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On almost EGS theorems

We show assuming small data that massless solutions to the reflection symmetric Einstein-Vlasov system with Bianchi VII0 symmetry which are not locally rotational symmetric, can be arbitrarily close to and will remain close to isotropy as regards the shear. However in general the shear will not tend to zero and the Hubble normalised Weyl curvature will blow up. We thus have generalised the work [10, 16] which considered a nontilted radiation fluid to the massless Vlasov case. This represents another example of the fact that almost-EGS theorems do not hold in general and that collisionless matter behaves differently than a perfect fluid.


Introduction
Since the microwave background is almost isotropic it is natural to consider an isotropic matter distribution. What are the consequences of this assumption for the space-time? The theorem of Ehlers, Geren and Sachs [5] gave an answer to this question proving that for collisionless matter the space-time has to be either stationary or Robertson-Walker. This was generalised later to the Boltzmann case [15].
What happens if the matter distribution is almost isotropic and the universe is non-stationary? One might think that the space-time has to be almost isotropic. Different results were obtained proving almost EGS-theorems following the research line initiated in [14].
However these results were obtained under certain assumptions which do not hold in general [11,12].
If a cosmological constant is present non-linear stability and isotropisation of solutions has been shown for a variety of matter models cf. [13,6] and references therein. What happens if no cosmological constant is present?
In [16] it was shown that Bianchi VII 0 solutions which are not locally rotational symmetric (LRS) with a non-tilted perfect fluid isotropise as regards the shear but do not isotropise as regards the Hubble normalised Weyl curvature. They also showed that self-similarity breaking occurs for any non-LRS Bianchi VII 0 solution. This was proven for any non-tilted fluid except for a radiation fluid. The latter case was proven in [10]. Afterwards these results were extended to a tilted fluid in [4,9].
One could argue that these results are special since the matter model is a perfect fluid, but more recently the massive case for reflection symmetric solutions to the Einstein-Vlasov system with Bianchi VII 0 symmetry was covered in [7]. Since the microwave background comes from massless particles it is of special interest to treat this case and in the present paper we obtain a similar result for the massless case assuming small data.
The massless case (radiation fluid) when treating a tilted or non-tilted fluid for a fluid with a linear equation of state P = (γ − 1)ρ was more complicated in the case γ = 4 3 . The reason is that in this case some eigenvalues of the linear part vanish and center manifold theory has to be used. As a consequence the shear variables have a polynomial decay for γ = 4 3 while they have an exponential decay for 2 3 < γ < 2 with γ = 4 3 . In the Vlasov case the massive and the massless case are also different. In the massive case the solutions tend to the same behaviour as in the dust case forming a system of differential equations where the linear part has negative eigenvalues. In the massless case the linear part has an eigenvalue which vanishes, but now instead of an equilibrium point there is an equilibrium line.
Nevertheless in both massive and massless Vlasov case there is an exponential decay. The behaviour at late times thus differs from that of a radiation fluid where a polynomial decay was found [10].
Moreover in [10,9] it was shown that the shear tends to zero both in the nontilted and the tilted case. We show here that for collisionless mattter although the shear will always remain small, it nevertheless does not tend to zero, which is another difference with respect to a fluid.
Finally the result of this paper extends the known results concerning the late time behaviour of massless solutions to the Einstein-Vlasov system with Bianchi symmetries where there has been some recent progress [1,2,8].

The massless Einstein-Vlasov system
In this section we introduce the massless Einstein-Vlasov system with Bianchi symmetry. Consider a four-dimensional oriented and time oriented Lorentzian manifold (M, 4 g) and a distribution function f , then the massless Einstein-Vlasov system is written as where G αβ is the Einstein tensor and T αβ is the energy-momentum tensor defined by Here, the integration is over the future pointing light-cone H at a given spacetime point which is defined by with the apex removed, and χ is the distribution function multiplied by the Lorentz invariant measure and L the Liouville operator. The basic equations we will use can be found in Sections 7.3-7.4 and Chapter 25 of [13]. We also refer to this book for an introduction to the Einstein-Vlasov system. Let Σ be a spacelike hypersurface in M with n its future directed unit normal. Let g be the Riemannian metric induced on Σ by 4 g. We define the second fundamental form as k(X, Y ) = ∇ X n, Y for vectors X and Y tangent to Σ, where ∇ is the Levi-Civita connection associated with 4 g. The Hamiltonian and momentum constraint equations are as follows: where k = k ab g ab the trace of the second fundamental form, R and ∇ the scalar curvature and the Levi-Civita connection of g respectively, and matter terms are given by ρ = T αβ n α n β and J i X i = −T αβ n α X β for X tangent to Σ. Here and throughout the paper we assume that Greek letters run from 0 to 3, while Latin letters from 1 to 3, and also follow the sign conventions of [13].

The massless Einstein-Vlasov system with Bianchi symmetry
A Bianchi spacetime is defined to be a spatially homogeneous spacetime whose isometry group possesses a three-dimensional subgroup that acts simply transitively on spacelike orbits. A Bianchi spacetime admits a Lie algebra of Killing vector fields. These vector fields are tangent to the group orbits, which are the surfaces of homogeneity. Using a left-invariant frame, the metric induced on the spacelike hypersurfaces depends only on the time variable. Let G be the three-dimensional Lie group, e i a basis of the Lie algebra, and ξ i the dual of e i . The metric of the Bianchi spacetime in the left-invariant frame is written as We will need equations (25.17)-(25.18) of [13] (without scalar field) with the notation T ab = S ab :ġ where the dot means the derivative with respect to time t and R ab is the Ricci tensor associated to the induced 3-metric. Note that in the massless case Since k does not depend on spatial variables, the constraint equations are as follows: where we have dropped the bar on the covariant derivative by a slight abuse of notation. Following conventions of [13] ∇ ej e l = Γ i jl e i and the connection coefficients can be expressed in terms of the structure constants, cf. (25.3) of [13]: Note that as a consequence from last equation since g mi C mi vanishes due to the symmetry of the metric and the antisymmetry of the structure constants. Moreover the only non-zero components of Γ α βγ are Γ a bc and cf. Section 25.1 of [13]. We wish to express the momentum constraint in terms of the connection coefficients. We have which means and that one can express J i in terms of the metric, second fundamental form and the structure constants. Using (7) and considering the Bianchi A case where C l ml = 0, the last equation turns to Now denote by ijk the standard permutation symbol. For Bianchi A spacetime we have that (cf. E.1 of [13] with a j = 0) C n mi = mil n ln (9) where n lk is a symmetric matrix also called the structure constant matrix which characterises the Bianchi type. We thus have: Below, we collect and derive several useful equations. Using the fact thaṫ we obtaink and by taking the trace of (12) we havė It is convenient to express the second fundamental form as where σ ab is the trace free part and H = 1 3 k is the Hubble parameter. Then (13) becomesḢ and (4) becomes whereR = R H 2 and F = σ ab σ ab H 2 . In terms of the trace free part (12) transforms intȯ where r a b and π a b are the trace free part of R a b and S a b respectively. By the constraint equation (4) one can eliminate the energy density such that (13) reads:k Using the trace free part of k ab and the Hubble variable we obtain where we have defined It is convenient to introduce a dimensionless time variable τ as follows: and denote derivation with respect to that variable by a prime. Sometimes it is also useful to use the variable q: where we have used (17) in the last equation. The evolution equation of Σ a b is then Using (17) and (14) we have Since Σ a b is trace free sometimes it is convenient to work with Σ + and Σ − as was done in [7] which are defined by Using H 2 = ρ 3Ω and define w ± analogously to Σ ± by Note that by definition π 2 2 = S 2 2 − 1 3 tr S, π 3 3 = S 3 3 − 1 3 tr S and using (3) we obtain Using the fact that 0 ≤ S 2 2 + S 3 3 ≤ tr S = ρ and 0 ≤ S 2 2 ≤ tr S, 0 ≤ S 3 3 ≤ tr S, we obtain the following bounds for w ± : Our evolution equations of Σ ± are thus

Vlasov equation with Bianchi symmetry
Since we use a left-invariant frame, f will not depend on x a . We will assume that f has compact support for simplicity. Moreover, since g 00 = g 00 = −1 and g 0a = 0, we have p 0 = −p 0 = p a p b g ab , ρ = T 00 , and J a = −T 0a . The frame components of the energy-momentum tensor are thus We have that which means that The particle current density is The conservation of particle current density implies that which for any Bianchi A spacetime, where C n nl = 0 holds, implies with (7) and (8) thaṫ Together with (22) we obtain that f dp is a conserved quantity: f (τ, p)dp = f (τ 0 , p)dp.
Using the expressions for the connection coefficients and the antisymmetry of the structure constants the Vlasov equations as expressed in (25.14) of [13] where f depends on p i turns into from which follows that the equation for the distribution function in terms of p i dropping the bar by a slight abuse of notation is When treating the Bianchi I case in [8] we considered derivatives of the following quantity: In the Bianchi I case f does not depend on t, but in general it does. Let us thus consider the terms where the time derivative of f is involved. Consider first the term related coming from the energy density. If we substitute the Vlasov equation (23) and integrate by parts we have where we have used again the fact that g ab is symmetric while C d ba is antisymmetric which implies that summing over both of them gives zero. Moreover for all Bianchi A spacetimes C a ba = 0 which means that the whole term vanishes in that case.
Consider now the other term which appears when taking the time derivative of the distribution function in (24) and call it V j i . Using again the Vlasov equation and integrating by parts If e is equal to a the term vanishes due to the antisymmetry of the structure constants and d = a the term vanishes for Bianchi A spacetimes. Let us assume we consider Bianchi A space times. In that case which also vanishes when summed over the rest due the antisymmetry of the structure constants. Thus The derivative of w j i with respect to τ is then: where where Consider the integral in the numerator and call it X jd ic . Doing a similar procedure as for W j i , i.e. using the Vlasov equation and integrating by parts: where the last equality is obtained by the same considerations as when treating W j i . We will consider that we are close to the case that f has some symmetries. In that case ξ jd ic takes some specific values which we will denote byξ jd ic . Let us consider the trace free partw i j = w i j − 1 3 δ i j and definẽ Then (27) turns into In particular for the terms w where we have used the notation 3 The equations for Bianchi VII 0 with reflection symmetry for massless Vlasov particles close to the isotropic case In the following we will deduce the relevant equations for Bianchi VII 0 with reflection symmetry. We will exclude the LRS case, since that cases reduces to Bianchi I, which presents a completely different behaviour and was already studied in [8].
We will assume that we are close to the isotropic case. This means that as in [8] the only non-vanishing expressions forξ ab cd are (suspending the Einstein summation convention for the next expressions ): Setting this into the equations for w ± we have the metric and the second fundamental form are diagonal, the evolution will preserve this symmetry. Everything will be diagonal and also J a = T 0a = 0 since it is an integral over an odd number of momenta which vanishes due to the reflection symmetry. Since J a = 0 we have from (10) that k i j and n ij commute, which means there exists a basis where one can simultaneously diagonalise both. If we thus choose eigenvectors of k i j (t 0 ) as the frame and choose g(t 0 ) to be diagonal since J a = 0, everything will remain diagonal. Now if n ij is diagonal we have that the right hand side of (10) is always zero, which comes from the fact that due to (9), the structure constants cannot have two equal indices in that case. As a consequence the momentum constraint will be trivially be satisfied for all time.
Moreover due to this symmetry the quantity W i j defined in (24) and the quantity Z jd ic defined in (29) vanish as well. The reason is the same as for J a . The integrals involved in the numerator for W i j and Z jd ic are V i j and X jd ic which are integrals over an odd number of momenta which due to the reflection symmetry vanish. As a consequence, for the reflection symmetric case, the equations forw and ξ jd ic will be identical to the equations for the Bianchi I case treated in [8]. In particular the equations for w ± in the reflection symmetric case using .

The constraint equation is
For the reflection symmetric massless Bianchi VII 0 case we have the same equations for the curvature variables as in the massive case [7]. Let (ijk) denote a cyclic permutation of (123) and let us suspend the Einstein summation convention for the next three formulas. Introduce ν i as the signs depending on the Bianchi type [ Table 1 of [3]]. Now define where this means e.g. n 1 = ν 1 g11 g22g33 . Using this notation the Ricci tensor is given by [(11a) of [3]] The curvature variables are In the Bianchi VII 0 case we have ν 1 = 0 and ν 2 = ν 3 = 1, which means that N 22 and N 33 are positive definite and the only non-vanishing structure constants are (cf. Appendix E, p. 695 of [13]): The curvature expressions are The relevant equations are where N 22 > 0 and N 33 > 0, which we transform into the following variables: As a consequence The relevant evolution equations are thus

WHU variables
Making the same transformations as in WHU [16] M = 1 N + > 0, Since Ω = ρ 3H 2 ≥ 0, we have that The relevant equations for Σ + , M , X and ψ are obtained from equations (23)-(26) of [7] by making the following replacements in terms of the relevant quantities we use here: The relevant equations in our case are thus We have put the equations (41)-(45) in such a way that they are easy to compare with the equations in [10,16]. They are identical to the equations in [16] if one sets in our equations w + = w − = 0. Define The equations for ξ jd ic are then In the reflection symmetric case and assuming that our metric is diagonal the relevant equations are Note that the system is not closed, since we could consider the evolution equations of higher order terms such as Φ jdb ica etc. However the considered evolution equations will be sufficient to obtain the desired future asymptotic behaviour.

Future asymptotic behaviour
In order to obtain the desired result it will be necessary to assume closeness to an equilibrium line given by the curve This will become clear when treating the truncated system in Section 4.3. This system is obtained by neglecting the oscillatory terms, the higher order momentum termsξ kl ij and focusing on the equations for Σ + , X and w + .
We begin by observing that the relevant quantities are bounded. The quantities Σ + , X and Ω are bounded due to (39)-(40), w + and w − are bounded due to (19). Moreover ξ ij ij can be bounded by w i i or w j j (no summation of the indices in the previous three quantities) which are bounded by 1. The quantitiesξ ij ij just differ a constant from ξ ij ij according to (31). As a consequence α, β, γ and δ defined in (48)-(51) are bounded as well. Similarly Φ ijk ijk is bounded which implies that also the derivatives of Σ + , X, Ω, w + , w − andξ ij ij are bounded.

Estimate of M
Since X and w + are small, we can assume that Y is small. We will use a bootstrap argument. Let us assume that there exists an interval [τ 0 , τ 1 ) where the following estimates hold: where ε M , ε Σ , ε Y are some small quantities all smaller than ε. The last assumption (66) will hold initially since X is assumed to be different from zero and C is some arbitrary constant, not necessarily small. DefineM

The truncated system
Before considering the full system we consider the truncated system where we neglect the oscillatory terms with M and the term α which is related to higher order momenta. Sincew − basically only appears in the evolution equation of w − we ignore that equation for the moment.
Proof. The proof is a straightforward computation.
The evolution equation forΣ + using the variableŶ is then From (84) we obtain the evolution equation forŶ which iŝ Note thatû is zero forΣ + =Ŷ = 0. In fact using Sylvester's criterion we see it is positive definite for any other value. The evolution equation forû is using (85) as follows: from which follows: which proves the lemma.
Using a Runge-Kutta-Fehlberg method one can solve easily the truncated system for some random numbers and from the graph one see that the variables converge rapidly to the equilibrium values.
Plot of the evolution of solutions to the truncated system

Closing the bootstrap argument
In order to close the bootstrap argument we generalise Lemma 1 which corresponds to the truncated system to the general system using the estimate for M and bounds for the matter terms.
Note that all the smallness assumptions for the different variables imply that the corresponding barred quantities are also small making M smaller if necessary.

Bounds for the matter terms
For the matter terms it will be sufficient to show that they can be bounded by small quantities. From (80) and since w + , w − , ξ ij ij , Φ ij ij are bounded we have using (59) and (64) As a consequence α and γ have a similar bound. From (79) we obtain the same type of bound having in mind that γ is bounded, obtaining:w

Estimate of the quotient of w − and X
Now consider the evolution equation of B =w − X using the evolution equations forX (76) andw − (79) : Now using the bootstrap assumptions and the estimate (94) we obtain that that u = O(ε 2 e (−1+Cε)τ ) andΣ Choosing ε smaller if necessary gives the estimates for Σ + and Y which closes that bootstrap argument and finishes the proof of Theorem 1.

The Weyl parameter and the shear at late times
We conclude with the following corollary: Corollary 1. Consider the same assumptions as in Theorem 1. Then and Proof. Using the estimate (93) and the estimate forw − (87) making ε smaller if necessary we obtain a lower bound for X Together with the estimate for M gives the desired conclusion for the Hubble normalised Weyl curvature (96). For the second part, note that the second derivatives of the barred quantitiesX,Σ + ,M ,w + andw − are bounded. The terms we have collected in O(M ) in the differential equations for the barred quantities are polynomials ofX,Σ + ,M ,w + ,w − , cos ψ and sin ψ, all multiplied by M . All the possible derivatives of these quantities will be bounded, since the only problematic term could come from the derivative of ψ. The quantity 1 M can be absorbed by the factor M and the quantity 1 X is now not problematic anymore since we have shown that it X has the lower bound (98). The limits forΣ + ,M are obtained from the estimates of the theorem. Taking the limit in the differential equations forX,w + andw − we obtain that these quantities also have a limit. Taking the limit of the definitions ofX,Σ + ,M ,w + ,w − and Y in terms of the unbarred quantities together with the limit of M we obtain the desired conclusion.

Discussion
Our results show that one can have a homogeneous cosmological model with massless particles where the shear is arbitrarily small and remains small, and the anisotropy of the matter distribution is arbitrarily small and remains small, but where the space-time is far from and remains far from an isotropic spacetime as the corollary shows.
In particular we have shown the asymptotic behaviour of reflection symmetric solutions to the Einstein-Vlasov system with Bianchi VII 0 symmetry. The solutions tend to the equilibrium solution of what we have called the truncated system assuming small data.
In the massive case it was shown that reflection symmetric solutions to the Einstein-Vlasov system with Bianchi VII 0 symmetry behave as Einstein-dust solutions assuming small data, cf. Theorem 1 of [7].
In the massless case, the relation between pressure (trace of energy momentum tensor) and energy density is fixed and given by (3). However here we have shown that in the massless case, the behaviour differs from that of a radiation fluid, in the sense that we obtain an exponential decay rate, while the decay rate for a radiation fluid, both tilted and non-tilted, is polynomial [9,10,16].
Another difference is that in the collisionless case the shear will always remain small, but will not tend to zero. If we come back to the question of what happens with the spacetime if the matter distribution is almost isotropic, the answer to this question would be in the case considered here given by our corollary. In particular the anisotropy of the Hubble normalised trace free part of the second fundamental form is related to the anisotropy of the energy momentum tensor by (97).