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Equilibrium points of the tilted perfect fluid Bianchi VI state h space Pantelis S. Apostolopoulos∗ University of Athens, Department of Physics, 5 Nuclear and Particle Physics Section, 0 Panepistemiopolis, Zografos 157 71, Athens, Greece 0 2 February 7, 2008 n a J 4 Abstract 2 v We present the full set of evolution equations for the spatially homogeneous cosmologies 0 of type VIh filled with a tilted perfect fluid and we provide the corresponding equilibrium 4 pointsoftheresultingdynamicalstatespace. Itisfoundthatonlywhenthegroupparameter 0 satisfies h > 1 a self-similar solution exists. In particular we show that for h > 1 there 7 − −9 40 exists a self-similar equilibrium point provided that γ ∈ 25(3++3√√−hh),23 whereas for h<−91 − c/0 sthelef-ssitmatielarpasroalumteiotnersbbeelloonnggssttootthheesiunbtcelravsaslnγαα∈=(cid:16)01h,a25(v3+i+n(cid:16)3√√g−−nhho)n(cid:17)-.zerTohvi(cid:17)sorftaimcitilyy. Ionf bnoewthecxaasecst q the equilibriumpoints haveafive dimensionalstablemanifold andmay actasfuture attrac- - α r tors atleastfor the models satisfying nα =0. Also we givethe exactformof the self-similar g metrics in terms of the state and group parameters. As an illustrative example we provide : v the explicit form of the corresponding self-similar radiationmodel (γ = 4), parametrisedby 3 i the group parameter h. Finally we show that there are no tilted self-similar models of type X III and irrotational models of type VIh. r a KEY WORDS: Exact Solutions; Perfect Fluid Models; Self-Similarity. 1 Introduction Althoughonasufficiently largeobservationalscale, thepresentstateoftheUniverseisdescribed by the Friedmann-Lemaître (FL) model which is isotropic and spatially homogeneous, there are potentialproblemsmainlyregardingtheobservedlocal structuresofour“lumpy”Universewhich cannotbeexplained withintheclass ofFL models. Thereforemoregeneral cosmological models, which in some dynamical sense, are “close” to FL but not isotropic in local scale, can be used in order to answer many open and important questions. For example it is of interest to understand the presence, the form and the evolution of small (local) density and expansion anisotropies in the Universe or to investigate the constraints that measurements of temperature anisotropies are able to impose on the curvature of space-time. In addition it is important to classify all possible asymptotic states near the cosmological initial singularity (i.e. near the Planck time) ∗E-mail: [email protected]. 1 and into the future that are permitted by the Einstein’s Field Equations (EFE) with a view to explaining how the real Universe may have evolved. Inthispointofviewthesimplest(anisotropic)generalisation oftheFLuniversesaretheSpa- tiallyHomogeneous(SH)cosmologiesadmittingaG groupofisometriesactingon3-dimensional 3 spacelike hypersurfaces . From cosmological point of view, it is of importance to study the C evolution of vacuum or models filled with a gamma-law perfect fluid matter source having an energy-momentum tensor of the form: T = (µ˜+p˜)u u +p˜g (1.1) ab a b ab where µ˜,p˜ = (γ 1)µ˜ are the energy density and the pressure measured by the observers − comoving with fluid velocity ua (uau = 1). Since in SH models there is a preferred unit a − timelike congruence na (nan = 1) normal to the spatial foliations we can divide them into a − C non-tilted [1] and tilted [2] models according to whether the fluid velocity ua is parallel or not to the timelike direction na. In the last two decades, the study of SH models is heavily based on the qualitative analysis of the resulting system of the (induced) first order ordinary differential equations. Using the so called orthonormal frame formalism (pioneered by Ellis [1]) which is based on choosing a frame tetrad invariant under the group of isometries (thus ensuring the spatial independency of the kinematical and dynamical quantities of the models) and a set of expansion-normalized variables,theevolutionandconstraintequations,followedfromtheEFE,becomeanautonomous system of decoupled first order differential equations which can be studied with the aid of the well-established theory of dynamical systems [3]. The evolution of a specific model is studied in the so called dynamical state space which represents the set of all the physical states (at some instant of time) of the corresponding model [3]. Under this perspective of studying SH models, equilibrium points (i.e. fixed points) and their stability, of the resulting dynamical system, play an important role in the description of the asymptotic behavior (into the past/future as well as in the intermediate times of their evolution) since they may represent past or future attractors for more general models. Vacuum and non-tilted perfect fluid SH models have been extensively studied in the liter- ature [3, 4, 5, 6, 7, 8, 9] revealing new and important features of the SH models like e.g. the asymptotically self-similarity breaking and the divergence of the Weyl curvature at late times for type VII models, providing a solid counterexample to the isotropisation conjecture (i.e. 0 shear isotropisation implies a corresponding result for the Weyl curvature scalar). On the other hand it is natural to expect that the behavior of SH models will be modified accordingly by the presence of a tilted fluid velocity leading also to new interesting phenomena. Thefirststep of qualitative analyzing tilted models hasbeen doneforBianchi type IImodels [10]. Inparticularithasbeenshownthatγ lawtiltedperfectfluidmodels,arefutureasymptotic − to the Collins-Stewart non-tilted model [11] when 2 < γ 10, consequently these models do 3 ≤ 7 not isotropise and the angle of tilt becomes negligible at late times. At the value γ = 10 the 7 tilt destabilise the Collins-Stewart model and there is an exchange of stability with the self- similar equilibrium point in which γ 10,14 . Furthermore at the value γ = 14 there is ∈ 7 9 9 a second bifurcation between the equilibr(cid:16)ia 10(cid:17), 14 and 14,2 and exhibits the property of 7 9 9 the asymptotically extreme tilt for models wh(cid:16)ere th(cid:17)e state(cid:16)param(cid:17)eter γ belongs to the interval 14,2 . 9 (cid:16) Re(cid:17)cently it was shown that the self-similar equilibrium points of Bianchi type VI0 models 2 play a similar role in the asymptotic behaviour of generic models. For example it was found [12] that at the value γ = 6 the tilt destabilise the Collins solution [13] and a family of models 5 satisfying nα = 0 are future asymptotic to the Rosquist and Jantzen self-similar model [12, 14] α for γ 6, 3 . However it has been shown that generic models (i.e. those satisfying nα = 0) ∈ 5 2 α 6 are not a(cid:16)sym(cid:17)ptotically self-similar [12] and may be extreme tilted at late times for 6 < γ < 2 5 [15]. In the case of class B tilted models less information is available due to the increased com- plexity of theevolution equations. Recently thewholefamily of Bianchi class Bmodelshas been studied and some results concerning the stability of the non-tilted equilibrium points have been given [16] (see also [17, 18]). Motivated from the above facts, the goal of this work is to present the full set of evolution equations,thevacuum,non-tiltedandtiltedequilibriumpointsfortheimportantclassofBianchi type VI models and interpret, for some of them, their geometric and dynamical properties. h Anoutlineofthepaperisasfollows: section 2reviewsandpresentsthebasicresultsconcern- ing the set of equations which describes the dynamics of the tilted perfect fluid SH models. By specialising to the case of Bianchi type VI models, we provide the complete set of the evolution h equations and we identify the resulting dynamical state space. In section 3 we findthe complete set of equilibrium points and for the case which are represented by self-similar models, we give all tilted perfect fluid models admitting a proper Homothetic Vector Field (HVF). Finally in section 4 we summarise and discuss the implications some of the obtained results. Throughout the following conventions have been used: spatial frame indices are denoted by lower Greek letters α,β,... = 1,2,3, lower Latin letters denote space-time indices a,b,... = 0,1,2,3 and we use geometrised units such that 8πG = c= 1. 2 Dynamical state space of tilted perfect fluid Bianchi type VI h models In SH tilted perfect fluid models, the autonomous differential equation governing their evolution can be written in the form: dx = f (x) (2.1) dτ where x is the state vector representing the set of all the physical variables that describe the dynamics of the corresponding model, f(x) is a polynomial function of the state vector and τ is the dimensionless time variable defined by: dt 1 dH = , = (1+q)H (2.2) dτ H dτ − where q,H are the deceleration and Hubble parameter respectively. In [10] and using the orthonormal frame approach, the EFE are reformulated in terms of the components of the shear tensor of the normal timelike congruence na, the spatial curvature of the orbits of the G isometry group and the spatial part of the tilted fluid velocity ua. The 3 evolution equations for the type VI models can be found by specialising the set of equations h given in [10]: ′ µν Σ = (2 q)Σ +2ǫ Σ R S +Π (2.3) αβ − − αβ (α β)µ ν − αβ αβ 3 ′ µ µν N = qN +2Σ N +2ǫ N R (2.4) αβ αβ (α β)µ (α β)µ ν A′ = qA Σ µA +ǫµν A R (2.5) α α− α µ α µ ν Ω′ = ΩG−1 2Gq (3γ 2) (2 γ)v2 γΣ vµvν +2γA vµ (2.6) µν µ − − − − − h i v v′ = α (3γ 4) 1 v2 +(2 γ)Σ vγvδ + α [1 (γ 1)v2]{ − − − γδ − − (cid:16) (cid:17) + (2 γ) (γ 1) 1 v2 A vβ Σβv + − − − − β }− α β h (cid:16) (cid:17)i +ǫ µν R +N δv v v2A (2.7) α − µ µ δ ν − α (cid:16) (cid:17) where a prime denotes derivative w.r.t. τ. The above system is subjected to the algebraic constraints: Ω = 1 Σ2 K (2.8) − − 3γG−1Ωv = 3Σ βA ǫ µνΣ βN (2.9) α α β − α µ βν where we have set G = 1+(γ 1)v2 (2.10) − and the deceleration parameter is given by the relation: 1 q = 2Σ2+ G−1Ω (3γ 2) 1 v2 +2γv2 = 2 − − 1 h (cid:16) (cid:17) i = 2(1 K) G−1Ω 3(2 γ) 1 v2 +2γv2 . (2.11) − − 2 − − h (cid:16) (cid:17) i Using the freedom of a time-dependent spatial rotation, we may choose the orthonormal tetrad to be the eigenframe of N therefore the contracted form of Jacobi identities N Aβ = 0 αβ αβ implies: 0 0 0 N = 0 N 0 , A = A δ1. (2.12) αβ  2  α 1 α 0 0 N 3     The evolution equation of N can be used to express the angular velocity R of the spatial αβ α frame in terms of the shear variables: N +N 2 3 R = Σ , R = Σ , R = Σ . (2.13) 1 23 2 13 3 12 N N − 2 3 − Inaddition equations (2.4)and(2.5)haveafirstintegral which isusedtoexpressthecomponent A in the well known form: 1 A2 = hN N . (2.14) 1 2 3 Following [10] we introduce the shear variables: 1 1 Σ+ = (Σ22+Σ33), Σ− = (Σ22 Σ33) (2.15) 2 2√3 − 4 1 1 1 Σ = Σ , Σ = Σ , Σ Σ . (2.16) 1 23 3 12 13 13 √3 √3 → √3 In the case of type VI models we have h < 0 and N N < 0. With these identifications h 2 3 we obtain the following set of evolution equations for the basic expansion-normalised variables x= (Σ+,Σ−,Σ1,Σ3,Σ13,N2,N3,vα): (N N )2 18 Σ2 +Σ2 Ωγ 2v2 v2 v2 Σ′ = (2 q)Σ 2− 3 − 13 3 1 − 2 − 3 (2.17) + − − +− 6 − 2G (cid:0) (cid:1) (cid:0) (cid:1) √3 N2 N2 4√3N Σ2 Σ′− = (2 q)Σ− 2 − 3 + 3 1 + − − − 6 N N (cid:0) (cid:1) 2 3 − √3Ωγ v2 v2 +√3 2Σ2 Σ2 +Σ2 + 2 − 3 (2.18) 1− 13 3 2G (cid:0) (cid:1) (cid:16) (cid:17) Σ′1 = −"4N√23N3NΣ3− −q+2 √3Σ−+1 #Σ1+ − (cid:16) (cid:17) √hN N (N N )+6Σ Σ √3v v Ωγ +√3 2 3 3− 2 13 3 + 2 3 (2.19) 3 G 2√3N Σ Σ √3v v Ωγ Σ′3 = q−√3Σ−−3Σ+−2 Σ3+ N 3 N13 1 + 1G2 (2.20) 2 3 (cid:16) (cid:17) − 2√3N Σ Σ √3v v Ωγ Σ′13 = q+√3Σ−−3Σ+−2 Σ13− N 2 N3 1 + 1G3 (2.21) 2 3 (cid:16) (cid:17) − N2′ = q+2√3Σ−+2Σ+ N2 (2.22) (cid:16) (cid:17) N3′ = q 2√3Σ−+2Σ+ N3 (2.23) − (cid:16) (cid:17) and the evolution equation (2.7) for the frame components of the tilted fluid velocity. The algebraic constraint (2.9) reads: 3v Ωγ √3Σ (N N )+6 hN N Σ + 1 = 0 (2.24) 1 3 2 2 3 + − G p 3v Ωγ √3Σ N +3√3 hN N Σ 2 = 0 (2.25) 13 3 2 3 3 − G p 3v Ωγ √3Σ N 3√3 hN N Σ + 3 = 0. (2.26) 3 2 2 3 13 − G p We note that the shear scalar Σ2 = ΣαβΣαβ and the spatial curvature K are: 6 Σ2 = Σ2++Σ2−+Σ21+Σ23+Σ213 (2.27) (N N )2 2 3 K = hN N + − (2.28) 2 3 12 therefore the inequality Ω 0 and the constraint (2.8) imply that the state space R7 is ≥ D ⊂ bounded (we recall that N N < 0). 2 3 5 3 Determination of the equilibrium points Equilibrium points of the autonomous differential equation (2.1) play an important role in the evolution of the SH models since they determine various stable and unstable invariant submani- foldsofthestatespace . Thesepointscanbefoundfromthesolutionofthealgebraicequations D f(x) = 0 and (2.24)-(2.26) which we now list in the following subsections. We note that the case of type III models is included by setting (whenever is appropriate) h = 1. − 3.1 Vacuum Equilibrium Points 1. Kasner Circle [10] K N = N = 0, vαv = 0, Σ2 = 1 2 3 α Σ2 +Σ2 = 1, Σ = Σ = Σ = 0, q = 2. + − 1 13 3 ± 2. Kasner Line with tilt [10] Ktilt N = N = 0, vαv = v2 < 1, v = v = 0, Σ2 = 1 2 3 α 3 1 2 √3(3γ 4) 3(2 γ)(3γ 2) Σ+ = √3Σ−+3γ 4, Σ− = − ± − − − − 4 p 2 Σ = Σ = Σ = 0, q = 2, γ 2. 1 3 13 3 ≤ ≤ 3. Kasner Circle with extreme tilt [10] extreme K N = N = 0, vαv = v2 = 1, v = v = 0, Σ2 = 1 2 3 α 3 1 2 Σ2 +Σ2 = 1, Σ = Σ = Σ = 0, q = 2, 0 < γ < 2. + − 1 3 13 4. Collins Vacuum Plane Wave Arc (VI ) (h = 1) [3] L h 6 −9 12hN N = 2√3 (N N )2+3+(N N )2 6, vαv = 0, 2 3 2 3 2 3 α − − − − − q 2 √3+ (N N )2+3 12hN N +(N N )2 2 3 2 3 2 3 − − − Σ2 = (cid:20) q 12(N (cid:21) hN )2 i 2 3 − √3 √3+ (N N )2+3 − 2− 3 2√3hN2N3Σ+ Σ+ = (cid:20) q (cid:21), Σ1 = , − 6 (N N ) 2 3 − √3 (N N )2+3+√3 2 3 − − Σ− = Σ13 = Σ3 = 0, q = (cid:18)q (cid:19) 3 6 0 < γ < 2. 5. Vacuum plane wave with tilt1 ± (VI ) (h = 1) Mtilt h 6 −9 2 (N N )2+3 √3(3γ 5) 2 3 − − ∓ − N2 = N2, N3 = N3, vαvα = (cid:20)q 12hN N (γ 1)2 (cid:21) , 2 3 − 2 (N N )2+3 √3 12hN N +(N N )2 2 3 2 3 2 3 − − ± − Σ2 = (cid:20)q 12(N (cid:21) hN )2 i 2 3 − √3 (N N )2+3 √3 2 3 ∓ − − − Σ+ = (cid:20) q (cid:21), Σ− = Σ13 = Σ3 = υ2 = υ3 = 0 6 6Σ √hN N + 2 3 Σ = 1 (N N )√3 2 3 − √3 (N N )2+3 √3(3γ 5) 2 3 ± − − − − v1 = (cid:20) q (cid:21), 6√hN N (1 γ) 2 3 − √3 √3 (N N )2+3 2 3 ± − − q = (cid:20) q (cid:21) 3 (N N )4 2 3 h= − . 12N N 2√3 (N N )2+3 (N N )2+6 2 3 2 3 2 3 ± − − − − (cid:20) q (cid:21) We remark that the state parameter γ is constrained via the inequality 1 v2 > 0. − 6. Vacuum plane wave with extreme tilt (VI ) (h = 1). ± Mextreme h 6 −9 Same as the case (VI ). However the state parameter can take any value in the interval Mtilt h (0,2). 1ItappearsthatthisformoftheCollinstypeVIhplanewavesolutionhasbeenalsogivenin[18]using,however, a different notation. 7 3.2 Non Vacuum Equilibrium Points 1. Flat Friedmann-Lemaître Equilibrium Point [10] F N = N = 0, vαv = 0, Σ2 = 0 2 3 α 3γ 2 Σ+ = Σ− = Σ1 = Σ13 = Σ3 = 0, q = − , Ω = 1 2 0 < γ < 2. 2. Collins-Stewart type II non-tilted Equilibrium Point (II) [3] CS 3 (2 γ)(3γ 2) (3γ 2)2 N = 0,N = − − , vαv = 0, Σ2 = − 2 3 α 4 64 p 2 3γ √3(3γ 2) Σ+ = − , Σ− = − , Σ1 = Σ13 = Σ3 = 0 16 16 3γ 2 2 3(6 γ) q = − , < γ < 2, Ω = − . 2 3 16 3. Hewitt type II tilted Equilibrium Point (II) [10] tilt P (γ 2)(3γ 4)(5γ 4) (3γ 4)(7γ 10) N = 0,N = 3 − − − , vαv = − − , 2 3 α s 18 17γ (11γ 10)(5γ 4) − − − (3γ 4) 9γ2 20γ +12 9γ 14 √3(5γ 6) Σ2 = − − , Σ+ = − , Σ− = − , 17γ 18 8 8 (cid:0) (cid:1) − 3(γ 2)(7γ 10)(11γ 10) Σ = − − − , Σ = Σ = 0, 13 1 3 s 16(18 17γ) − 3(2 γ) 21γ2 24γ +4 Ω = − − 4(17γ 18) (cid:0) (cid:1) − (3γ 4)(7γ 10) 3γ 2 10 v = − − , q = − , < γ < 2. 2 s(11γ 10)(5γ 4) 2 7 − − 4. Type II Line of tilted Equilibrium Points (II) [10] tilt L 2(27b2 +2)(17 54b2) 6 27b2+1 27b2+2 N = − ,N = 0, vαv = , 2 s 57 3 α (54b2 17)(81b2 32) (cid:0) (cid:1)(cid:0) (cid:1) − − 2 3b2 27b2+2 (27b2 +1)(32 81b2) Σ2 = − , Σ = − , Σ = b, 3 13 (cid:0) (cid:1)1(cid:0)9 (cid:1) −p 3√57 2√3 6(27b2+1)(27b2+2) Σ− = − 9 , Σ+ = Σ1 = 0, v3 = s(17 54b2)(32 81b2) − − 4 2916b4 1215b2 +236 2 14 q = , Ω = − , b < , γ = . 3 342 | | 3√3 9 8 5. Type II Extreme tilted Equilibrium Point (II) [10] extreme P 6√19 N = ,N = 0, vαv =1, 2 3 α 19 28 10√57 2√3 Σ2 = , Σ = , Σ = , 3 13 57 − 171 3 2√3 Σ− = , Σ+ = Σ1 = 0, v3 = 1 − 9 4 20 q = , Ω = , 0< γ < 2. 3 57 ± 6. Collins Type VI non-tilted Equilibrium Point (VI ) [3] h h C 3 (2 γ)(3γ 2) N = − − ,N = N , vαv = 0, 2 3 2 α 4 − p (3γ 2)2(1 3h) Σ2 = − − , Σ3 = Σ13 = Σ− = 0, 16 2 3γ √3(2 3γ)√ h Σ = − , Σ = ± − − , + 1 4 4 3γ 2 3[h(3γ 2) γ+2] 2 2(1 h) q = − , Ω = − − , γ − . 2 4 3 ≤ ≤ (1 3h) − 7. Type VI tilted Equilibrium Point (VI ) (h = h2 and h = 1/3): h Ctilt h − 1 1 6 − 1/2 2h γ[ (3h +1)]1/2 3h2(4qγ β) 4h [q(5γ 4)+2(γ 2)]+3β N − 1 − 1 1 − − 1 − − 3 n o = √3 6h4γ[(q+1)(2qγ β)(6q(γ 1)+5γ 6)] h3[γ3 120q3 +16q2 28q+49 + { 1 − − − − 1 − (cid:16) (cid:17) +(q+1) 2γ2(144q2 +32q+23)+60γ(q+1)(4q +1) 72(q+1)2 ]+ − − (cid:16) (cid:17) +h2β[γ2(8q2 66q 41)+(q+1)(4γ(q+18) 12(q+1))]+ 1 − − − h β2[2q(4γ 5)+13γ 10] β3 1/2 1 − − − } 3β2 h2[6q(γ 1)+5γ 6] 2h [q(3γ 2)+γ 2]+β vαv = 1 − − − 1 − − α 4h2N2γ2 3h2(4qγ β) 4h [q(5γ 4)+2(γ 2)]+3β 1 (cid:8)3 1 − − 1 − − (cid:9) (cid:8) (cid:9) √3 2h N ζ +v2γ2(γ 1) +2N ζ +v γ2(β+2 q) Σ = 1 3 1 − 3 1 − , 3 − 12ζ1/2γ (cid:8) (cid:2) (cid:3) (cid:9) 9 √3[h (q+1)(5γ 6) β] q 1 Σ13 = Σ3, Σ1 = − − , Σ+ = , Σ− = 0 6h γ −2 1 3h2qγ+h (q+1)(6 5γ)+β 4h2N2 2ζ(γ 1)+γ2 +(γ 1)β2 Ω = 1 1 − 1 3 − − − 6h2γ3β (cid:2) (cid:3)(cid:8)1 (cid:2) (cid:3) (cid:9) β v = , v = v = ζ1/2γ−1 1 2 3 −2h N γ 1 3 A+B q = Λ where we have set: A = 3h γ 1+3h h (3γ 2) 5γ +6 1 1 1 | || − − |× (γ 1)[h2(γ 1)(7γ 6)2+2h (γ 2) 27γ2 37γ +6 + ×{ − 1 − − 1 − − (cid:16) (cid:17) +(γ 2)2(9γ 1)] 1/2 − − } B = 18h4γ2(γ 1)(3γ 2)+3h3 66γ4 427γ3 +808γ2 588γ +144 1 − − 1 − − − (cid:16) (cid:17) 3h2 106γ4 595γ3 +1200γ2 1052γ +336 − 1 − − − (cid:16) (cid:17) 3h 90γ4 499γ3 +984γ2 844γ +272 +(3γ 2)(35γ 36)(γ 2)] 1 − − − − − − (cid:16) (cid:17) Λ = 2[27h4γ(γ 1)2(3γ 2) 18h3 15γ4 62γ3 +93γ2 58γ +12 + 1 − − − 1 − − (cid:16) (cid:17) +3h2 75γ4 384γ3 +704γ2 560γ +168 1 − − − (cid:16) (cid:17) 6h 30γ3 127γ2 +166γ 68 +(35γ 36)(γ 2)] 1 − − − − − (cid:16) (cid:17) β = 2q 3γ +2, − 10