Future stability of expanding spatially homogeneous FLRW solutions of the spherically symmetric Einstein--massless Vlasov system with spatial topology $\mathbb{R}^3$

Spatially homogeneous FLRW solutions constitute an infinite dimensional family of explicit solutions of the Einstein--massless Vlasov system with vanishing cosmological constant. Each member expands towards the future at a decelerated rate. These solutions are shown to be nonlinearly future stable to compactly supported spherically symmetric perturbations, in the case that the spatial topology is that of $\mathbb{R}^3$. The decay rates of the energy momentum tensor components, with respect to an appropriately normalised double null frame, are compared to those around Minkowski space. When measured with respect to their respective $t$ coordinates, certain components decay faster around Minkowski space, while others decay faster around FLRW.


Introduction
Standard homogeneous isotropic cosmological models in general relativity are described by the Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes where I ⊂ R is an open interval, (Σ, g Σ ) is a constant curvature manifold, and a : I → (0, ∞) is an appropriate scale factor. See Section 5.3 of [21] for more on FLRW spacetimes. This article concerns radiation filled FLRW cosmologies in which the constant curvature manifold is Euclidean space, (Σ, g Σ ) = (R 3 , g Eucl ), with radiation described by spatially homogenous solutions of the massless Vlasov equation, and their stability properties.

The Einstein-massless Vlasov system
Consider a 3 + 1 dimensional Lorentzian manifold (M, g) and let denote the mass shell of (M, g). Consider some local coordinates {t, x 1 , x 2 , x 3 } on M, and let {t, x i , p µ } denote the corresponding conjugate coordinate system for T M, i.e. (t, x i , p µ ) describes the point p µ ∂ x µ | (t,x) ∈ T M. The massless Vlasov equation on (M, g) takes the form where p 0 is defined by the mass shell relation g µν p µ p ν = 0. (1. 3) The Einstein-massless Vlasov system consists of equation (1.2) coupled to the Einstein equations Ric(g) µν − 1 2 R(g)g µν = T µν , (1.4) where the energy momentum tensor takes the form f (t, x, p)p µ p ν √ − det g −p 0 dp 1 dp 2 dp 3 , where indices are raised and lowered with respect to the metric g (so that, for example, p 0 = g 0µ p µ ). Here t is also denoted x 0 , Greek indices, such as µ, ν, range over 0, 1, 2, 4, and lower case Latin indices, such as i, j, k range over 1, 2, 3.

The spatially homogeneous FLRW family of solutions
The spatially homogeneous FLRW family constitutes an infinite dimensional family of explicit solutions of the Einstein-massless Vlasov system (1.2)-(1.5).
Remark 1.3 (T 3 spatial topology). Each (1.7)-(1.8), and more generally the solutions of Remark 1.2, also define a spatially homogeneous FLRW solution of the Einstein-massless Vlasov system on the manifold (0, ∞) × T 3 . This article will only be concerned with solutions on (0, ∞) × R 3 . (See also Remark 3.5 for a further comment on the solutions on (0, ∞) × T 3 .) The components of the energy momentum tensor of (1.7)-(1.8), with respect to the Cartesian (t, x 1 , x 2 , x 3 ) coordinate system, take the form . The spacetimes (M • , g • ) expand from a spacelike singularity at t = 0 (with Kretchmann scalar |Rm • | 2 g• = 3 2t 4 ) and are future geodesically complete. See the Penrose diagram of Figure 1. Such solutions feature in the Ehlers-Geren-Sachs Theorem [12], which in particular ensures that all solutions of the Einstein-massless Vlasov system for which f is isotropic and irrotational are either stationary or described by an FLRW metric as above.

First version of the main theorem
The main result of the present work concerns the future stability of the family (1.7)-(1.8).
Theorem 1.4 (Future stability of FLRW). Each spatially homogeneous FLRW solution is future nonlinearly stable, as a solution of the Einstein-massless Vlasov system (1.2)-(1.5), to compactly supported spherically symmetric perturbations. More precisely, consider an FLRW solution of the form (1.7)-(1.8) for some smooth, non-identically vanishing µ (decaying suitably so that ̺ is defined). For all compactly supported spherically symmetric initial data sufficiently close to that of (1.7)-(1.8) on Σ 1 = {t = 1}, the resulting solution is future geodesically complete, is isometric to (1.7)-(1.8) after some retarded time and, in an appropriately normalised double null gauge, the (appropriately normalised) components of the energy momentum tensor decay to zero as (1.9) to leading order, and each metric component and Christoffel symbol either remains close or decays to its FLRW value to the past of this retarded time, with quantitative polynomial rates.
A more precise version of Theorem 1.4 is stated below in Section 4. See also the Penrose diagram in The assumption of compact support can be relaxed, and is made for convenience in order to localise the proof. Indeed, following [9], the perturbation of f , in the coupled problem, vanishes close to the centre of spherical symmetry at late times. The absence of singularities in future evolution therefore follows from the aforementioned quantitative decay properties, along with comparatively soft arguments including an extension principle around non-central points (see [11] or Theorem 2.3 below). Remark 1.6 (Birkhoff-type theorem). The proof of Theorem 1.4 in particular contains a Birkhoff-type theorem for the system (1.2)-(1.5) which ensures that any spherically symmetric solution with a regular centre, for which f is equal to its FLRW value f • (defined by (1.7)), is locally isometric to the FLRW spacetime (M • , g • ). See the final step in the proof of Theorem 5.1 (from which a general theorem can be extracted). Recall also the Ehlers-Geren-Sachs Theorem [12].
for unknowns g, four velocity u µ -satisfying g µν u µ u ν = −1 -pressure p, and density ρ, with p and ρ related by the radiation equation of state p = ρ 3 .
The fluid variables for the FLRW metric take the form, in Cartesian coordinates, (1.10) It is known that the solution (1.10) is unstable to formation of shock waves for the Euler equations ∇ µ T µν = 0 on the fixed FLRW background (M • , g • ) (see Chapter 9 of [34]).
Note also the linear Jeans instability for the Einstein-Euler system linearised around FLRW [24] (see also the discussion in [6]).

Previous works
There have been previous works on the Einstein-Vlasov system in cosmological settings for Einstein equations with a positive cosmological constant [31], and also for perturbations of the vacuum Milne solution [1,3,13]. Note also related works [15,19,32,33] on the Einstein-Euler system. See also [2,16,17]. Each of these works considers perturbations of spacetimes with metric of the form (1.1) undergoing accelerated expansion or linear expansion (see Remark 1.1). Contrast with the solutions (1.7)-(1.8) of the present work in which undergo slow decelerated expansion.
In the asymptotically flat setting, Minkowski space has been shown to be nonlinearly stable for both the massless and massive Einstein-Vlasov systems, first in spherical symmetry [9,30], and later to general perturbations [14,26,35]. The present work is based on the approach of [9]. Anti-de Sitter space has also been shown to be nonlinearly unstable as a solution of the Einstein-massless Vlasov system with a negative cosmological constant [28].
Though there is no direct non-relativistic analogue of the present problem, there is an infinite dimensional family of explicit, spatially homogenous, stationary solutions of the Vlasov-Poisson system. The works [5,20,22] on Landau damping, and also the celebrated works [4,18,29] on the torus T 3 , concern the stability of these families of solutions. Note however that one must apply the Jeans swindle to make sense of the problems considered in these works in the gravitational setting. See also [23].

Outline of the paper
Section 2 contains preliminaries on the Einstein-massless Vlasov system in double null gauge. Section 3 concerns further properties of the FLRW spacetimes, introduced in Section 1.2. A double null gauge in FLRW is defined in Section 3.1 and, as a precursor to the proof Theorem 1.4, properties of the massless Vlasov equation (1.2)-(1.3) on the FLRW background spacetimes are presented in Section 3.2. In Section 4 a more precise version of Theorem 1.4 is formulated, and in Section 5 its proof is given.

Spherical symmetry
acts by isometry on (M, g) and preserves f . More precisely, it is assumed that there is a smooth isometric action O : M × SO(3) → M on (M, g) such that, for each p ∈ M, the orbit of the action satisfies either Orb(p) ≃ S 2 , or Orb(p) = {p} and, moreover, for each of the generators Ω 1 , Ω 2 , Ω 3 of the SO(3) action, the corresponding flows Φ i : for all (x, p) ∈ P. It is assumed that the centre consists of a single timelike geodesic, and that M Γ splits diffeomorphically as for a smooth 2-manifold Q = (M Γ)/SO(3). The quotient space M/SO (3), also denoted Q, is viewed a manifold with boundary, with boundary identified with the centre Γ.

Double null gauge
A double null gauge consists of functions u, v : Q → R such that the level hypersurfaces of u foliate Q by outgoing lines which are null with respect to the induced metric on Q, and the level hypersurfaces of v foliate Q by ingoing null lines. The level hypersurfaces of u and v lift to outgoing and incoming null cones of M and such a double null gauge can be complemented with local coordinates (θ 1 , θ 2 ) on S 2 to local coordinates (u, v, θ 1 , θ 2 ) for M. For a given double null gauge, the metric g can be written in double null form where γ is the unit round metric on S 2 , where Ω is a function on Q and R : Q → R is the area radius function The area radius R extends regularly to 0 on the centre Γ. Define preserves the double null form (2.2). Under such a change, the metric takes the form It thus follows that The inverse metric of (2.2) takes the form where A and B range over 1 and 2. The nonvanishing Christoffel symbols of g are where / Γ A BC are the Christoffel symbols of (S 2 , γ).
For a given spherically symmetric double null gauge (u, v, θ 1 , θ 2 ) for (M, g), one defines a corresponding Such a coordinate system induces a coordinate system (u, v, θ 1 , θ 2 , p v , p 1 , p 2 ) on the mass shell P , with p u defined by the mass shell relation (1.3), which takes the form where the angular momentum L : The volume form on in this induced coordinate system takes the form The condition (2.1) that f is spherically symmetric means that, in a given double null gauge, f can be written as a function -which, abusing notation slightly, is also denoted f : The energy momentum tensor T on M takes the form In double null gauge, the components take the form . It moreover follows from the mass shell relation (2.7) that

The spherically symmetric Einstein-massless Vlasov system in double null gauge
The Einstein equations (1.4) in spherical symmetry take the form of the following system of equations for (Ω, R, f ), where / g = R 2 γ. The mass shell relation (1.3) takes the form In particular, Ω 2 T uv = R 2 γ AB T AB , or 4Ω −2 T uv = R −2 γ AB T AB , and so equation (2.14) can be rewritten The Vlasov equation (1.2) in spherical symmetry takes the form The See [27] for more on the spherically symmetric Einstein-massless Vlasov system.

The functions t and r
Define functions t and r by When the FLRW spacetime (1.7)-(1.8) is expressed in the double null gauge introduced in Section 3.1 below, the functions t and r coincide with those of Section 1.2.

The Cauchy problem
The Einstein-massless Vlasov system (1.2)-(1.5) admits the following local existence theorem. Initial data consists of a Riemannian 3 manifold (Σ, g), along with a symmetric (0, 2) tensor K on Σ, and a function f 1 : T Σ → [0, ∞), satisfying constraint equations R − |K| 2 g + (trK) 2 = 2 det g f 1 |p| g dp 1 dp 2 dp 3 , divK i − ∇ i trK = det g f 1 p i dp 1 dp 2 dp 3 , where R is the scalar curvature and ∇ is the Levi-Civita connection of g. A development of initial data (Σ, g, K, f 1 ) is a globally hyperbolic spacetime (M, g) which admits Σ as a Cauchy hypersurface, with induced first and second fundamental form g and K respectively, such that the restriction of f to the mass shell over Σ, P | Σ , coincides with f 1 , when T Σ is appropriately identified with P | Σ . See [31] for more on the Cauchy problem for the Einstein-Vlasov system. See also [27] for a version of Theorem 2.2 in a more general spherically symmetric setting.
The following extension principle for spherically symmetric solutions of the system (1.2)-(1.5), concerning non-central points, will be used in what follows. It is assumed that (M, g, f ) is the maximal development of a smooth spherically symmetric initial data set (Σ, g, K, f 1 ) for the Einstein-massless Vlasov system (1.2)-(1.5), and that Q is the quotient manifold of M by the action of the SO(3) isometry, as in Section 2.1 above. Theorem 2.3 (Extension principle, around non-central points, for the spherically symmetric Einstein-massless Vlasov system [11]). Let Q be as above. Let (u * , v * ) be such that there exists U < u * and V < v * such that the characteristic diamond D u * ,v * (2.21) Assume also that there exists 0 < R 0 < R 1 such that and that f (u, v, ·, ·) is compactly supported for all (u, v) ∈ D u * ,v * U,V . Then (u * , v * ) ∈ Q. For a proof of Theorem 2.3 see Section 4.3 of [11] (where one has to replace the massive mass shell relation with its massless analogue (2.17), which does not affect the proof). Theorem 2.3 is preferred to the softer extension principle of [10] in view of the presence of the anti-trapped surfaces in the spacetimes under consideration (see Remark 3.1 below).

The FLRW spacetimes
Recall the FLRW spacetimes introduced in Section 1.2. In Section 3.1 a double null gauge is introduced in each of these spacetimes. Section 3.2 concerns properties of the massless Vlasov equation on these spacetimes. Though the main theorem of Section 3.2, Theorem 3.2, is not used in the proof of Theorem 1.4, its proof is presented as a simple precursor to that of Theorem 1.4. There is no symmetry assumption required, however, for Theorem 3.2.

The FLRW metrics in double null gauge
Recall that the FLRW spacetime (M • , g • ) takes the form Define double null coordinates and note that, since t ≥ 0 and The FLRW metric g • in the above double null gauge takes the form defined on the quotient manifold The metric g • can be written and moreover The non-vanishing Christoffel symbols of (3.1) take the form where / Γ A BC are the Christoffel symbols of (S 2 , γ). It follows from (1.9) that the null components of the energy momentum tensor of f • satisfy The mass shell relation (2.17) takes the form Recall the expression (2.7) for p u , and definep Where there is no ambiguity, for example in Section 3.2,p u will also be denoted p u . Given a solution (Q, Ω 2 , R, f ) of the spherically symmetric Einstein-massless Vlasov system in double null gauge (2.
Note in particular that, if f = f • , then The asymmetry in u and v arises from the choice of parameterising the mass shell by p v , rather than p u .

The massless Vlasov equation on an FLRW background
The proof of Theorem 1.4 is based on the following proof of decay of components of solutions of the massless Vlasov equation on a fixed FLRW backround.
The geodesic equations for p u (s) and p v (s) take the forṁ One has equations for t(s) = t(γ(s)) and r(s) = r(γ(s)) along the geodesiċ t(s) = t(s)  (1)) ∈ B and let v 0 be sufficiently large. There exist 0 < c < 1 < C, L 0 ≥ 0, and s 0 ∈ [1, ∞) such that v(γ(s 0 )) = v 0 and the components of the tangent vector to γ satisfy, for all s ≥ s 0 , Moreover, there exists retarded times U 0 < 0 < U 1 such that the u component of γ satisfies Proof. Provided v 0 is suitably large, the existence of s 0 ∈ [1, ∞) such that v(γ(s 0 )) = v 0 follows from the compactness of B.
Consider now the bounds (3.10). For simplicity, suppose that r(γ(1)) = 0 (otherwise replace 1 in the following with 1 + ǫ for some ǫ > 0). By the conservation of angular momentum (2.20), one has the conservation law for all s ≥ 1, (3.12) from which the first of (3.10) trivially follows. For the remaining inequalities of (3.10), suppose first that γ is radial, so that the conserved quantity L(s) vanishes. It follows from the mass shell relation It follows that the there is a time s * ≥ 1 at which the geodesic hits the centre, i.e. a time at which r(s * ) = 0. The geodesic then becomes outgoing, with p u (s) = 0 for s > s * , and lim s↓s * The bounds (3.10) then again trivially hold, as before. Suppose now that γ is not radial, so that L = 0, where L is defined by (3.12). Recall (see (3.9)) that If p v (1) ≥ p u (1) then it follows that p v (s) ≥ p u (s) for all s ≥ 1 and, by the latter of (3.8), r(s) is nondecreasing in s and the geodesic remains away from the centre. If p v (1) < p u (1) then, since ǫ 0 r −3 dr = ∞ for all ǫ > 0, it follows that there exists a time s * ≥ 1 such that p v (s * ) = p u (s * ), and moreover r(s) > 0 for all 1 ≤ s ≤ s * . As above r(s) is then non-decreasing for s ≥ s * and so the geodesic again remains away from the centre. Using now the mass shell relation (3.3) and the equations (3.6)-(3.7), d ds and so (defining s * = 1 if p v (1) ≥ p u (1) and integrating from s = s * ), Consider the latter of (3.10). It follows from (3.7) and (3.8) that (3.14) The lower bound follows from the sign of the right hand side of (3.14). For the upper bound, (3.13) guarantees that, if v 0 is sufficiently large (and hence s 0 ≥ s * is sufficiently large), then p v (s 0 ) ≥ 2p u (s 0 ). Equation (3.9) implies that and so equation (3.14) implies that and the upper bound follows. The estimate for p u (s) in (3.10) finally follows from returning to (3.13). Consider now (3.11). The existence of U 0 follows trivially from the fact thaṫ The existence of U 1 follows from the fact that, by (3.10) and (3.15).
The proof of Theorem 3.2 can now be given.

Remark 3.4 (Massless Vlasov equation on Minkowski space). Consider Minkowski space
with respect to which the metric takes the form the components of the energy momentum tensor (1.5) for solutions of the massless Vlasov equation arising from compactly supported initial data decay with the rates for t ≥ 1 [35]. Compare with the rates (3.4) which, with respect to the double null frame e 3 = t − 1 2 ∂ u , e 4 = t − 1 2 ∂ v normalised so that g • (e 3 , e 4 ) = −2, take the form

Cauchy stability and the domain of dependence property
Recall the setting of Theorem 4.1. Let (M, g) denote the unique maximal development of (Σ 1 , g, K, f 1 ), and let (Q, Ω 2 , R, f ) denote the associated maximal solution of the reduced system (2.13)-(2.19) (see Section 2.3), in which the residual gauge freedom, discussed in Remark 2.1, is normalised so that so that, in particular, r ≥ δ 0 and t ≥ 1 in U δ0,L0,v0 . By the domain of dependence property, in the region {u ≤ U 0 }, for some appropriate U 0 < 0, the solution coincides with the FLRW solution (1.7)-(1.8). In particular, and let δ 0 > 0 be such that r(U 1 , v 0 ) = v 0 − U 1 > δ 0 . By Cauchy stability (for both the Einstein-massless Vlasov system (1.2)-(1.5) and the geodesic equations) the solution exists up to {t = t 0 } and, by Proposition 3.3, for any future-maximal null geodesic γ : if v 0 is sufficiently large and ε 0 > 0 is sufficiently small. Moreover The remainder of the proof will then be concerned with the region {v ≥ v 0 } ∩ {u ≥ U 0 }.

The renormalised equations
Rather than considering the equations (2.13)-(2.16) directly, in the proof of Theorem 4.1 it is more convenient to consider the following renormalisation of the equations with their respective FLRW quantities, where the solution is compared with the FLRW solution using the double null gauge of Section 3.1, as discussed at the end of Section 3.1. For any null geodesic γ : [1, S) → M, write the tangent vector to γ aṡ The geodesic equations for p u (s) and p v (s) take the forṁ The following renormalisation of (5.8), will also be used. Equation (5.9) is obtained from (5.8) using the mass shell relation (2.17) and the fact thaṫ t = t 1 2 (p v + p u ).

The bootstrap theorem
The main content of the proof of Theorem 4.1 is contained in the following bootstrap theorem. Given T > t 0 and U 1 > 0, define the region Theorem 5.1 (Bootstrap theorem). Suppose that T > t 0 is such that the maximal development exists up to time T -in the sense that (u(t, r), v(t, r)) ∈ Q for all t 0 ≤ t < T , 11) and, for some fixed ε > 0 and U 1 > 0, the solution moreover satisfies, for all (u, v) ∈ Q U1,T , Then, if U 1 is suitably large and ε is suitably small, there exists a constant C > 0 (independent of ε and U 1 ) such that the inequalities (5.12)-(5.14) hold in Q U1,T with ε replaced by Cε 0 and (5.11) holds with U 1 replaced by U 1 /2. Moreover, the solution is isometric to the spatially homogeneous FLRW solution (1.7)-(1.8) in the Proof. The proof is divided into several steps. First, the size of the support of f − f • is estimated. These estimates are then used to obtain estimates on the components of the energy momentum tensor. The metric quantities are then estimated in the region {U 0 ≤ u ≤ U 1 }. Finally, the solution is shown to be isometric to the spatially homogeneous FLRW solution (1.7)-(1.8) in the region {u ≥ U 1 }. Throughout the proof the notation A B is used when there exists a constant K, which may depend on U 1 − U 0 , such that A ≤ KB. Constants 0 < c ≪ 1 ≪ C are always independent of U 1 − U 0 . Estimates for the support of f − f • : Let δ 0 , L 0 and v 0 be as in Section 5.1, and recall that, for each null geodesic γ : [1, S) → M emanating from supp(f 1 ), if S is sufficiently large there exists a time s 0 ∈ [1, S) such that (γ(s 0 ),γ(s 0 )) ∈ U δ0,L0,v0 .
For such a null geodesic, writeγ By the bootstrap assumption (5.11), the u coordinate of γ satisfies for all s 0 ≤ s < S. Recall moreover that v 0 > U 1 and so R(v 0 , U 1 ) > 0. It follows in particular that There exist constants 0 < c ≪ 1 ≪ C such that the components of the tangent vectorγ of any such null geodesic γ : [1, S) → M emanating from supp(f 1 ) satisfy for all s ≥ s 0 . Indeed, let γ : [s 0 , S) → M be a null geodesic such that (γ(s 0 ),γ(s 0 )) ∈ U δ0,L0,v0 . The first of (5.16) follows from the conservation of angular momentum (2.20), which implies that along with the fact that t 2 r 4 ≤ 2R 4 , which follows from the bootstrap assumption (5.14). It follows from the geodesic equation (5.9) that In order to obtain the lower bound of the latter of (5.16), consider the set The set E is manifestly a closed, connected, non-empty subset of [s 0 , S). Moreover, if s ∈ E, then by the fact that along with the former and latter of (5.16). It in particular follows from (5.16) that Consider now the support property (5.11). For any null geodesic γ : [s 0 , S) → M as above, it follows from (5.16) that, Thus, since t(s 0 ), v 0 ≥ 1, supp(π(f − f • )) ⊂ {U 0 ≤ u ≤ C}, (5.22) and so (5.11) in fact holds with U 1 replaced by U 1 /2, provided that U 1 ≥ 2C.
Estimates for the energy momentum tensor components: Recall the expressions (2.10)-(2.12) for the components of the energy momentum tensor, along with the expression (3.2) for the components of the energy momentum tensor of f • . By (5.22), for any (u, v, p v , L), It follows from the support property (5.21) that Noting that Ω 2 R −2 p u = Ω 2 • R −2 •p u , it similarly follows that

(5.26)
Estimates for the metric quantities in the region u ≤ U 1 : For the metric quantities, consider the region U 0 ≤ u ≤ U 1 and recall that Let u and v be such that U 0 ≤ u ≤ U 1 , v ≥ v 0 and t 0 ≤ t(u, v) < T . Note that Equation (5.3) and the estimate (5.25) then imply that Thus, integrating and using the boundary condition (5.1), , it follows from equation (5.5) and the estimate (5.25) and the boundary condition (5.1) that and so (5.28) and the Grönwall inequality imply that Returning to (5.27) and (5.28), one then has and Integrating in u and v, using the boundary condition (5.1) and (5.2), and inserting (5.29)-(5.31), it follows that log Ω 2 Ω 2 Now since, for ε suitably small, it follows from the Grönwall inequality, after integrating (5.33) once more in u, that Returning to (5.29)-(5.31), it then follows that