aa r X i v : . [ g r- q c ] J a n Tilted two-fluid Bianchi type I models
Patrik Sandin ∗ Department of Physics, University of Karlstad,S-651 88 Karlstad, Sweden
November 4, 2018
Abstract
In this paper we investigate expanding Bianchi type I models with two tilted fluids with the samelinear equation of state, characterized by the equation of state parameter w . Individually the fluidshave non-zero energy fluxes w.r.t. the symmetry surfaces, but these cancel each other because of theCodazzi constraint. We prove that when w = 0 the model isotropizes to the future. Using numericalsimulations and a linear analysis we also find the asymptotic states of models with w >
0. We findthat future isotropization occurs if and only if w ≤ . The results are compared to similar modelsinvestigated previously where the two fluids have different equation of state parameters. PACS numbers: 04.20.-q, 04.20.Dw, 04.20.Ha, 98.80.-k, 98.80.Bp, 98.80.Jk ∗ Electronic address: [email protected] INTRODUCTION There have been numerous investigations of spatially homogeneous (SH) cosmological models with aperfect fluid as a matter source. In the study of these models one usually distinguish between twodifferent scenarios: when the fluid congruence is normal to the hypersurfaces of homogeneity and whenit is not. The models of the first kind are called orthogonal or non-tilted , and the second kind are called tilted models. The orthogonal models have been studied extensively over the past few decades (see [1]and references therein), while the tilted have received detailed investigations only more recently [2]-[11].In investigations of tilted Bianchi models, the models of type I have been ignored since they are notcompatible with any net energy flux with respect to the homogeneous hypersurfaces, something a tiltedfluid will induce. However, it is possible to consider models of type I with tilted fluids if one have twoor more fluids; the fluids can then all be tilted and induce energy flux individually, but the total energyflux must vanish. This was done for two tilted perfect fluids with linear equations of state in [12]. Thefluids had equations of state p (1) = w (1) ρ (1) and p (2) = w (2) ρ (2) , where ρ ( i ) and p ( i ) are the energydensities and pressures of the respective fluids in their own rest frames, and where it was assumed that0 ≤ w (2) < w (1) <
1. The case w (1) = w (2) was left due to lack of space and because of its qualitativelydifferent behavior – this will be the topic of the present paper.As an example of a model with two different perfect fluids with the same equation of state one canimagine a universe filled with two kinds of dust (the pressure of both fluids is zero). This is a reasonablemodel for both the baryonic and cold dark matter components of the total matter content of the universetoday according to observations. The study of Bianchi type I models with two tilted fluids with thesame equation of state can be viewed as the simplest anisotropic tilted model generalizing the standardisotropic cosmological scenario close to flatness.The SH Bianchi models admit a three dimensional group of isometries acting simply transitively onthe spacelike hypersurfaces that define the surfaces of homogeneity. The models are classified by theLie algebra of the isometry group, which results in a hierarchy of models of different complexity wherethe Bianchi type I model can be found on the bottom, its algebra being Abelian and obtainable fromall the other Bianchi models by Lie algebra contractions [1]. When the Einstein field equations areformulated as a dynamical system the Bianchi type I models appear as a boundary of a larger state spacedescribing the other Bianchi models. Describing the dynamics of this boundary is an important first stepto understanding the dynamics of the more general models.The Bianchi models are sometimes interesting even when studying general inhomogeneous cosmolog-ical models. This is the case for example in the very early universe near an initial singularity wherehorizons form and asymptotically shrink towards the singularity which result in the equations asymptot-ically approaching those of homogenous models, something referred to as asymptotic silence and locality [13].In this paper we analyze the system of autonomous DEs derived in [12] describing the time evolutionof the spatially homogeneous Bianchi type I models with two perfect fluids, but in contrast to [12] westudy the case when the two equations of state are the same. The paper is organized as follows: In section2 the system of equations and constraints is derived. In section 3 the future evolution of dust models isdescribed, and in section 4 the evolution of models with other equation of state is investigated. Section5 consists of a summary of the results and a discussion of their implication. Appendix A contains adescription of the equilibrium points of the system and their stability properties, and appendix B studiesthe dynamics on the vacuum subset. We begin by deriving the evolution equations for the tilted two-fluid Bianchi type I models. We first givethe evolution equations of a general Bianchi model, in terms of expansion-normalized variables definedrelative to the timelike congruence normal to the group orbits. This derivation is described in [1], or [14],but for completeness we give a short description of it here.First one introduces a group-invariant orthonormal frame { e , e α } , where e = n is the normal tothe group orbits and α = 1 , ,
3. The commutation functions, γ abc ( t ) are the basic variables:[ e b , e c ] = γ abc e a . THE DYNAMICAL SYSTEM e , e α ] = − [ H δ αβ + σ αβ + ǫ αβγ Ω γ ] e β , (1a)[ e α , e β ] = c γαβ e γ = 2 a [ α δ β ] γ + ǫ αβδ n δγ . (1b)where H is the Hubble scalar, which is related to the expansion θ of the normal congruence n accordingto H = θ ; σ αβ is the shear associated with n ; Ω α is the Fermi rotation which describes how the spatialtriad rotates with respect to a gyroscopically fixed so-called Fermi frame; n αβ and a α describe the Liealgebra of the 3-dimensional simply transitive Lie group and determine the spatial three-curvature, seee.g. [1].The energy-momentum tensor is similarly decomposed, T ab = ρ n a n b + 2 q ( a n b ) + p ( g ab + n a n b ) + π ab , (2)and is described by the source terms relative to the orthonormal frame, { ρ, p, q α , π αβ } . To obtain a regular, dimensionless system of equations the commutation functions and the sourceterms are normalized with the hubble scalar according toΣ αβ = σ αβ H , N αβ = n αβ H , A α = a α H , R α = Ω α H ,
Ω = ρ H , P = p H , Q α = q α H , Π αβ = π αβ H . (3)We also choose a new dimensionless time coordinate τ according to dτdt = H. (4)The evolution of H is determined by the deceleration parameter q , H ′ = − (1 + q ) H , (5)where ′ denotes differentiation with respect to τ . Raychaudhuri’s equation gives an expression for q interms of the variables (3): q = 2Σ + (Ω + 3 P ) , (6)where Σ = Σ αβ Σ αβ . The Einstein field equations and the Jacobi identities then yields the followingsystem of equations: Evolution equations : Σ ′ αβ = − (2 − q )Σ αβ + 2 ǫ γδ h α Σ β i δ R γ − R h αβ i + 3Π αβ , (7a) A ′ α = [ q δ αβ − Σ αβ − ǫ αβγ R γ ] A β , (7b)( N αβ ) ′ = [ q δ γ ( α + 2Σ γ ( α + 2 ǫ γ ( αδ R δ ] N β ) γ . (7c) Constraint equations : 0 = 1 − Σ +
16 3
R − Ω , (8a)0 = (3 δ αγ A β + ǫ αδγ N δβ ) Σ βγ − Q α , (8b)0 = A β N βα . (8c)where R h αβ i and R are the trace-free and scalar parts of the Hubble-normalized three-curvature,respectively, according to: THE DYNAMICAL SYSTEM R h αβ i = B h αβ i + 2 ǫ γδ h α N β i δ A γ , R = − B αα − A α A α ; B αβ = 2 N αγ N γβ − N γγ N αβ , and where h .. i denotes trace-free symmetrization of the indices, i.e. A h αβ i = A ( αβ ) − δ αβ A α A β . Thecontracted Bianchi identities yields evolution equations for the total source variables:Ω ′ = (2 q −
1) Ω − P + 2 A α Q α − Σ αβ Π αβ , (9a) Q ′ α = − [2(1 − q ) δ αβ + Σ αβ + ǫ αβγ R γ ] Q β + (3 δ αγ A β + ǫ αδγ N δβ ) Π βγ . (9b)These equations are a consequence of the evolution and constraint equations (7) and (8) and give noadditional information, but are useful as auxiliary equations.The system of equations (7) and (8) with (6) is not fully determined, there are no evolution equationsfor the variables R α that represent the angular velocity of the spatial frame { e α } . We can freely specify R α in any way most convenient. Also the evolution of the source variables P and Π αβ are not determineduntil we specify a source. We consider the source to be two tilted perfect fluids, only interacting witheach other gravitationally. We can then split the energy-momentum tensor into two parts, each satisfyingthe conservation equation separately T ab = X i T ab ( i ) , ∇ a T ab ( i ) = 0 , ( i = 1 , , (10)where T ab ( i ) = (˜ ρ ( i ) + ˜ p ( i ) )˜ u a ( i ) ˜ u b ( i ) + ˜ p ( i ) g ab . (11)We now impose the same linear equation of state for the two fluids, i.e., ˜ p ( i ) = w ˜ ρ ( i ) , where w = const .The four velocities u ( i ) can be written in the form˜ u a ( i ) = 1 q − v a ( i ) v a ( i ) ( n a + v a ( i ) ) ; n a v a ( i ) = 0 . (12)Since we have two separate conservation equations we now have two sets of source variables { Ω ( i ) , P ( i ) , Q ( i ) α , Π ( i ) α } , ( i = 1 , , where each Ω ( i ) and Q ( i ) α satisfies equations (9). The source variables can all be expressed in terms of thenormalized energy density and the three-velocity of the respective fluid as Q α ( i ) = (1 + w )( G ( i )+ ) − v α ( i ) Ω ( i ) , (13a) P ( i ) = w Ω ( i ) + (1 − w ) Q ( i ) α v α ( i ) , (13b)Π ( i ) αβ = Q ( i ) h α v ( i ) β i , (13c)where G ( i ) ± = 1 ± w v i ) and v i ) = v α ( i ) v α ( i ) . One can use equations (9) and (13) to obtain evolutionequations for the source terms Ω ( i ) and v α ( i ) :Ω ′ ( i ) = (2 q − − w ) Ω ( i ) + [(3 w − v ( i ) α − Σ αβ v β ( i ) + 2 A α ] Q α ( i ) , (14a) v ′ α ( i ) = ( G ( i ) − ) − h (1 − v i ) )(3 w − − w A β v β ( i ) ) + (1 − w )( A β + Σ γβ v γ ( i ) ) v β ( i ) i v α ( i ) − [Σ αβ + ǫ αβγ ( R γ + N γδ v δ ( i ) )] v β ( i ) − A α v i ) . (14b)For the Bianchi type I models we have A α = N αβ = 0. The Codazzi constraint (8b) then becomes, Q α = Q (1) α + Q (2) α = 0, which, taken in combination with (13), forces the 3-velocities of the two fluids tobe anti-parallel. Kinematically the situation is similar to that of Bianchi type I with a general magneticfield studied in [15], and it is therefore natural to exploit the same mathematical structures in the present THE DYNAMICAL SYSTEM e , i.e. v α ( i ) = (0 , , v ( i ) ). Demanding that these conditions on v α ( i ) holdfor all times lead to the following conditions R = − Σ , R = Σ . (15)This leaves R undetermined, however, we still have the freedom of arbitrary rotations in the 1-2-plane,which we use to set R = 0 . (16)Following [15], we introduce the variables Σ + , Σ A , Σ B , Σ C according toΣ + = (Σ + Σ ) , Σ + i Σ = √ A e i φ , Σ − + i √ = (Σ B + i Σ C ) e φ , (17)where Σ − = (Σ − Σ ) / (2 √ = Σ + Σ A + Σ B + Σ C . (18)The angular variable φ decouples from the other equations, φ ′ = − Σ C , which reduces the dimension ofthe dynamical system by one. The resulting dynamical system is the following: Evolution equations : Σ ′ + = − (2 − q )Σ + + 3Σ A − Q (1) v (1) − Q (2) v (2) , (19a)Σ ′ A = − (2 − q + 3Σ + + √ B )Σ A , (19b)Σ ′ B = − (2 − q )Σ B + √ A − √ C , (19c)Σ ′ C = − (2 − q − √ B )Σ C , (19d) v ′ ( i ) = ( G ( i ) − ) − (1 − v i ) )(3 w ( i ) − + ) v ( i ) , (19e)Ω ′ ( i ) = (2 q − − w ( i ) )Ω ( i ) + (3 w ( i ) − + ) Q ( i ) v ( i ) . (19f) Constraint equations : 0 = 1 − Σ − Ω (1) − Ω (2) , (20a)0 = Q (1) + Q (2) , (20b)where q = 2Σ + (Ω m + 3 P m ) = 2 − (Ω m − P m ) ; Ω m = Ω (1) + Ω (2) , P m = P (1) + P (2) . (21)The assumption of non-negative energy densities, Ω ( i ) ≥
0, together with (21) and (20a), yields that ≤ q ≤ The state space consists of S = { Σ + , Σ A , Σ B , Σ C , v (1) , v (2) , Ω (1) , Ω (2) } subject to the constraints (20a),(20b). Both v ( i ) = 0 and v i ) = 1 defines invariant subsets which means that v i ) , contained in the range(0 , < Ω (1) Ω (2) , < v i ) < , (22)but asymptotically the orbits of the system may approach the boundary and hence we consider its closure,¯ S , which means that we consider the set defined by Σ ≤ , ≤ v i ) ≤
1; 0 ≤ Ω ( i ) ≤
1, in such a way thatthe constraints (20) are satisfied.
THE DYNAMICAL SYSTEM A → − Σ A , Σ C → − Σ C ; ( v (1) , v (2) ) → − ( v (1) , v (2) ) . (23)We therefore assume without loss of generality that Σ A ∈ [0 , , Σ C ∈ [0 , , v (1) ∈ [0 , v (2) ∈ [ − , v (1 − v ) (1 − w ) v (1 − v ) (1 − w ) = k. (24)where k is a positive real constant. The submanifolds defined by (24) foliates the state space and aredependent on w . Projections onto v (1) × v (2) -space for dust and almost stiff equations of state are shownin Figure 1.PSfrag replacements k → ... ∞ ... v (1) v (2) (a) Dust, w = 0. PSfrag replacements0 k → . . . ∞ ... v (1) v (2) (b) Almost stiff, w = 0 . Figure 1: The leaves of foliation in v (1) × v (2) -space.The constant of motion (24) correlates the velocities of the fluids such that if one of them becomesextremely tilted or orthogonal asymptotically so must the other. Note also that it is not possible that oneof the fluids will dominate over the other asymptotically; if the normalized energy density of one of themvanishes then this is also the case for the other, which follows from considering the constraints (24) and(20b) simultaneously. This excludes a large part of the boundary of ¯ S from consideration since it may notbe approached. When examining invariant subsets and fix points on the boundary in the following wewill only consider those that can be approached from the interior, i.e. not those where Ω ( i ) = 0 = Ω ( j ) , v ( i ) = 0 = v ( j ) , or v i ) = 1 = v j ) , where i = j . The dynamical system (19), (20), admits a number of invariant subsets, conveniently divided into twoclasses: (i) ‘geometric subsets’, i.e., sets associated with conditions on the shear and hence the metric;(ii) invariant sets associated with conditions on the tilt or energy densities. We will introduce a notationwhere the kernel suggests the type of subset and where a subscript, when existent, suggests the values of v (1) and v (2) . Geometric subsets • T W : The ‘twisting’ subset, characterized by Σ C = 0 , Σ A = 0, which leads to that the decoupled φ -variable satisfies φ = const and hence Σ ∝ Σ − Σ . THE DYNAMICAL SYSTEM ≤ w <
13 13 < w <
12 12 < w <
59 59 59 < w < source/saddleK source/saddleKS ± v (1) v (2) center-saddleF Ω (1) Ω (2) sink saddleLRSL v (1) v (2) does not exist sink saddleTW saddleTWL v (1) v (2) does not exist sink (TWL v (1) v (2) ⊂ GS v (1) v (2) ) saddleGS v (1) v (2) does not exist sink does not existG saddle (G ⊂ GS v (1) v (2) ) sinkTable 1: A list of fix points when 0 ≤ w <
1. The fix points are denoted by a kernel that is related to asubset of which the fix point belong in combination with a subscript and sometimes also a superscript.The subscript indicates the fix point values of v (1) and v (2) . The superscript of the Friedmann fix pointsF ∗∗ indicates of the values of Ω (1) and Ω (2) while it denotes the sign of Σ B in the case of the Kasnersurface KS ±∗∗ . A complete characterization of the fix points is given in Appendix A; here we have givena description in terms of their stability properties. • RD : The constantly rotated diagonal subset, given by Σ A = 0 , Σ C = 0 ( R α = 0). This subset isthe diagonal subset, discussed next, rotated with a constant angle around e . • D : The diagonal subset, defined by Σ A = Σ C = 0; Σ B = Σ − , and hence R α = 0. • LRS : The locally rotationally symmetric subset. This plane symmetric subset of the diagonalsubset is characterized by the additional condition Σ B = Σ − = 0. This is the simplest subsetcompatible with two tilted fluids. Matter subsets • O : The orthogonal subset for which v (1) = v (2) = 0. In this subset Ω (1) ∝ Ω (2) and the distinctionbetween the two fluids becomes artificial. The orthogonal subset describes a single orthogonal fluid.In general this subset is expressed in a non-Fermi frame for which Σ A Σ C = 0, however, usuallywhen dealing with this case one makes a rotation to a Fermi frame in which the shear and themetric are diagonal so that O belongs to D . • ET : The double extreme tilt subset where both fluids propagate with the speed of light, v (1) =1 = − v (2) ⇒ Ω (1) = Ω (2) = 3 P (1) = 3 P (2) . • K : The vacuum subset is called the Kasner subset and is defined by Ω m = 0; Σ = 1; it describesthe Kasner solutions, but in general in a non-Fermi propagated frame, and with v ( i ) as test fields.Other invariant subsets can be obtained by taking intersections of the ones described above.Apart from the above subsets there are also a number of fix points. These, and their eigenvalues,are given in Appendix A, but we summarize them in Table 1 along with their stability properties. IfΩ (1) and Ω (2) are zero then v (1) and v (2) act as test fields. This is the case for the Kasner fix pointsK ∗∗ which all describe the same Kasner solutions; the KS ± v (1) v (2) describe a particular Kasner solutiondetermined by w ; the line of Friedmann fix points F Ω (1) Ω (2) describe the flat Friedmann-Lemaˆıtre solutionfor a single orthogonal perfect fluid with a linear equation of state. However, all the remaining fix pointsare physically distinct. FUTURE ISOTROPIZATION OF TILTED MULTI-FLUID DUST MODELS Arguably the most physically relevant equation of state is the case when w = 0. This describes two perfectfluids with zero pressure, commonly referred to as dust , and is a good description of the actual mattercontent of the universe during the matter dominated epoch. For this case we can prove the followingproposition Proposition 3.1.
If w=0, then the future asymptotic state of the system (19) is the Friedmann-Lematresolution.Proof.
For w = 0 we have [ln (1 − v i ) )Ω i ) ] ′ = 2 (2 q − , (25)which is a monotonically increasing function since q ≥ . Since it is also bounded from above it mustapproach a limit value and hence we have q → . This implies P m →
0, Ω m →
1, which in turn implies v ( i ) →
0. The constraint (20a) ensures isotropization and we have reduced the system to the fix pointsFL Ω (1) Ω (2) .Observations of galaxies and galaxy clusters correlates the distribution of dark matter with the bary-onic matter visible in the galaxies, which implies that the velocity of dark matter is aligned with thevelocity of visible matter [16]. The Bianchi type I model is an example of how a universe where the dustflows initially are non-aligned can evolve into a state where they become aligned asymptotically to thefuture.The linear analysis of appendix A, and the dynamics on the Kasner subset, described in AppendixB and numerical simulations suggests that asymptotically to the past the system approaches a doublyextremely tilted Kasner model described by one of the fix points in (44a), as conjectured in section 4.2. As was shown in [12] no tilted two-fluid Bianchi type I models with Q ( i ) > v i ) < w > isotropize to the future. The theorem does not tell us what the asymptotic state is other than that itis anisotropic. The conclusions we make about the future global attractors for w < rests on the localstability analysis of the fix points and numerical simulations, which makes us confident of the followingconjectures: Conjecture 4.1.
The ω -limit for all orbits that have Σ A , Σ C = 0 initially is contained in the set FL Ω (1) Ω (2) if w ≤ , LRSL v (1) v (2) if < w ≤ , TWL v (1) v (2) if < w < , GS v (1) v (2) if w = , and G if < w < . Conjecture 4.2.
The ω -limit for all orbits that have Σ A = 0 , Σ C = 0 initially is contained in the set FL Ω (1) Ω (2) if w ≤ , LRSL v (1) v (2) if < w ≤ , TWL v (1) v (2) if < w < , and T W if ≤ w < . Conjecture 4.3.
The ω -limit for all orbits that have Σ A = 0 initially is contained in the set FL Ω (1) Ω (2) if w ≤ and in the set LRSL v (1) v (2) if < w < . The conjectures about the future attractors can conveniently be summarized by three diagrams show-ing the attractors for different values of w , for the general Bianchi type I set, the ’twisting’ subset andthe diagonal subset. See figure 2. Numerical calculations, the local analysis of the fix points in appendix A, and the dynamics on the Kasnersubset described in Appendix B, supports the following conjecture:
Conjecture 4.4.
The α -limit for every orbit with Q (1) > , v < , v < initially on the generalgeometric set with Σ A Σ C = 0 is one of the fix points on the global past attractor A {∗∗} for the Kasnersubset K given in equation (44) . SUMMARY AND DISCUSSION
13 12 59 FL Ω (1) Ω (2) LRSL v (1) v (2) TWL v (1) v (2) GS v (1) v (2) G w TW LRSL v (1) v (2) (a) The general case PSfrag replacements0
13 1259 FL Ω (1) Ω (2) LRSL v (1) v (2) TWL v (1) v (2) GS v (1) v (2) G w TW LRSL v (1) v (2) (b) The T W subset
PSfrag replacements0 FL Ω (1) Ω (2) LRSL v (1) v (2) TWL v (1) v (2) GS v (1) v (2) G w TW LRSL v (1) v (2) (c) The RD , D , LRS subsets
Figure 2: Future global attractor bifurcation diagrams for the various geometric subsets.All of the tilted models approach a vacuum dominated Kasner singularity, where the influence ofthe matter becomes negligible. The fluids may either become aligned with each other and the normalcongruence to the homogeneous hypersurfaces, or anti-aligned and extremely tilted, depending on thevalue of w and which of the Kasner states that is approached. We have studied Bianchi type I models with two tilted fluids with the same linear equation of state,parameterized by the equation of state parameter w , using dynamical systems methods. The absenceof spatial curvature in these models forces the fluids to be anti-aligned with each other. The paper waswritten with the modest ambition of completing the analysis of multi-fluid Bianchi type I models in[12] where it was assumed that one the fluids were stiffer than the other (i.e. there were two differentequations of state parameters such that w (1) > w (2) ). As expected the models described here are in manyways similar to those in [12], and do, for example, exhibit rather similar bifurcation structure to thefuture, however , there are differences. One important difference is the existence of a constant of motionthat correlates the velocities and energy densities of the fluids. This prohibits the asymptotic states tobecome single fluid cosmologies with an extra test field as was the case in many situations when the twofluids had different equations of state.We proved that for models with w = 0 the system isotropizes to the future and approaches a futureasymptotic Friedmann universe where both fluids become orthogonal to the homogenous hypersurfaces.For models with w > w = 1 / w = 1 / R α = 0 , φ = const. , the asymptotic state is described relative to a constantly rotating frame. The frameis tied to the fluid three-velocity and hence indicates that the fluid is rotating relative to a Fermi frame. Afinal bifurcation occurs at w = 5 / C = 0. This implies that the decoupled variable φ is linearly decreasing ( φ ′ = − Σ C ) and hence that the frame rotation vector R α is rotating in the planespanned by e and e . Both fluid becomes extremely tilted (i.e. | v ( i ) | = 1).The past asymptotic state is for all non-self similar solutions a fix point located on the vacuum subset,generically either of zero or extreme tilt, depending on the asymptotic value of one of the shear variables.A subset of measure zero can asymptotically to the past have intermediate tilt if the shear variable Σ + tends to a specific value.One can ask wether the anti-alignment of the fluids is a physically realistic scenario. Is there somemechanism which asymptotically would produce such a state from more general models? For past asymp- FIX POINTS AND LOCAL STABILITY ANALYSIS
A Fix points and local stability analysis
Kasner fix points : There are two circles of Kasner points and two surfaces of fix points when 0 ≤ w .The Kasner circles are characterized by Σ + = ˆΣ + , Σ B = ˆΣ − , Σ A = Σ C = 0 , Ω (1) = Ω (2) = 0, where ˆΣ ± are constants that satisfy ˆΣ + ˆΣ − = 1, and the following values of v ( i ) :K : v (1) = v (2) = 0 , K : v (1) = − v (2) = 1 . (26a)The eigenvalues for the two cases are:K : 0 ; λ Σ A ; λ Σ C ; λ v (1) ; λ v (2) ; 3(1 − w ) ; 3(1 − w ) , (27a)K : 0 ; λ Σ A ; λ Σ C ; λ v (1) ; λ v (2) ; 2(1 + ˆΣ + ) , (27b)where λ Σ A = − (3 ˆΣ + + √ − ) , λ Σ C = 2 √ − , (28a) λ v ( i ) = 3 w − + , λ v ( i ) = − w − + ) / (1 − w ) . (28b)In the K case the Codazzi constraint (20b) is singular and hence it cannot be locally solved; in theother case (20b) has been used to eliminate Ω (1) . The zero eigenvalue corresponds to that one has aone-parameter set of fixed points. The two surfaces of Kasner fix points are characterized byKS ± v (1) v (2) : Σ A = Σ C = Ω (1) = Ω (2) = 0 , Σ + = (1 − w ) , Σ B = ± q − Σ , (29) (cid:18) v (1) v (2) (cid:19) · − v − v ! (1 − w ) = k . where the superscript denotes the sign of Σ B , and k is the constant defined in (24). The relation between v (1) , v (2) and k constrains the three free parameters and thus gives a surface of fix points. After eliminatingΩ (1) locally by means of the Codazzi constraint (20b), the eigenvalues for the Kasner surfaces are:KS ± v (1) v (2) : 0 ; 0 ; 0 ; λ Σ A ; λ Σ C ; 3(1 − w ) , (30a)where again λ Σ A = − (3 ˆΣ + + √ − ) , λ Σ C = 2 √ − , where ˆΣ + , ˆΣ − take the fix point values for therelevant line of fix points. Here two zero eigenvalues corresponds to that one has a surface of fix pointswhile the third is associated with the existence of a one parameter set of solutions that are anti-parallelw.r.t. each other on each side of the surface of fix points. Friedmann fix points : On the Friedmann subset there exists one line of fix points parameterized bythe constant of motion k (24):FL Ω (1) Ω (2) : Σ + = Σ A = Σ B = Σ C = 0 , Ω m = 1 v (1) = 0 , v (2) = 0 , Ω (1) Ω (2) = √ k , (31a)where the superscript refers to the values of Ω (1) and Ω (2) . The associated eigenvalues are:FL Ω (1) Ω (2) : λ , , , = − (1 − w ) ; 3 w − , (32a)were we have used the Codazzi constraint (20b) to eliminate the variable v (2) . Two of the eigenvalues of λ , , , refer to λ Σ A and λ Σ C . FIX POINTS AND LOCAL STABILITY ANALYSIS Fix points on
LRS : When < w there is an additional line of fix points, LRSL v (1) v (2) , which enter thephysical state space when w = , and move into the LRS -subset with increasing values of w . The lineintersects the foliation determined by (24) and can be parametrised by k . We have here chosen to use v (1) as a parameter instead for reasons of computational simplicity. In the stiff perfect fluid limit ( w = 1)the line merge with the coalesced Kasner surfaces. The line of fix points is characterized by:LRSL v (1) v (2) :Σ A = Σ B = Σ C = 0 , Σ + = − (3 w − , v (2) v (1) = − w − w + 1 , w − w + 1 ≤ v (1) ≤ , Ω (1) = 34 (1 − w )(5 w + 1)(3 w − wv )(1 + w )[(5 w + 1) v + (3 w − , Ω (2) = 34 (1 − w )[(5 w + 1) v + w (3 w − ](1 + w )[(5 w + 1) v + (3 w − . (33a)After eliminating Ω (1) locally the eigenvalues for the LRS-line are:LRSL v (1) v (2) : λ Σ A = 3(2 w −
1) ; λ Σ B = λ Σ C = − (1 − w ) ; 0 ; − (1 − w ) (cid:16) ± q A ( w, v (1) ) (cid:17) , (34a)where Re A ( w, v (1) ) <
1; since the expression for A ( w ( i ) ) is rather messy we will refrain from giving it. Fix points on
T W :TW : Σ + = − , Σ C = 0 , Σ A = Σ B = √ , v (1) = 1 , v (2) = − , Ω (1) = Ω (2) = . (35)Local elimination of Ω (1) by means of the Codazzi constraint (20b) yields the eigenvalues: λ Σ C = ; − ; − (1 ± i √
39) ; 6(3 − w )5(1 − w ) ; 6(3 − w )5(1 − w ) . (36)When < w < there exists one more line of fix points on T W : TWL v (1) v (2) . This line comes intoexistence when the line LRSL v (1) v (2) bifurcate into two at w = ; it then wanders away from D when w increases and eventually leaves the physical state space through TW when w = . Like LRSL v (1) v (2) italso can be parameterized by k but we choose v (1) here also for simplicity. The fix points are characterizedby TWL v (1) v (2) : Σ + = − (3 w (1) − , Σ A = q (1 − w )(2 w − , Σ B = √ w − , Σ C = 0 , v (1) v (2) = (1 − w )(15 w − − w + 18 w − , (1 − w )(15 w − − w + 18 w − ≤ v (1) ≤ , Ω (1) = B ( w, v (1) ) , Ω (2) = 1 − (3 w − w − − B ( w, v (1) ) , (37a)where B ( w, v (1) ) = 34 (1 − w )(7 − w )(25 w − w + 1)(1 + wv )(1 + w )[( − w + 18 w − v + (1 − w )(7 − w )] . (37b)Local elimination of Ω (1) yields the following eigenvalues:TWL v (1) v (2) : λ Σ C = − (5 − w ) ; 0 ; λ ( w, v (1) ) ; λ ( w, v (1) ) ; λ ( w, v (1) ) ; λ ( w, v (1) ) , (38a)where λ , , , exhibit extremely messy expressions, which we therefore refrain from giving. They all havethe property that the real part of the eigenvalues always are negative in the domain of definition of thefix point set, thus making the entire line a local attractor in the range 1 / < w < / Fix point in the generic geometric manifold : There exists one fix point G for which all theoff-diagonal components of the shear are non-zero. It thus exists on the generic ‘geometric’ manifold, buton the ‘matter boundary’ ET where both fluids are extremely tilted. It is characterized by:G : Σ + = − , Σ A = √ , Σ B = Σ C = √ , v (1) = 1 , v (2) = − , Ω (1) = Ω (2) = . (39) THE K SUBSET (1) yields the eigenvalues: λ , , , = − (cid:18) ± i q ± √ (cid:19) ; λ , = − w − − w ) . (40)At w = there exists a triangular surface of fix points, GS v (1) v (2) , connecting TWL v (1) v (2) with G .GS v (1) v (2) is given by:GS v (1) v (2) : Σ + = − , Σ A = √ √ s v (1) v (2) + 34 v (1) v (2) − , Σ B = √ , Σ C = 13 √ s v (1) v (2) + 64 v (1) v (2) − − ≤ v (1) v (2) ≤ − , Ω (1) = 13 − v (2) (9 + 5 v )( v (1) − v (2) )(3 − v (1) v (2) ) , Ω (2) = 13 v (1) (9 + 5 v )( v (1) − v (2) )(3 − v (1) v (2) ) . (41a)Local elimination of Ω (1) yields two zero eigenvalues and four others with complicated dependence on v (1) and v (2) but which all have negative real part on the entire set GS v (1) v (2) .The local stability analysis of the fix points can be summarized in a table showing how the localattractor is transferred from set to set with increasing values of w - from the isotropic Friedmann solutionsfor sub-radiation equation of states to the increasingly anisotropic solutions when the fluid becomes stiffer.invariantsubset: FLO LRS T W
GENERIC MANIFOLDlocalsink: FL Ω (1) Ω (2) → LRSL v (1) v (2) → TWL v (1) v (2) → GS v (1) v (2) → G w : ∈ [0 , ) ∈ ( , ] ∈ ( , ) ∈ ( , B The K subset We here discuss the Kasner subset K with the state space K = { Σ + , Σ A , Σ B , Σ C , v (1) , v (2) } , subjectedto the constraints (20a) and (24). The equations for the test fields v (1) ∈ [0 , v (2) ∈ [ − ,
0] decouplefrom those of the shear but are still coupled to each other through (24). The state space therefore canbe written as the following Cartesian product: K = KP × { v (1) , v (2) } , KP = { Σ + , Σ A , Σ B , Σ C } , (42)where KP is the projected Kasner state space, which of course is subjected to Σ = 1. By determiningthe α - and ω -limits for solutions on KP one can then determine the asymptotic states of v (1) and v (2) ,and thus the α - and ω -limits for solutions on K . Let us therefore first turn to the equations on KP :Σ ′ + = 3Σ A ; Σ ′ A = − (3Σ + + √ B )Σ A ; Σ ′ B = √ A − √ C ; Σ ′ C = 2 √ B Σ C . (43)This system is defined on the compact space Σ = 1. Since Σ + is monotonically increasing on a compactspace it must approach a constant value, hence we have Σ A ∝ Σ ′ + → + − √ B ) ′ = − √ C ≤
0, Σ C will by the same argument alsovanish asymptotically to both the future and to the past. Σ A = Σ C = 0 defines a circle of fix points,the projected Kasner circle: KP , see Figure 3(a). It is described by Σ + = ˆΣ + , Σ B = ˆΣ − , where theconstants ˆΣ + , ˆΣ − satisfy ˆΣ + ˆΣ − = 1.From this we conclude that the the α -limits for all solutions with Σ A Σ C = 0 on KP resides on thelocal source of KP , yielding a segment on KP characterized by − ≤ Σ + = ˆΣ + ≤ − , ≤ ˆΣ − ≤ √ ,i.e., the segment consists of sector (213) together with the fix points Q and T on KP . The ω -limitresides on the local sink, which consists of segment (312) together with the fix points Q and T , seeFigure 3(b)The α -limits for solutions on K are determined by the α -limits on KP which determine the asymptoticlimits for v ( i ) . The equation for | v ( i ) | ∈ [0 ,
1] on KP is given by: | v ( i ) | ′ = ( G ( i ) − ) − (1 − v i ) )(3 w − EFERENCES + Σ Σ B (321)(231)(213)(123) (132) (312)0T T T Q Q Q (a) Kasner sectors PSfrag replacements Σ + Σ Σ B (321)(231)(213)(123)(132) (312)0T T T Q Q Q (b) The α - and ω -limits on KP Figure 3: The projected Kasner circle KP is divided into sectors ( i, j, k ), defined by Σ i < Σ j < Σ k ,where i, j, k is a permutation of 1 , ,
3, and where Σ = ˆΣ + + √ − , Σ = ˆΣ + − √ − , Σ = − + , andthe points Q α , corresponding to the non-flat plane symmetric Kasner solution, and T α , corresponding tothe Taub form for the Minkowski spacetime. The global past attractor for the general geometric set withΣ A Σ B = 0 on KP consists of sector (213) together with Q and T on KP . The global future attractorconsists of sector (312) together with Q and T on KP .2 ˆΣ + ) | v ( i ) | . It follows that the α -limits for all orbits on K on the general geometric set with Σ A Σ C = 0resides on the global past attractor A {∗∗} , where the subscript denotes the range of values of w , given by A { w< } = { K : ˆΣ + ∈ (cid:2) − , − (cid:3) } , (44a) A { w = 23 } = { K : ˆΣ + ∈ (cid:2) − , − (cid:1) } ∪ { KS + v (1) v (2) : ˆΣ + = − } , (44b) A { Dynamical systems in cosmology , Cambridge: Cambridge Univer-sity Press (1997).[2] C. G. Hewitt, R. Bridson, and J. Wainwright. The Asymptotic Regimes of Tilted Bianchi IICosmologies. Gen. Rel. Grav. 65 (2001).[3] J. D. Barrow and S. Hervik. The future of tilted Bianchi universes. Class. Quantum Grav. universes. Class. Quantum Grav. 579 (2005).[6] S. Hervik, R. J. van den Hoogen, and A. Coley. Future asymptotic behaviour of tilted Bianchi modelsof type IV and VII h . Class. Quantum Grav. 607 (2005). EFERENCES cosmologies with vorticity. Class. Quantum Grav. 845 (2006).[8] S. Hervik and W. C. Lim. The late time behaviour of vortic Bianchi type VIII universes. Class.Quantum Grav. h models. Class. Quantum Grav. − / models. Class. Quantum Grav. : 103502 (2003).[14] G. F. R. Ellis and H. van Elst. Cosmological models (Carg`ese lectures 1998) in Theoretical andObservational Cosmology , edited by M. Lachi`eze-Rey, Dordrecht: Kluwer (1999) p. 1.[15] V. G. LeBlanc. Asymptotic states of magnetic Bianchi I cosmologies. Class. Quantum Grav.14