | Q − ¯ Q | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) tt n (cid:16) ˙ Q ( s ) − ˙¯ Q ( s ) (cid:17) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) tt n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) − ¯ Σ − ¯ Ω (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:124) (cid:123)(cid:122) (cid:125) ≤ L D (cid:124)(cid:123)(cid:122)(cid:125) ≤ D n | ∆ Q ( s ) | ds + M D n ( t − t n ) + O ( D n ) ≤ LD n (cid:90) tt n | ∆ Q ( s ) | ds + M D n ( t − t n ) + O ( D n ) , | ∆Φ ( t ) | = | Φ ( t ) − ¯ Φ ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) tt n (cid:16) ˙ Φ ( s ) − ˙¯ Φ ( s ) (cid:17) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) tt n D (cid:40) (cid:0) µ − ω (cid:1) ¯ Ω b µ ω +
32 sin ( ( Φ − s ω )) g + Q (cid:0) Ω + (cid:1) cos ( Φ − s ω ) cos ( ( Φ − s ω )) ω + Q cos ( Φ − s ω ) ω (cid:32) Q (cid:16) − γ (cid:0) Σ + Ω − (cid:1) − Q Σ + Σ + Ω (cid:17) + Σ (cid:33)(cid:41) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O ( H ) ≤ M D n ( t − t n ) + O ( D n ) , whereby assuming C i ( Ω , Σ , Q , Φ ) , i = , . . . , g i , i = , . . . , t ∈ [ t n , t n + ] . By continuity of¯ Ω , Ω , ¯ Σ , Σ , Φ , ¯ Ω k , Q the following constants L , M , M , M and M are finite: L = max t ∈ [ t n , t n + ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) − ¯ Ω − ¯ Σ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ , M = max t ∈ [ t n , t n + ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:32) ¯ Q (cid:16) ( γ − ) ¯ Σ + ( γ − ) ¯ Ω − γ + (cid:17) − ¯ Q (cid:16) Σ + Σ + Ω − (cid:17) + Q ¯ Σ + ( − γ + Q + ) ¯ Σ + ( − γ + Q + ) ¯ Ω + γ − Q − (cid:33) ∂ g ∂ Q + (cid:32) Ω (cid:0) ¯ Σ (cid:0) ( γ − ) ¯ Q − (cid:1) + Σ (cid:1) + (cid:16) ¯ Σ − (cid:17)(cid:16) Σ (cid:0) ( γ − ) ¯ Q − (cid:1) + Q + Σ − (cid:17)(cid:33) ∂ g ∂ Σ + (cid:32) Σ (cid:0) ¯ Ω (cid:0) ( γ − ) ¯ Q − (cid:1) + Ω (cid:1) + (cid:16) ¯ Ω − (cid:17)(cid:0) ¯ Ω (cid:0) ( γ − ) ¯ Q − (cid:1) + Ω (cid:1) + (cid:16) ¯ Q − (cid:17) ¯ Σ ¯ Ω (cid:33) ∂ g ∂ Ω + (cid:32) Q (cid:16) γ (cid:16) − Σ − Ω (cid:17) − Q Σ + Σ + Ω − (cid:17) + Σ + Q (cid:16) Ω − (cid:17) cos ( ( Φ − t ω )) (cid:33) g − Ω (cid:16) Q + ( γ − ) Q Σ − (cid:17) g + (cid:32) Ω (cid:16) γ (cid:16) − Σ − Ω (cid:17) − Q Σ + Σ + Ω − (cid:17) + Ω (cid:16) Ω − (cid:17) cos ( ( Φ − t ω )) (cid:33) g − Q Ω (cid:0) Ω − (cid:1) sin ( Φ − t ω ) cos ( Φ − t ω ) ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ . M = max t ∈ [ t n , t n + ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:32) ¯ Q (cid:16) ( γ − ) ¯ Σ + ( γ − ) ¯ Ω − γ + (cid:17) − ¯ Q (cid:16) Σ + Σ + Ω − (cid:17) + Q ¯ Σ + ( − γ + Q + ) ¯ Σ + ( − γ + Q + ) ¯ Ω + γ − Q − (cid:33) ∂ g ∂ Q + (cid:32) Ω (cid:0) ¯ Σ (cid:0) ( γ − ) ¯ Q − (cid:1) + Σ (cid:1) + (cid:16) ¯ Σ − (cid:17)(cid:16) Σ (cid:0) ( γ − ) ¯ Q − (cid:1) + Q + Σ − (cid:17)(cid:33) ∂ g ∂ Σ + (cid:32) Σ (cid:0) ¯ Ω (cid:0) ( γ − ) ¯ Q − (cid:1) + Ω (cid:1) + (cid:16) ¯ Ω − (cid:17)(cid:0) ¯ Ω (cid:0) ( γ − ) ¯ Q − (cid:1) + Ω (cid:1) + (cid:16) ¯ Q − (cid:17) ¯ Σ ¯ Ω (cid:33) ∂ g ∂ Ω + ( Q Σ Ω cos ( ( Φ − t ω )) − ( γ − ) Q Σ Ω ) g + (cid:32) Q Ω cos ( ( Φ − t ω )) + Σ − Q (cid:16) γ (cid:16) Σ + Ω − (cid:17) + Σ ( Q − Σ ) − Ω + (cid:17)(cid:33) g + (cid:32) (cid:16) − ( γ − ) Σ Ω − (cid:16) Σ − (cid:17) ( ( γ − ) Σ + Q ) (cid:17) + Σ Ω cos ( ( Φ − t ω )) (cid:33) g − Q Σ Ω sin ( Φ − t ω ) cos ( Φ − t ω ) ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ , M = max t ∈ [ t n , t n + ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:32) ¯ Q (cid:16) ( γ − ) ¯ Σ + ( γ − ) ¯ Ω − γ + (cid:17) − ¯ Q (cid:16) Σ + Σ + Ω − (cid:17) + Q ¯ Σ + ( − γ + Q + ) ¯ Σ + ( − γ + Q + ) ¯ Ω + γ − Q − (cid:33) ∂ g ∂ Q + (cid:32) Ω (cid:0) ¯ Σ (cid:0) ( γ − ) ¯ Q − (cid:1) + Σ (cid:1) + (cid:16) ¯ Σ − (cid:17)(cid:16) Σ (cid:0) ( γ − ) ¯ Q − (cid:1) + Q + Σ − (cid:17)(cid:33) ∂ g ∂ Σ + (cid:32) Σ (cid:0) ¯ Ω (cid:0) ( γ − ) ¯ Q − (cid:1) + Ω (cid:1) + (cid:16) ¯ Ω − (cid:17)(cid:0) ¯ Ω (cid:0) ( γ − ) ¯ Q − (cid:1) + Ω (cid:1) + (cid:16) ¯ Q − (cid:17) ¯ Σ ¯ Ω (cid:33) ∂ g ∂ Ω + (cid:16) (cid:16) Q − (cid:17) Ω cos ( ( Φ − t ω )) − ( γ − ) (cid:16) Q − (cid:17) Ω (cid:17) g − (cid:16) Q − (cid:17) ( ( γ − ) Σ + Q ) g + (cid:32) Q (cid:16) − γ (cid:16) Σ + Ω − (cid:17) − Q Σ + Σ + Ω − (cid:17) + Σ + Q Ω cos ( ( Φ − t ω )) (cid:33) g − Q (cid:0) Q − (cid:1) Ω sin ( Φ − t ω ) cos ( Φ − t ω ) ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ . M = max t ∈ [ t n , t n + ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) µ − ω (cid:1) ¯ Ω b µ ω +
32 sin ( ( Φ − t ω )) g + Q (cid:0) Ω + (cid:1) cos ( Φ − t ω ) cos ( ( Φ − t ω )) ω + Q cos ( Φ − t ω ) ω (cid:32) Q (cid:16) − γ (cid:16) Σ + Ω − (cid:17) − Q Σ + Σ + Ω (cid:17) + Σ (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ . Using Gronwall’s Lemma 6, we have for t ∈ [ t n , t n + ] : (cid:12)(cid:12)(cid:12) ∆Ω ( t ) (cid:12)(cid:12)(cid:12) ≤ M D n ( t − t n ) (cid:104) + LD n ( t − t n ) e LD n ( t − t n ) (cid:105) (cid:46) M ( t − t n ) D n + O ( D n ) ≤ M D n + O ( D n ) , (cid:12)(cid:12)(cid:12) ∆Σ ( t ) (cid:12)(cid:12)(cid:12) ≤ M D n ( t − t n ) (cid:104) + LD n ( t − t n ) e LD n ( t − t n ) (cid:105) (cid:46) M ( t − t n ) D n + O ( H n ) ≤ M D n + O ( D n ) , (cid:12)(cid:12)(cid:12) ∆Ω k ( t ) (cid:12)(cid:12)(cid:12) ≤ M D n ( t − t n ) (cid:104) + LH n ( t − t n ) e LD n ( t − t n ) (cid:105) (cid:46) M ( t − t n ) D n + O ( H n ) ≤ M H n + O ( D n ) , | ∆Φ ( t ) | ≤ M D n , due to t − t n ≤ t n + − t n = D n .Finally, taking the limit as n → ∞ , we obtain D n →
0. Then, as D n →
0, it follows the functions Ω , Σ , Q , Φ and ¯ Ω , ¯ Σ , ¯ Q , ¯ Φ have the same limit as τ → ∞ . (cid:3)
1. Initial layer and exterior solution forKantowski-Sachs or closed FLRW model
According to Remark 1, Theorem 3 is valid on a timescale t < t ∗ , where D remains close to zero, but at a crit-ical time t ∗ : lim t → t ∗ D ( t ) = ∞ . A lower bound for t ∗ isestimated as ¯ t = sup { t > − Q ( t ) + Q ( t ) Σ ( t ) > } . If ¯ t = ∞ this would means that 1 − Q ( t ) + Q ( t ) Σ ( t ) > D → t → ∞ , η → ∞ , in contradiction to lim t → t ∗ D ( t ) = ∞ .Then, generically ¯ t < ∞ . Then, the vector of solu-tions can be expressed as the sum of an exterior so-lution as D → + (which corresponds to the solution ofthe time-averaged system) and an inner solution when D → ∞ , t → t ∗− . By inner solution we mean the correc-tion of the solution in the interval ( t ∗− , t ∗ ) which is theinterval of non-uniform-continuity of D . Therefore, wepropose the expansion: Ω ( t ) = ¯ Ω ( t ) + D ( t ) Ω in (cid:18) t ∗ − tD ( t ) (cid:19) , Q ( t ) = ¯ Q ( t ) + D ( t ) Q in (cid:18) t ∗ − tD (( t ) (cid:19) , Σ ( t ) = ¯ Σ ( t ) + D ( t ) Σ in (cid:18) t ∗ − tD ( t ) (cid:19) , Φ ( t ) = ¯ Φ ( t ) + D ( t ) Φ in (cid:18) t ∗ − tD ( t ) (cid:19) , (B11)where T = t ∗ − tD is the slow-time when D → ∞ , t → t ∗− and the fast variable as t (cid:28) t ∗ and D → + . The initialconditions are: Ω in ( + ) = , Q in ( + ) = , Σ in ( + ) = , Φ in ( + ) = . For the inner solution we propose theasymptotic expansion as D → ∞ : Ω in ( T ) = Ω in ( T ) + N ∑ j = Ω jin ( T ) D ( t ( T )) j + O (cid:18) D N + (cid:19) , Q in ( T ) = Q in ( T ) + N ∑ j = Q jin ( T ) D ( t ( T )) j + O (cid:18) D N + (cid:19) , Σ in ( T ) = Σ in ( T ) + N ∑ j = Σ jin ( T ) D ( t ( T )) j + O (cid:18) D N + (cid:19) , Φ in ( T ) = Φ in ( T ) + N ∑ j = Φ jin ( T ) D ( t ( T )) j + O (cid:18) D N + (cid:19) , (B12)where t ( T ) is found by inversion of t ∗ − tD ( t ) = T . (B13)The initial conditions are: Ω in ( + ) = , Σ in ( + ) = , Q in ( + ) = , Φ in ( + ) = , and Ω jin ( + ) = , Σ jin ( + ) = , Q jin ( + ) = , Φ jin ( + ) = , j > . Defining x = ( Ω , Q , Σ , Φ ) T , ¯ x = ( ¯ Ω , ¯ Q , ¯ Σ , ¯ Φ ) T and x in = ( Ω in , Q in , Σ in , Φ in ) T . Equation (B11) can be ex-pressed as x ( t ) = ¯ x ( t ) + D ( t ) x in (cid:18) t ∗ − tD ( t ) (cid:19) + x in (cid:18) t ∗ − tD (cid:19) + N ∑ j = x jin (cid:16) t ∗ − tD (cid:17) D ( t ) j − + . . . , t < t ∗ . (B14)It implies: d x ( t ) dt = d ¯ x ( t ) dt + ˙ D ( t ) x in (cid:18) t ∗ − tD ( t ) (cid:19) − d x in ( T ) dT (cid:18) + ( t ∗ − t ) ˙ D ( t ) D ( t ) (cid:19) − N ∑ j = d x jin ( T ) dT (cid:18) + ( t ∗ − t ) ˙ D ( t ) D ( t ) (cid:19) D ( t ) j − N ∑ j = ( j − ) x jin ( T ) ˙ D ( t ) D ( t ) j + . . . . (B15)Then, d x in ( T ) dT = D ( t ) (cid:16) d ¯ x ( t ) dt − d x ( t ) dt (cid:17) D ( t ) + ( t ∗ − t ) ˙ D ( t ) + ˙ D ( t ) D ( t ) D ( t ) + ( t ∗ − t ) ˙ D ( t ) x in ( T ) − D ( t ) x in ( T ) − N ∑ j = (cid:34) d x jin ( T ) dT + ( j − ) x jin ( T ) ˙ D ( t ) D ( t ) (cid:0) D ( t ) + ( t ∗ − t ) ˙ D ( t ) (cid:1) (cid:35) D ( t ) j + . . . . (B16)Defining¯ f ( ¯ x )= ¯ Ω (cid:32) ¯ Q (cid:16) − γ (cid:0) ¯ Σ + ¯ Ω − (cid:1) − Q ¯ Σ + Σ + Ω − (cid:17) + Σ (cid:33) − (cid:0) ¯ Q − (cid:1) (cid:16) γ (cid:0) ¯ Σ + ¯ Ω − (cid:1) + Σ ( Q − Σ ) − Ω + (cid:17) (cid:16) (cid:0) ¯ Σ − (cid:1) (cid:0) − Q − ( γ − ) ¯ Q ¯ Σ + (cid:1) − ( γ − ) ¯ Q ¯ Σ ¯ Ω (cid:17) , f − ( t , x , D )= − b µ √ cos ( t ω − Φ ) sin (cid:16) √ D sin ( t ω − Φ ) Ω f ω (cid:17) − b f γ Q Ω µ sin (cid:32) (cid:113) D sin ( t ω − Φ ) Ω f ω (cid:33) − b f γ (cid:0) Q − (cid:1) µ sin (cid:32) (cid:113) D sin ( t ω − Φ ) Ω f ω (cid:33) − b f γ Q Σ µ sin (cid:32) (cid:113) D sin ( t ω − Φ ) Ω f ω (cid:33) − b µ √ Ω sin ( t ω − Φ ) sin (cid:16) √ D sin ( t ω − Φ ) Ω f ω (cid:17) , f ( t , x ) = ( ω − µ ) Ω sin ( t ω − Φ ) ω ( ω − µ ) sin ( t ω − Φ ) ω , f ( t , x ) = Ω (cid:16) Q (cid:0) Ω − (cid:1) ( − γ + ( Φ − t ω )) − Σ (cid:0) Q + ( γ − ) Q Σ − (cid:1) (cid:17) (cid:0) Q − (cid:1) (cid:16) − ( γ − ) Σ + γ − Q Σ + Ω ( − γ + ( Φ − t ω )) − (cid:17) (cid:16) (cid:0) Σ − (cid:1) (cid:0) − Q − ( γ − ) Q Σ + (cid:1) + Q ΣΩ ( − γ + ( Φ − t ω )) (cid:17) − Q sin ( ( t ω − Φ )) , q ( t , x , D ) = b f µ γ Q sin (cid:113) D sin ( t ω − Φ ) Ω f ω , q ( t , x ) = ( γ − ) Q Ω cos ( t ω − Φ )+ (cid:16) Σ Q − Σ + ( γ − ) Σ Q + γ (cid:18) µ sin ( t ω − Φ ) Ω ω − (cid:19) Q (cid:17) , such that lim D → (cid:2) f − ( t , x , D ) D − + f ( t , x ) (cid:3) = , lim D → q ( x ( t ) , D ) = , and f − ( t , x , D ) D − + f ( t , x ) , q ( t , x , D ) are bounded as D → ∞ . We obtain d ¯ x ( t ) dt = ¯ f ( ¯ x ( t )) D ( t ) , d x ( t ) dt = f − ( t , x ( t ) , D ( t )) D ( t ) − + f ( t , x ( t ))+ f ( t , x ( t )) D ( t ) , ˙ D ( t ) = q ( t , x ( t ) , D ( t )) + q ( t , x ( t )) D ( t ) . Using t ∗ − t = T D ( t ) . We obtain d x in ( T ) dT = q ( t , x ( t ) , D ( t )) + q ( t , x ( t )) D ( t ) + T ( q ( t , x ( t ) , D ( t )) + q ( t , x ( t )) D ( t ) ) x in ( T )+ + T ( q ( t , x ( t ) , D ( t )) + q ( t , x ( t )) D ( t ) ) × (cid:32)(cid:16) ¯ f ( ¯ x ( t )) − f ( t , x ( t )) (cid:17) D ( t ) − f − ( t , x ( t ) , D ( t )) D ( t ) − − f ( t , x ( t )) (cid:33) − d x in ( T ) dT D ( t ) − N ∑ j = (cid:34) d x jin ( T ) dT + ( j − ) x jin ( T ) × q ( t , x ( t ) , D ( t )) + q ( t , x ( t )) D ( t ) + T ( q ( t , x ( t ) , D ( t )) + q ( t , x ( t )) D ( t ) ) (cid:35) D ( t ) j + . . . . (B17)We obtain for large D : d x in ( T ) dT = T x in ( T ) + (cid:16) ¯ f ( ¯ x ( t )) − f ( t , x ( t )) (cid:17) T q ( x ( t )) D ( t ) − d x in ( T ) dT D ( t ) − N ∑ j = (cid:34) d x jin ( T ) dT + T ( j − ) x jin ( T ) (cid:35) D ( t ) j + . . . . (B18)Collecting the coefficients of same power in D − j , we3obtain d x in ( T ) dT = T x in ( T ) , (B19) d x in ( T ) dT = (cid:16) ¯ f ( ¯ x ( t )) − f ( t , x ( t )) (cid:17) T q ( t , x ( t )) , (B20) d x jin ( T ) dT = − T ( j − ) x jin ( T ) , j ≥ x in ( T ) = T C , (B22)where C is a vector of integration constants. The thirdequation admits the general solution x jin ( T ) = T j − C j , j ≥ C j is a vector of integration constants. The sec-ond equation can be written as T d x in ( T ) dT = (cid:16) ¯ f ( ¯ x ( t )) − f ( t , ¯ x ( t ) (cid:17) q ( t , x ( t ))+ (cid:16) f ( t , ¯ x ( t )) − f ( t , x ( t )) (cid:17) q ( t , x ( t )) , (B24)Expanding x ( t ) using (B14) and previous facts: x ( t ) − ¯ x ( t ) = ( t ∗ − t ) C + x in (cid:18) ( t ∗ − t ) D ( t ) (cid:19) + N ∑ j = C j ( t ∗ − t ) j − + O (cid:32) ( t ∗ − t ) N + (cid:33) . (B25)Suppose that (cid:107) f (cid:107) L ∞ t , x , (cid:107) q (cid:107) L ∞ t , x , (cid:13)(cid:13)(cid:13) q (cid:13)(cid:13)(cid:13) L ∞ t , x < ∞ , and that f ( t , x ) is Lipschitz continuous and q is con-tinuous with respect to x for all 0 ≤ t ≤ t ∗ − δ . Wehave (cid:90) T + s x in (cid:48) ( s ) ds = − (cid:90) t ∗ − δ t (cid:16) ¯ f ( ¯ x ( s )) − f ( s , ¯ x ( s ) (cid:17) q ( s , x ( s )) ds − (cid:90) t ∗ − δ t (cid:16) f ( ¯ x ( s )) − f ( s , x ( s )) (cid:17) q ( s , x ( s )) ds . s x in (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T + − (cid:90) T + x in ( s ) ds = − (cid:90) t ∗ − δ t (cid:16) ¯ f ( ¯ x ( s )) − f ( s , ¯ x ( s ) (cid:17) q ( s , x ( s )) ds − (cid:90) t ∗ − δ t (cid:16) f ( ¯ x ( s )) − f ( s , x ( s )) (cid:17) q ( s , x ( s )) ds . T x in ( T ) − (cid:90) T + x in ( s ) ds = − (cid:90) t ∗ − δ t (cid:16) ¯ f ( ¯ x ( s )) − f ( s , ¯ x ( s ) (cid:17) q ( s , x ( s )) ds − (cid:90) t ∗ − δ t (cid:16) f ( ¯ x ( s )) − f ( s , x ( s )) (cid:17) q ( s , x ( s )) ds . T x in ( T ) = − (cid:90) t ∗ − δ t x in (cid:18) t ∗ − sD ( s ) (cid:19) (cid:18) t ∗ − sD ( s ) (cid:19) (cid:48) ds − (cid:90) t ∗ − δ t (cid:16) ¯ f ( ¯ x ( s )) − f ( s , ¯ x ( s ) (cid:17) q ( s , x ( s )) ds − (cid:90) t ∗ − δ t (cid:16) f ( ¯ x ( s )) − f ( s , x ( s )) (cid:17) q ( s , x ( s )) ds . T x in ( T ) = (cid:90) t ∗ − δ t x in (cid:18) t ∗ − sD ( s ) (cid:19) (cid:18) + T ( s ) ˙ D ( s ) D ( s ) (cid:19) ds − (cid:90) t ∗ − δ t (cid:16) ¯ f ( ¯ x ( s )) − f ( s , ¯ x ( s ) (cid:17) q ( s , x ( s )) ds − (cid:90) t ∗ − δ t (cid:16) f ( ¯ x ( s )) − f ( s , x ( s )) (cid:17) q ( s , x ( s )) ds . D we have T x in ( T ) = (cid:90) t ∗ − δ t x in (cid:18) t ∗ − sD ( s ) (cid:19) ( t ∗ − s ) q ( s , x ( s )) ds − (cid:90) t ∗ − δ t (cid:16) ¯ f ( ¯ x ( s )) − f ( s , ¯ x ( s ) (cid:17) q ( s , x ( s )) ds − (cid:90) t ∗ − δ t (cid:16) f ( ¯ x ( s )) − f ( s , x ( s )) (cid:17) q ( s , x ( s )) ds . Using (B25) we have T x in ( T ) = (cid:90) t ∗ − δ t ( t ∗ − s ) q ( s , x ( s )) × (cid:34) x ( s ) − ¯ x ( s ) − ( t ∗ − s ) C − N ∑ j = C j ( t ∗ − s ) j − + O (cid:32) ( t ∗ − s ) N + (cid:33) (cid:35) ds − (cid:90) t ∗ − δ t (cid:16) ¯ f ( ¯ x ( s )) − f ( s , ¯ x ( s ) (cid:17) q ( s , x ( s )) ds − (cid:90) t ∗ − δ t (cid:16) f ( ¯ x ( s )) − f ( s , x ( s )) (cid:17) q ( s , x ( s )) ds . Then, (cid:107) T x in ( T ) (cid:107) ≤ (cid:107) q (cid:107) L ∞ t , x (cid:90) t ∗ − δ t ( t ∗ − s ) (cid:107) x ( s ) − ¯ x ( s ) (cid:107) ds + (cid:107) q (cid:107) L ∞ t , x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t ∗ − δ t (cid:34) ( t ∗ − s ) (cid:107) C (cid:107) + (cid:107) C (cid:107) + N ∑ j = (cid:107) C j + (cid:107) ( t ∗ − s ) j + O (cid:32) ( t ∗ − s ) N (cid:33) (cid:35) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:13)(cid:13)(cid:13) q (cid:13)(cid:13)(cid:13) L ∞ t , x (cid:90) t ∗ − δ t (cid:107) ¯ f ( ¯ x ( s )) − f ( s , ¯ x ( s ) (cid:107) ds + c L (cid:13)(cid:13)(cid:13) q (cid:13)(cid:13)(cid:13) L ∞ t , x (cid:90) t ∗ − δ t (cid:107) x ( s ) − ¯ x ( s ) (cid:107) ds . Let be t ∗ ∈ R , t < t ∗ , < δ < t ∗ − t . Then, (cid:107) q (cid:107) L ∞ t , x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t ∗ − δ t (cid:34) N ∑ j = (cid:107) C j + (cid:107) ( t ∗ − s ) j + O (cid:32) ( t ∗ − s ) N (cid:33) (cid:35) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) q (cid:107) L ∞ t , x max ≤ j ≤ N (cid:107) C j + (cid:107) (cid:90) t ∗ − δ t (cid:34) − ( t ∗ − s ) − N t ∗ − s − (cid:35) ds + (cid:107) q (cid:107) L ∞ t , x O (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ − N − ( t ∗ − t ) − N N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) = M ( t ∗ , t , δ , N ) , where M ( t ∗ , t , δ , N ) : = (cid:107) q (cid:107) L ∞ t , x max ≤ j ≤ N (cid:107) C j + (cid:107)× (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( − ) N (cid:16) B t − t ∗ + ( N , − N ) − B − δ ( N , − N ) (cid:17) + ln (cid:18) t − t ∗ + − δ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:107) q (cid:107) L ∞ t , x O (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ − N − ( t ∗ − t ) − N N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) and B z ( a , b ) = (cid:82) z t a − ( − t ) b − dt gives the incompletebeta function. By construction, ¯ M exists such that (cid:107) ¯ f ( ¯ x ( s )) − f ( s , ¯ x ( s ) (cid:107) ≤ ¯ M . Then, (cid:13)(cid:13)(cid:13) T x in ( T ) (cid:13)(cid:13)(cid:13) ≤ (cid:107) q (cid:107) L ∞ t , x (cid:90) t ∗ − δ t ( t ∗ − s ) (cid:107) x ( s ) − ¯ x ( s ) (cid:107) ds + c L (cid:13)(cid:13)(cid:13) q (cid:13)(cid:13)(cid:13) L ∞ t , x (cid:90) t ∗ − δ t (cid:107) x ( s ) − ¯ x ( s ) (cid:107) ds + L ( t ∗ , t , δ , n ) ≤ max (cid:26) (cid:107) q (cid:107) L ∞ t , x , c L (cid:13)(cid:13)(cid:13) q (cid:13)(cid:13)(cid:13) L ∞ t , x (cid:27)(cid:124) (cid:123)(cid:122) (cid:125) K × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t ∗ − δ t ( t ∗ + − s ) (cid:107) x ( s ) − ¯ x ( s ) (cid:107) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + L ( t ∗ , t , δ , N ) where L ( t ∗ , t , δ , N ) : = (cid:107) q (cid:107) L ∞ t , x (cid:16) (cid:107) C (cid:107) (cid:0) ( t ∗ − t ) − δ (cid:1) + (cid:107) C (cid:107) ( t ∗ − δ − t ) (cid:17) + M ( t ∗ , t , δ , N ) + (cid:13)(cid:13)(cid:13) q (cid:13)(cid:13)(cid:13) L ∞ t , x ¯ M ( t ∗ − t − δ ) According to Theorem 3 for any s with t < s < t ∗ − δ , < δ < t ∗ − t for which D ( s ) is finite.We have (cid:107) x ( s ) − ¯ x ( s ) (cid:107) ≤ (cid:107) M (cid:107) max t ≤ s ≤ t ∗ − δ D ( s ) where M = ( M , M , M , M ) T and M i , i = , , , ∀ s : t < s < t ∗ − δ , < δ < t ∗ − t : (cid:13)(cid:13)(cid:13) T x in ( T ) (cid:13)(cid:13)(cid:13) ≤ L ( t ∗ , t , δ , N )+ K (cid:107) M (cid:107) max t < s < t ∗ − δ D ( s ) (cid:90) t ∗ − δ t ( t ∗ + − s ) ds = L ( t ∗ , t , δ , N ) + K (cid:107) M (cid:107) max t ≤ s ≤ t ∗ − δ D ( s ) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( − δ + t − t ∗ − )( δ + t − t ∗ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We have the asymptotics: L ( t ∗ , t , δ , N ) ∼ − ln ( − δ ) ∼ δ , max t < s < t ∗ − δ D ( s ) ∼ D ( t ∗ − δ ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( − δ + t − t ∗ − )( δ + t − t ∗ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ δ ( δ + ) ∼ δ as δ → t → t ∗ . Hence (cid:13)(cid:13)(cid:13) T x in ( T ) (cid:13)(cid:13)(cid:13) ≤ δ [ + K (cid:107) M (cid:107) D ( t ∗ − δ )] + O (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ − N N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) . (B26)Finally, defining r = tt ∗ , we have x ( t ) − ¯ x ( t ) = t ∗ ( − r ) C + C t ∗ D ( t )( − r )+ N ∑ j = C j t ∗ j − ( − r ) j − + O (cid:32) t ∗ N ( − r ) N (cid:33) . (B27) Appendix C: Numerical simulation
In this section we present the numerical evidencethat support the main theorems presented in the sec- tion 3, solving numerically the full and time-averagedsystems obtained for Kantowski-Sachs and FLRW withpositive curvature metrics. For this purpose, it waselaborated an algorithm in the programming language
Python where the systems of differential equationswere solved using the solve ivp code provided by the
SciPy open-source
Python -based ecosystem. The in-tegration method used was
Radau that is an implicitRunge-Kutta method of the Radau IIa family of order5 with a relative and absolute tolerances of 10 − and10 − , respectively. All systems of differential equa-tions were integrated with respect to η , instead of t with an integration range of − ≤ η ≤
10 for theoriginal systems and − ≤ η ≤
100 for the time-averaged systems. All of them partitioned in 10000data points. Furthermore, each full and time-averagedsystems were solved considering only one matter com-ponent. These are cosmological constant ( γ = γ = γ = / γ = Ω = Ω m ≡ Ω m ≡ µ = √ / b = √ / ω = √
2, that leads to a value of f = b µ ω − µ = /
10 which fulfills the condition f ≥ V ( φ ) = φ + ( − cos ( φ )) . (C1)
1. Kantowski-Sachs
For the Kantowski-Sachs metric we integrate:1. The full system given by (A2).2. The time-averaged system:6 ∂ η ¯ Ω = ¯ Ω (cid:32) ¯ Q (cid:16) − γ (cid:0) ¯ Σ + ¯ Ω − (cid:1) − Q ¯ Σ + Σ + Ω − (cid:17) + Σ (cid:33) , ∂ η ¯ Q = − (cid:0) ¯ Q − (cid:1) (cid:16) γ (cid:0) ¯ Σ + ¯ Ω − (cid:1) + Σ ( Q − Σ ) − Ω + (cid:17) , ∂ η ¯ Σ = (cid:16) (cid:0) ¯ Σ − (cid:1) (cid:0) − Q − ( γ − ) ¯ Q ¯ Σ + (cid:1) − ( γ − ) ¯ Q ¯ Σ ¯ Ω (cid:17) , ∂ η ¯ Φ = , ∂ η D = − D (cid:16) Σ ( − ¯ Q + Q ¯ Σ ) + Q ¯ Ω + γ ¯ Q ( − ¯ Σ − ¯ Ω ) (cid:17) , ∂ η t = / D , (C2)where as an initial condition we use the seven dataset presented in the Table I for a better comparison ofboth systems.In Figures 6(a)-13(b) are presented projections ofsome solutions of the full system (A2) and time-averaged system (C2) in the ( Σ , D , Ω ) and ( Q , D , Ω ) space with their respective projection when D = γ = γ = γ = / γ =
2. FLRW metric with positive curvature
For the FLRW metric with positive curvature ( k =+
1) we integrate:1. The full system given by (A6).2. The time-averaged system: ∂ η ¯ Ω = ( γ − ) ¯ Q ¯ Ω (cid:0) − ¯ Ω (cid:1) , ∂ η ¯ Q = − (cid:0) − ¯ Q (cid:1) (cid:0) γ (cid:0) − ¯ Ω (cid:1) + Ω − (cid:1) , ∂ η ¯ Φ = , ∂ η D = − ¯ Q (cid:0) γ ( − ¯ Ω ) + ¯ Ω (cid:1) D , ∂ η t = / D , (C3) except for the γ = Ω in the time-averaged system (C3) be-comes trivial. In this case we integrate the followingtime-averaged system: ∂ η ¯ Ω = D ¯ Q ¯ Ω ( ω − µ ) b µ ω ∂ η ¯ Q = − (cid:0) − ¯ Q (cid:1) (cid:18) D ¯ Ω ( ω − µ ) b µ ω + (cid:19) , ∂ η ¯ Φ = D ¯ Ω ( ω − µ ) b µ ω − D Ω ( ω − µ ) b µ ω , ∂ η D = − D ¯ Q − D ¯ Q ¯ Ω ( ω − µ ) b µ ω , ∂ η t = / D . (C4)Independent of γ value we use as initial conditions theten data set presented in the Table II, where the datasets I , II and V II are the symmetrical counterpart withrespect to Q of the data sets i , ii and vii .In figures 14(a)-17(b) are presented projections ofsome solutions of the full system (A6) and time-averaged system (C3) for γ (cid:54) = γ = Q , D , Ω ) space with their respective projectionwhen D = γ = γ = γ = / γ = k = + D i ii iiiiv vvivii (a) Projections in the space ( Σ , D , Ω ) . The surface is given by the constraint Ω = − Σ . i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.50.750.800.850.900.95 ivii ivii ivii ivii ivii ivii ivii0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.5 ii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vvi 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.580.01800.01850.01900.01950.02000.02050.0210 iv viv viv viv viv viv viv v (b) Projection in the space ( Σ , Ω ) . The black line represent the constraint Ω = − Σ . Figure 6: Some solutions of the full system (A2) (blue) and time-averaged system (C2) (orange) for the Kantowski-Sachsmetric when γ =
0, in the projection Q =
0. We have used for both systems the initial data sets presented in the Table I. Q D i iiiii iv vvi vii (a) Projections in the space ( Q , D , Ω ) . The surface is given by the constraint Q = Q i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii 0.2 0.0 0.2 0.4 0.6 0.8 Q iiii vii iiii vii iiii vii iiii vii iiii vii iiii vii iiii vii0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 Q iiiiiiiiiiiiii 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 Q ivvivvivvivvivvivvivv (b) Projection in the space ( Q , Ω ) . The black line represent the constraint Q = Figure 7: Some solutions of the full system (A2) (blue) and time-averaged system (C2) (orange) for the Kantowski-Sachsmetric when γ =
0, in the projection Σ =
0. We have used for both systems the initial data sets presented in the Table I. D i ii iiiiv vvivii (a) Projections in the space ( Σ , D , Ω ) . The surface is given by the constraint Ω = − Σ . i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.50.750.800.850.900.95 ivii ivii ivii ivii ivii ivii ivii0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.5 ii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vvi 0.44 0.46 0.48 0.50 0.52 0.54 0.560.01800.01850.01900.01950.02000.02050.0210 iv viv viv viv viv viv viv v (b) Projection in the space ( Σ , Ω ) . The black line represent the constraint Ω = − Σ . Figure 8: Some solutions of the full system (A2) (blue) and time-averaged system (C2) (orange) for the Kantowski-Sachsmetric when γ =
1, in the projection Q =
0. We have used for both systems the initial data sets presented in the Table I. Q D i iiiiiiv vvivii (a) Projections in the space ( Q , D , Ω ) . The surface is given by the constraint Q = Q i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii 0.2 0.0 0.2 0.4 0.6 0.8 Q iiii vii iiii vii iiii vii iiii vii iiii vii iiii vii iiii vii0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 Q iiiiiiiiiiiiii 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 Q iv viv viv viv viv viv viv v (b) Projection in the space ( Q , Ω ) . The black line represent the constraint Q = Figure 9: Some solutions of the full system (A2) (blue) and time-averaged system (C2) (orange) for the Kantowski-Sachsmetric when γ =
1, in the projection Σ =
0. We have used for both systems the initial data sets presented in the Table I. D i ii iiiiv vvivii (a) Projections in the space ( Σ , D , Ω ) . The surface is given by the constraint Ω = − Σ . i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.50.750.800.850.900.95 ivii ivii ivii ivii ivii ivii ivii0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.5 ii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vvi 0.44 0.46 0.48 0.50 0.52 0.540.01800.01850.01900.01950.02000.02050.0210 iv viv viv viv viv viv viv v (b) Projection in the space ( Σ , Ω ) . The black line represent the constraint Ω = − Σ . Figure 10: Some solutions of the full system (A2) (blue) and time-averaged system (C2) (orange) for the Kantowski-Sachsmetric when γ = /
3, in the projection Q =
0. We have used for both systems the initial data sets presented in the Table I. Q D i iiiiiiv vvivii (a) Projections in the space ( Q , D , Ω ) . The surface is given by the constraint Q = Q i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii 0.2 0.0 0.2 0.4 0.6 0.8 Q iiii vii iiii vii iiii vii iiii vii iiii vii iiii vii iiii vii0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 Q iiiiiiiiiiiiii 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 Q iv viv viv viv viv viv viv v (b) Projection in the space ( Q , Ω ) . The black line represent the constraint Q = Figure 11: Some solutions of the full system (A2) (blue) and time-averaged system (C2) (orange) for the Kantowski-Sachsmetric when γ = /
3, in the projection Σ =
0. We have used for both systems the initial data sets presented in the Table I. D i ii iiiiv vvivii (a) Projections in the space ( Σ , D , Ω ) . The surface is given by the constraint Ω = − Σ . i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii i ii iiiiv-vvivii 0.2 0.1 0.0 0.1 0.2 0.3 0.40.750.800.850.900.95 ivii ivii ivii ivii ivii ivii ivii0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.5 ii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vviii iiiiv-vvi 0.46 0.48 0.50 0.52 0.54 0.560.0160.0170.0180.0190.0200.021 iv viv viv viv viv viv viv v (b) Projection in the space ( Σ , Ω ) . The black line represent the constraint Ω = − Σ . Figure 12: Some solutions of the full system (A2) (blue) and time-averaged system (C2) (orange) for the Kantowski-Sachsmetric when γ =
2, in the projection Q =
0. We have used for both systems the initial data sets presented in the Table I. Q D i iiiiiiv vvivii (a) Projections in the space ( Q , D , Ω ) . The surface is given by the constraint Q = Q i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii i iiiii iv-vvi vii 0.2 0.0 0.2 0.4 0.6 0.8 Q iiii vii iiii vii iiii vii iiii vii iiii vii iiii vii iiii vii0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 Q iiiiiiiiiiiiii 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 Q iv viv viv viv viv viv viv v (b) Projection in the space ( Q , Ω ) . The black line represent the constraint Q = Figure 13: Some solutions of the full system (A2) (blue) and time-averaged system (C2) (orange) for the Kantowski-Sachsmetric when γ =
2, in the projection Σ =
0. We have used for both systems the initial data sets presented in the Table I. Q D i iiiiiiv vvi viiIII VII (a) Projections in the space ( Q , D , Ω ) . The surface is given by the constraint Q = Q i iiiii iv-vvi viiIII VII i iiiii iv-vvi viiIII VII i iiiii iv-vvi viiIII VII i iiiii iv-vvi viiIII VII i iiiii iv-vvi viiIII VII i iiiii iv-vvi viiIII VII i iiiii iv-vvi viiIII VII i iiiii iv-vvi viiIII VII i iiiii iv-vvi viiIII VII i iiiii iv-vvi viiIII VII 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Q ivii ivii ivii ivii ivii ivii ivii ivii ivii ivii0.0 0.2 0.4 0.6 0.8 1.0 Q iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Q ivv ivv ivv ivv ivv ivv ivv ivv ivv ivv (b) Projection in the space ( Q , Ω ) . The black line represent the constraint Q = Figure 14: Some solutions of the full system (A6) (blue) and time-averaged system (C3) (orange) for the FLRW metric withpositive curvature ( k = +
1) when γ =
0. We have used for both systems the initial data sets presented in the Table II. Q D i iiiiiiv vvivii (a) Projections in the space ( Q , D , Ω ) . The surface is given by the constraint Q = Q i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Q ivii ivii ivii ivii ivii ivii ivii0.0 0.2 0.4 0.6 0.8 1.0 Q iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Q ivv ivv ivv ivv ivv ivv ivv (b) Projection in the space ( Q , Ω ) . The black line represent the constraint Q = Figure 15: Some solutions of the full system (A6) (blue) and time-averaged system (C4) (orange) for the FLRW metric withpositive curvature ( k = +
1) when γ =
1. We have used for both systems the initial data sets presented in the Table II. Q D i iiiiiiv vvivii (a) Projections in the space ( Q , D , Ω ) . The surface is given by the constraint Q = Q i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Q ivii ivii ivii ivii ivii ivii ivii0.0 0.2 0.4 0.6 0.8 1.0 Q iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Q ivv ivv ivv ivv ivv ivv ivv (b) Projection in the space ( Q , Ω ) . The black line represent the constraint Q = Figure 16: Some solutions of the full system (A6) (blue) and time-averaged system (C3) (orange) for the FLRW metric withpositive curvature ( k = +
1) when γ = /
3. We have used for both systems the initial data sets presented in the Table II. Q D i iiiiiiv vvivii (a) Projections in the space ( Q , D , Ω ) . The surface is given by the constraint Q = Q i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii i iiiii iv-vvivii 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Q ivii ivii ivii ivii ivii ivii ivii0.0 0.2 0.4 0.6 0.8 1.0 Q iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi iiiii iv-vvi 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Q ivv ivv ivv ivv ivv ivv ivv (b) Projection in the space ( Q , Ω ) . The black line represent the constraint Q = Figure 17: Some solutions of the full system (A6) (blue) and time-averaged system (C3) (orange) for the FLRW metric withpositive curvature ( k = +
1) when γ =
2. We have used for both systems the initial data sets presented in the Table II. Table I: Seven initial data sets for the simulation of the full system (A2) and time-averaged system (C2) for the Kantowski-Sachs metric. All the conditions are chosen in order to fulfill the inequalities Σ ( ) + Ω ( ) ≤ ≤ Q ≤ Sol. D ( ) Σ ( ) Ω ( ) Q ( ) ϕ ( ) t ( ) i 0 .
03 0 . . .
65 0 0ii 0 .
03 0 . . .
95 0 0iii 0 .
03 0 . . .
006 0 .
48 0 .
02 0 .
25 0 0v 0 .
03 0 .
48 0 .
02 0 .
25 0 0vi 0 .
03 0 . .
25 0 .
15 0 0vii 0 .
03 0 0 .
76 0 . Table II: Ten initial data sets for the simulation of full system (A6) and time-averaged system (C3) for γ (cid:54) = γ =
1, for the FLRW metric with positive curvature ( k = + ≤ Ω ≤ ≤ Q ≤ Sol. D ( ) Ω ( ) Q ( ) ϕ ( ) t ( ) .i 0 . . .
65 0 0I 0 . . − .
65 0 0ii 0 . . .
95 0 0II 0 . . − .
95 0 0iii 0 . . .
02 0 .
02 0 .
25 0 0v 0 . .
02 0 .
25 0 0vi 0 . .
25 0 .
15 0 0vii 0 . .
76 0 . . . − . Table III: Three initial data sets for the simulation of full system (A6) and time-averaged system (C3) for the FLRW metricwith positive curvature ( k = + Ω > ≤ Q ≤ Sol. D ( ) Ω ( ) Q ( ) ϕ ( ) t ( ) i 0 . . .
05 1 . .
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