Spatially homogeneous Friedmann–Lemaître–Robertson–Walker (FLRW) solutions constitute an infinite dimensional family of explicit solutions of the Einstein–massless Vlasov system with vanishing cosmological constant. Each member expands toward 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 . 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.
I. INTRODUCTION
A. The Einstein–massless Vlasov system
B. 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).
(Decelerated expansion). Spacetimes with metric of the form (1.1) are said to undergo accelerated expansion if the scale factor a(t) satisfies a″(t) > 0 for all t, linear expansion if a″(t) = 0 for all t, or decelerated expansion if a″(t) < 0 for all t. Note that the solutions (1.7) and (1.8) undergo decelerated expansion. Previous stability works in cosmological settings have typically considered perturbations of spacetimes undergoing linear or accelerated expansion. See Sec. I D below.
( spatial topology). Each (1.7) and (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 . This article will only be concerned with solutions on . (See also Remark 3.5 for a further comment on the solutions on .)
The spacetimes expand from a spacelike singularity at t = 0 (with Kretchmann scalar ) and are future geodesically complete. See the Penrose diagram of Fig. 1. Such solutions feature in the Ehlers–Geren–Sachs Theorem,2 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.
Penrose diagram of the FLRW spacetime. Solutions of the massless Vlasov equation arising from compactly supported initial data on {t = 1} are supported in the darker shaded region. The solutions, of the coupled system, of Theorem 1.4 admit a similar Penrose diagram, where now the perturbation is supported in the darker shaded region.
Penrose diagram of the FLRW spacetime. Solutions of the massless Vlasov equation arising from compactly supported initial data on {t = 1} are supported in the darker shaded region. The solutions, of the coupled system, of Theorem 1.4 admit a similar Penrose diagram, where now the perturbation is supported in the darker shaded region.
C. First version of the main theorem
(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) and (1.8) for some smooth, non-identically vanishing μ (decaying suitably so that ϱ is defined). For all compactly supported spherically symmetric initial data sufficiently close to that of (1.7) and (1.8) on Σ1 = {t = 1}, the resulting solution is future geodesically complete, is isometric to (1.7) and (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 Sec. IV. See also the Penrose diagram in Fig. 1.
The proof of Theorem 1.4 is based on an understanding of decay properties of solutions of the massless Vlasov equation. As such, in addition to Theorem 1.4, decay properties of solutions of the massless Vlasov equation on a fixed FLRW background of the form (1.7) and (1.8) are given. See Theorem 3.2.
The assumption of compact support can be relaxed, and is made for convenience in order to localise the proof. Indeed, following,3 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 Ref. 4 or Theorem 2.3 below).
(Asymptotic stability of FLRW). Theorem 1.4, from the spacetime (as opposed to mass shell) perspective, can be viewed as the asymptotic stability of the FLRW solution (1.7) and (1.8) to spherically symmetric perturbations. There is residual double null gauge freedom (see Remark 2.1) which can be appropriately renormalised so that each metric component, Christoffel symbol and energy momentum tensor component is equal to its (possibly decaying) FLRW value, plus a remainder which decays strictly faster than its corresponding FLRW value. Such a renormalisation is not required in the Proof of Theorem 1.4 however, and thus is not further considered here.
(Comparison with decay rates on Minkowski space). The decay rates of solutions of the massless Vlasov equation on FLRW, obtained in Theorem 3.2, are later compared with those of the massless Vlasov equation on Minkowski space. When measured with respect to their respective t coordinates, certain components of the energy momentum tensor, with respect to an appropriately normalised double null frame, decay faster in Minkowski space, while others decay faster in FLRW. See Remark 3.4. Note also that solutions of the linear wave equation with localised initial data decay at the same rate on each of these two spacetimes.5
(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 . 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.2
It is known that the solution (1.10) is unstable to formation of shock waves for the Euler equations ∇μTμν = 0 on the fixed FLRW background (see Chap. 9 of Ref. 6).
Note also the linear Jeans instability for the Einstein–Euler system linearised around FLRW7 (see also the discussion in Ref. 8).
D. Previous works
There have been previous works on the Einstein–Vlasov system in cosmological settings for Einstein equations with a positive cosmological constant,9 and also for perturbations of the vacuum Milne solution.10–12 Note also related works13–17 on the Einstein–Euler system. See also Refs. 18–21. 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) and (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,3,22 and later to general perturbations.23–25 The present work is based on the approach of Ref. 3. 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.26
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 works27–29 on Landau damping, and also the celebrated works30–32 on the torus , 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 Ref. 33.
E. Outline of the paper
Section II contains preliminaries on the Einstein–massless Vlasov system in double null gauge. Section III concerns further properties of the FLRW spacetimes, introduced in Sec. I B. A double null gauge in FLRW is defined in Sec. III A and, as a precursor to the proof of Theorem 1.4, properties of the massless Vlasov equation (1.2) and (1.3) on the FLRW background spacetimes are presented in Sec. III B. In Sec. IV A more precise version of Theorem 1.4 is formulated, and in Sec. V its proof is given.
II. THE SPHERICALLY SYMMETRIC EINSTEIN–MASSLESS VLASOV SYSTEM
This section concerns facts about the Einstein–massless Vlasov system in spherical symmetry. In Sec. II A the spherical symmetry assumption is introduced. In Sec. II B spherically symmetric double null gauges are introduced, and the residual spherically symmetric double null freedom is discussed. In Sec. II C the Einstein–massless Vlasov system in a spherically symmetric double null gauge is presented. In Sec. II D functions t and r are introduced. Finally, in Sec. II E the Cauchy problem is discussed and an extension principle, which will be used in the Proof of Theorem 1.4, is stated.
A. Spherical symmetry
B. Double null gauge
A double null gauge consists of functions such that the level hypersurfaces of u foliate by outgoing lines which are null with respect to the induced metric on , and the level hypersurfaces of v foliate by ingoing null lines. The level hypersurfaces of u and v lift to outgoing and incoming null cones of and such a double null gauge can be complemented with local coordinates (θ1, θ2) on S2 to local coordinates (u, v, θ1, θ2) for .
C. The spherically symmetric Einstein–massless Vlasov system in double null gauge
The system (2.13)–(2.19) is equivalent to the Einstein–massless Vlasov system (1.2)–(1.5) in the sense that any spherically symmetric solution of (2.13)–(2.19) defines a solution of the reduced system (2.13)–(2.19) and, conversely, any solution of the reduced system (2.13)–(2.19) defines a solution of (1.2)–(1.5) with and g defined by (2.2).
See Ref. 34 for more on the spherically symmetric Einstein–massless Vlasov system.
D. The functions t and r
E. The Cauchy problem
(Local well posedness of the Cauchy problem for the Einstein–massless Vlasov system35,36). For any smooth initial data set for the Einstein–massless Vlasov system, as above, there exists a unique maximal development solving the system (1.2)–(1.5). Moreover, if is spherically symmetric then the maximal development is spherically symmetric.
See also Ref. 34 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 is the maximal development of a smooth spherically symmetric initial data set for the Einstein–massless Vlasov system (1.2)–(1.5), and that is the quotient manifold of by the action of the SO(3) isometry, as in Sec. II A above.
For a proof of Theorem 2.3 see Sec. IV C of Ref. 4 [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 Ref. 37 in view of the presence of the anti-trapped surfaces in the spacetimes under consideration (see Remark 3.1 below).
III. THE FLRW SPACETIMES
Recall the FLRW spacetimes introduced in Sec. I B. In Sec. III A 1 double null gauge is introduced in each of these spacetimes. Section III B concerns properties of the massless Vlasov equation on these spacetimes. Though the main theorem of Sec. III B, 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.
A. The FLRW metrics in double null gauge
(Anti-trapped spheres). Note that the FLRW spacetime contains anti-trapped spheres of symmetry: if is such that u < 0, then λ◦(u, v) > 0 and ν◦(u, v) > 0.
B. 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 support of such solutions are depicted in Fig. 1.
Provided v0 is suitably large, the existence of s0 ∈ [1, ∞) such that v(γ(s0)) = v0 follows from the compactness of .
The proof of Theorem 3.2 can now be given.
IV. THE MAIN THEOREM
Note that, since the estimates (4.2) imply that Ω4R2 ∼ t4, Ω2R4 ∼ t5 and R6 ∼ t6 in supp(f − f◦) ∩ {v ≥ v0}, the rates (4.1) should be viewed as exactly the rates of Theorem 3.2.
The remainder of the paper is concerned with the Proof of Theorem 4.1.
V. THE PROOF OF THE MAIN THEOREM
This section concerns the Proof of Theorem 4.1. In Sec. V A the proof is reduced so that the main region of consideration arises from an appropriate characteristic initial value problem. In Sec. V B 1 more convenient renormalisation of the equations of Sec. II C are given. Section V C concerns Theorem 5.1, which forms the main content of the Proof of Theorem 4.1. In Sec. V D the Proof of Theorem 4.1 is completed.
A. Cauchy stability and the domain of dependence property
Recall the setting of Theorem 4.1. Let denote the unique maximal development of , and let denote the associated maximal solution of the reduced system (2.13)–(2.19) (see Sec. II C), in which the residual gauge freedom, discussed in Remark 2.1, is normalised so that λ|{t=1} = λ◦|{t=1} and ν|{t=1}∪Γ = ν◦|{t=1}∪Γ.
The remainder of the proof will then be concerned with the region {v ≥ v0} ∩ {u ≥ U0}.
B. The renormalised equations
C. The bootstrap theorem
The main content of the Proof of Theorem 4.1 is contained in the following bootstrap theorem.
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 {U0 ≤ u ≤ U1}. Finally, the solution is shown to be isometric to the spatially homogeneous FLRW solution (1.7) and (1.8) in the region {u ≥ U1}.
Throughout the proof the notation A ≲ B is used when there exists a constant K, which may depend on U1 − U0, such that A ≤ KB. Constants 0 < c ≪ 1 ≪ C are always independent of U1 − U0.
Estimates for the support of f − f◦: Let δ0, L0 and v0 be as in Sec. V A, and recall that, for each null geodesic emanating from supp(f1), if S is sufficiently large there exists a time s0 ∈ [1, S) such that .
D. The conclusion of the proof of the main theorem
Recall the set defined by (5.10). Consider the set of times T ∈ [t0, ∞) such that the solution exists up to time T — in the sense that for all t0 ≤ t < T, r ≥ 0 — and moreover satisfies the estimates (5.12)–(5.14) for all for some ɛ > 0 sufficiently small so that the conclusion of Theorem 5.1 holds. The set is manifestly connected and open, and is moreover non-empty by local existence and Cauchy stability, provided ɛ0 is sufficiently small. By Theorem 2.3, the bounds (5.13)–(5.14) imply that for all r ≥ 0 such that u(T, r) ≤ U1. By Theorem 5.1, the solution is isometric to the spatially homogeneous FLRW solution (1.7) and (1.8) in the region {t0 ≤ t(u, v) < T}∩{u ≥ U1} and thus for all r ≥ 0. Theorem 5.1 moreover guarantees that the bounds (5.12)–(5.14) cannot be saturated, and thus the set is moreover closed, and hence equal to [t0, ∞).
ACKNOWLEDGMENTS
I acknowledge support through Royal Society Tata University Research Fellowship Grant No. URF∖R1∖191409. I am grateful to G. Fournodavlos, H. Masaood, and J. Speck for helpful discussions.
AUTHOR DECLARATIONS
Conflict of Interest
The author has no conflicts to disclose.
Author Contributions
Martin Taylor: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.