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 $R3$. 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

*g*

_{Σ}) is a constant curvature manifold, and

*a*:

*I*→ (0, ∞) is an appropriate

*scale factor*. See Sec. 5.3 of Ref. 1 for more on FLRW spacetimes. This article concerns radiation filled FLRW cosmologies in which the constant curvature manifold is Euclidean space, $(\Sigma ,g\Sigma )=(R3,gEucl)$, with radiation described by spatially homogenous solutions of the massless Vlasov equation, and their stability properties.

### A. The Einstein–massless Vlasov system

*mass shell*of $(M,g)$. Consider some local coordinates {

*t*,

*x*

^{1},

*x*

^{2},

*x*

^{3}} on $M$, where

*t*is a time function, and let {

*t*,

*x*

^{i},

*p*

^{μ}} denote the corresponding conjugate coordinate system for $TM$, i.e., (

*t*,

*x*

^{i},

*p*

^{μ}) describes the point $p\mu \u2202x\mu |(t,x)\u2208TM$. The massless Vlasov equation on $(M,g)$, for a particle density

*f*:

*P*→ [0, ∞), takes the form

*p*

^{0}is defined by the mass shell relation

*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, 3, and lower case Latin indices, such as

*i*,

*j*,

*k*range over 1, 2, 3.

### 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).

*μ*: [0, ∞) → [0, ∞) such that

*μ*≢ 0, the metric

*g*

_{◦}and particle density

*f*

_{◦}defined by

(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.*

*The spatially homogeneous FLRW family of solutions*(1.7)

*and*(1.8)

*lie inside a more general family of anisotropic spatially homogeneous FLRW solutions. Given any smooth, sufficiently decaying function*$F:R3\u2192[0,\u221e)$

*such that*

*F*≢ 0

*and*

*the metric and particle density*

*where*

*also defines a solution of*(1.2)–(1.5)

*on*(1.6)

*. Since these solutions are, in general, not spherically symmetric (see Sec. II A*)

*, consideration here is restricted to the solutions of the form*(1.7)

*and*(1.8).

($T3$ 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* $(0,\u221e)\xd7T3$*. This article will only be concerned with solutions on* $(0,\u221e)\xd7R3$*. (See also Remark 3.5 for a further comment on the solutions on* $(0,\u221e)\xd7T3$*.)*

The spacetimes $(M\u25e6,g\u25e6)$ expand from a spacelike singularity at *t* = 0 (with Kretchmann scalar $|Rm\u25e6|g\u25e62=32t4$) 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.

### 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* Σ_{1} = {*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* $(M\u25e6,g\u25e6)$*. 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} *

**.**

*The metric*

*g*

_{◦}

*, defined by the former of (1.7) with*$a(t)=t12$

*, also arises as a solution of the Einstein–Euler system, i.e., Eq. (1.4) with*

*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*

*The fluid variables for the FLRW metric take the form, in Cartesian coordinates,*

*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* $(M\u25e6,g\u25e6)$ *(see Chap. 9 of Ref. 6).*

*Note also the linear Jeans instability for the Einstein–Euler system linearised around FLRW ^{7} (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 works^{13–17} 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 works^{27–29} on Landau damping, and also the celebrated works^{30–32} on the torus $T3$, 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

*spherically symmetric*if

*SO*(3) acts by isometry on $(M,g)$ and preserves

*f*. More precisely, it is assumed that there is a smooth isometric action $O:M\xd7SO(3)\u2192M$ on $(M,g)$ such that, for each $p\u2208M$, 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 $\Phi i:(0,2\pi )\xd7M\u2192M$ satisfy

### B. Double null gauge

A double null gauge consists of functions $u,v:Q\u2192R$ 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 $M$.

*g*can be written in double null form

*γ*is the unit round metric on

*S*

^{2}, where Ω is a function on $Q$ and $R:Q\u2192R$ is the area radius function

*R*extends regularly to 0 on the centre Γ. Define

*There is residual double null gauge freedom present in (2.2). If*$u\u0303,v\u0303:I\u2192R$

*are increasing functions, with*$I\u2282R$

*a suitable interval, then the change*$u\u21a6u\u0303(u)$

*,*$v\u21a6v\u0303(v)$

*preserves the double null form (2.2). Under such a change, the metric takes the form*

*It thus follows that*

*A*and

*B*range over 1 and 2. The nonvanishing Christoffel symbols of

*g*are

*S*

^{2},

*γ*).

*u*,

*v*,

*θ*

^{1},

*θ*

^{2}) for $(M,g)$, one defines a corresponding conjugate coordinate system (

*u*,

*v*,

*θ*

^{1},

*θ*

^{2},

*p*

^{u},

*p*

^{v},

*p*

^{1},

*p*

^{2}) for $TM$, whereby (

*u*,

*v*,

*θ*

^{1},

*θ*

^{2},

*p*

^{u},

*p*

^{v},

*p*

^{1},

*p*

^{2}) defines the point

*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

*L*:

*P*→ [0, ∞) is defined by

*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 $f:Q\xd7[0,\u221e)\xd7[0,\u221e)\u2192[0,\u221e)$ — of

*u*,

*v*,

*p*

^{v}, and $L=(R4\gamma ABpApB)12$,

*T*on $M$ takes the form

*p*

^{u}=

*p*

^{u}(

*u*,

*v*,

*p*

^{v},

*L*) defined by (2.7). It moreover follows from the mass shell relation (2.7) that

### C. The spherically symmetric Einstein–massless Vlasov system in double null gauge

*R*,

*f*),

^{2}

*T*

^{uv}=

*R*

^{2}

*γ*

_{AB}

*T*

^{AB}, or 4Ω

^{−2}

*T*

_{uv}=

*R*

^{−2}

*γ*

^{AB}

*T*

_{AB}, and so Eq. (2.14) can be rewritten

*L*:

*P*→ [0, ∞) defined by (2.8) is preserved by the geodesic flow

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 $(M,g,f)$ of (2.13)–(2.19) defines a solution $(Q,\Omega 2,R,f)$ of the reduced system (2.13)–(2.19) and, conversely, any solution $(Q,\Omega 2,R,f)$ of the reduced system (2.13)–(2.19) defines a solution of (1.2)–(1.5) with $M=Q\xd7S2$ 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

*K*on Σ, and a function

*f*

_{1}:

*T*Σ → [0, ∞), satisfying constraint equations

*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 Ref. 9 for more on the Cauchy problem for the Einstein–Vlasov system.

(Local well posedness of the Cauchy problem for the Einstein–massless Vlasov system^{35,36})**.** *For any smooth initial data set* $(\Sigma ,g\u0304,K,f1)$ *for the Einstein–massless Vlasov system, as above, there exists a unique maximal development* $(M,g,f)$ *solving the system* (1.2)–(1.5)*. Moreover, if* $(\Sigma ,g\u0304,K,f1)$ *is spherically symmetric then the maximal development* $(M,g,f)$ *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 $(M,g,f)$ is the maximal development of a smooth spherically symmetric initial data set $(\Sigma ,g\u0304,K,f1)$ 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 Sec. II A above.

*non-central points*, for the spherically symmetric Einstein–massless Vlasov system

^{4})

**.**

*Let*$Q$

*be as above. Let*(

*u*

_{*},

*v*

_{*})

*be such that there exists*

*U*<

*u*

_{*}

*and*

*V*<

*v*

_{*}

*such that the characteristic diamond*$DU,Vu*,v*={U\u2264u<u*,V\u2264v<v*}$

*is contained in*$Q$

*,*

*Assume also that there exists*0 <

*R*

_{0}<

*R*

_{1}

*such that*

*and that*

*f*(

*u*,

*v*, ⋅, ⋅)

*is compactly supported for all*$(u,v)\u2208DU,Vu*,v*$

*. Then*$(u*,v*)\u2208Q$.

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

*t*∈ (0, ∞), define

*t*≥ 0 and

*r*≥ 0 in $M\u25e6$,

*g*

_{◦}in the above double null gauge takes the form

*g*

_{◦}can be written

(Anti-trapped spheres). *Note that the FLRW spacetime* $(M\u25e6,g\u25e6)$ *contains anti-trapped spheres of symmetry: if* $(u,v)\u2208Q\u25e6$ *is such that* *u* < 0*, then* *λ*_{◦}(*u*, *v*) > 0 *and* *ν*_{◦}(*u*, *v*) > 0.

*S*

^{2},

*γ*).

*f*

_{◦}satisfy

*p*

^{u}, and define

*p*

^{u}.

*u*,

*v*) coordinates for $Q$, and the values of the coordinates (

*u*,

*v*,

*p*

^{v},

*L*) for $Q\xd7[0,\u221e)\xd7[0,\u221e)$, along with the above double null gauge for the spatially homogeneous FLRW solutions. For example

*f*=

*f*

_{◦}, then

*u*and

*v*arises from the choice of parameterising the mass shell by

*p*

^{v}, rather than

*p*

^{u}.

### 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.

**.**

*Let*

*f*

*be a solution of the massless Vlasov equation*(1.2)

*on*$((0,\u221e)\xd7R3,g\u25e6)$

*, where*

*g*

_{◦}

*is the FLRW metric (3.1), such that*

*f*

_{1}=

*f*|

_{{t=1}}

*is compactly supported. The components of the energy momentum tensor satisfy*

*for*

*t*≥ 1

*. Moreover, there exist*

*U*

_{0}≤

*U*

_{1}

*such that*

*where*$\pi :P\u2192M$

*denotes the natural*

*projection.*

The support of such solutions are depicted in Fig. 1.

*γ*as

*p*

^{u}(

*s*) and

*p*

^{v}(

*s*) take the form

*t*(

*s*) =

*t*(

*γ*(

*s*)) and

*r*(

*s*) =

*r*(

*γ*(

*s*)) along the geodesic

*Let*$B\u2282P|\Sigma 1$

*be a compact subset of the mass shell over*Σ

_{1}

*and let*$\gamma :[1,\u221e)\u2192M$

*be a future directed null geodesic such that*$(\gamma (1),\gamma \u0307(1))\u2208B$

*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*

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 $B$.

*r*(

*γ*(1)) ≠ 0 (otherwise replace 1 in the following with 1 +

*ϵ*for some

*ϵ*> 0). By the conservation of angular momentum (2.20), one has the conservation law

*γ*is radial, so that the conserved quantity

*L*(

*s*) vanishes. It follows from the mass shell relation (3.3) that either

*p*

^{u}(1) = 0 or

*p*

^{v}(1) = 0. The geodesic Eqs. (3.6) and (3.7) take the form

*t*(

*s*)

*p*

^{u}(

*s*) =

*p*

^{u}(1) and

*t*(

*s*)

*p*

^{v}(

*s*) =

*p*

^{v}(1). If

*p*

^{u}(1) = 0 then, by (3.8), $r\u0307(s)>0$ for all

*s*≥ 1 and so the solution remains away from the centre. The bounds (3.10) then trivially hold. If

*p*

^{v}(1) = 0 then Eq. (3.8) imply

*s*

_{*}≥ 1 at which the geodesic hits the centre, i.e., a time at which

*r*(

*s*

_{*}) = 0. The geodesic then becomes outgoing, with

*γ*is not radial, so that

*L*≠ 0, where

*L*is defined by (3.12). Recall [see (3.9)] that

*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 non-decreasing in

*s*and the geodesic remains away from the centre. If

*p*

^{v}(1) <

*p*

^{u}(1) then, since $\u222b0\u03f5r\u22123dr=\u221e$ for all

*ϵ*> 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 Eqs. (3.6) and (3.7),

*s*

_{*}= 1 if

*p*

^{v}(1) ≥

*p*

^{u}(1) and integrating from

*s*=

*s*

_{*}),

*v*

_{0}is sufficiently large (and hence

*s*

_{0}≥

*s*

_{*}is sufficiently large), then

*p*

^{v}(

*s*

_{0}) ≥ 2

*p*

^{u}(

*s*

_{0}). Equation (3.9) implies that

*p*

^{u}(

*s*) in (3.10) finally follows from returning to (3.13).

The proof of Theorem 3.2 can now be given.

*v*

_{0}be sufficiently large so that

*R*

_{◦}(

*U*

_{1},

*v*

_{0}) > 0. The inclusion (3.5) in particular implies that,

**.**

*Consider Minkowski space*$(R3+1,m)$

*, where*$m=\u2212dt2+(dx1)2+(dx2)2+(dx3)2$

*. In standard double null coordinates*

*u*=

*t*−

*r*

*,*

*v*=

*t*+

*r*

*,*$r=(x1)2+(x2)2+(x3)2$

*, 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

*.*

^{25}

*Compare with the rates (3.4) which, with respect to the double null frame*$e3=t\u221212\u2202u$

*,*$e4=t\u221212\u2202v$

*normalised so that*

*g*

_{◦}(

*e*

_{3},

*e*

_{4}) = −2

*, take the form*

**.**

*The FLRW metric*

*g*

_{◦}

*, along with the function*

*f*

_{◦}

*, defined by (1.7) and (1.8), also define a solution of the Einstein–massless Vlasov system on the manifold*$(0,\u221e)\xd7T3$

*. The components of the energy momentum tensor (1.5) for solutions of the massless Vlasov equation arising from compactly supported initial data on such a background decay with the rates*

*for all*

*t*

*, for a suitable norm*‖ ⋅‖

*. Compare again with the rates*(3.4).

## IV. THE MAIN THEOREM

*k*suitably large.

**.**

*Let*

*μ*: [0, ∞) → [0, ∞)

*be a smooth function, decaying suitably so that*$\u222bR3|p|\mu (|p|2)dp$

*is finite, such that*

*μ*≢ 0

*. Let*

*g*

_{◦}

*and*

*f*

_{◦}

*be defined by (1.7), (1.8). Let*$(\Sigma 1,g\u0304,K,f1)$

*be a spherically symmetric initial data set for the Einstein–massless Vlasov system on*$\Sigma 1={t=1}\u2282M\u25e6$

*such that*$f1\u2212f\u25e6|\Sigma 1$

*is compactly supported on*$P|\Sigma 1$

*and*$g\u0304\u2212g\u25e6|\Sigma 1$

*and*$K\u221212\u2202tg\u25e6|\Sigma 1$

*are compactly supported on*Σ

_{1}

*. There exists*

*ɛ*

_{0}> 0

*such that, if*

*then the resulting maximal developement of Theorem 2.2 is future geodesically complete and there exists*

*U*

_{0}< 0 <

*U*

_{1}

*such that, in an appropriately normalised double null gauge, the associated*$(Q,\Omega 2,R,f)$

*satisfies the estimates*

*in the region*{

*U*

_{0}<

*u*<

*U*

_{1}}

*, where*$t=14(v+u)2$

*and*

*r*=

*v*−

*u*

*. Moreover,*

*where*$\pi :P\u2192M$

*is the natural projection, and the solution is isometric to the spatially homogeneous FLRW solution (1.7) and (1.8) in the region*{

*u*≥

*U*

_{1}}.

Note that, since the estimates (4.2) imply that Ω^{4}*R*^{2} ∼ *t*^{4}, Ω^{2}*R*^{4} ∼ *t*^{5} and *R*^{6} ∼ *t*^{6} in supp(*f* − *f*_{◦}) ∩ {*v* ≥ *v*_{0}}, 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 $(M,g)$ denote the unique maximal development of $(\Sigma 1,g\u0304,K,f1)$, and let $(Q,\Omega 2,R,f)$ 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}.

*v*

_{0}>

*U*

_{1}, where

*U*

_{1}> 0 is to be determined later. Given

*δ*

_{0}> 0,

*L*

_{0}≥ 0, define the set

*r*≥

*δ*

_{0}and

*t*≥ 1 in $U\delta 0,L0,v0$.

*L*

_{0}> 0 be such that

*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 $\gamma :[1,S)\u2192M$ such that $(\gamma (1),\gamma \u0307(1))\u2208supp(f1)$, there exists

*s*

_{0}∈ [1, ∞) such that

*v*

_{0}is sufficiently large and

*ɛ*

_{0}> 0 is sufficiently small. Moreover

The remainder of the proof will then be concerned with the region {*v* ≥ *v*_{0}} ∩ {*u* ≥ *U*_{0}}.

### B. The renormalised equations

*γ*as

*p*

^{u}(

*s*) and

*p*

^{v}(

*s*) take the form

### C. The bootstrap theorem

The main content of the Proof of Theorem 4.1 is contained in the following bootstrap theorem.

*T*>

*t*

_{0}and

*U*

_{1}> 0, define the region

**.**

*Suppose that*

*T*>

*t*

_{0}

*is such that the maximal development exists up to time*

*T*

*— in the sense that*$(u(t,r),v(t,r))\u2208Q$

*for all*

*t*

_{0}≤

*t*<

*T*

*,*

*r*≥ 0

*— the difference*

*f*−

*f*

_{◦}

*is supported in the region*

*and, for some fixed*

*ɛ*> 0

*and*

*U*

_{1}> 0

*, the solution moreover satisfies, for all*$(u,v)\u2208QU1,T$

*,*

*Then, if*

*U*

_{1}

*is suitably large and*

*ɛ*

*is suitably small, there exists a constant*

*C*> 0

*(independent of*

*ɛ*

*and*

*U*

_{1}

*) such that the inequalities*(5.12)–(5.14)

*hold in*$QU1,T$

*with*

*ɛ*

*replaced by*

*Cɛ*

_{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) and (1.8) in the region*{

*t*

_{0}≤

*t*(

*u*,

*v*) <

*T*} ∩ {

*u*≥

*U*

_{1}}.

*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) and (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 Sec. V A, and recall that, for each null geodesic $\gamma :[1,S)\u2192M$ emanating from supp(*f*_{1}), if *S* is sufficiently large there exists a time *s*_{0} ∈ [1, *S*) such that $(\gamma (s0),\gamma \u0307(s0))\u2208U\delta 0,L0,v0$.

*u*coordinate of

*γ*satisfies

*s*

_{0}≤

*s*<

*S*. Recall moreover that

*v*

_{0}>

*U*

_{1}and so

*R*(

*v*

_{0},

*U*

_{1}) > 0. It follows in particular that

*c*≪ 1 ≪

*C*such that the components of the tangent vector $\gamma \u0307$ of any such null geodesic $\gamma :[1,S)\u2192M$ emanating from supp(

*f*

_{1}) satisfy

*s*≥

*s*

_{0}. Indeed, let $\gamma :[s0,S)\u2192M$ be a null geodesic such that $(\gamma (s0),\gamma \u0307(s0))\u2208U\delta 0,L0,v0$. The first of (5.16) follows from the conservation of angular momentum (2.20), which implies that

*t*

^{2}

*r*

^{4}≤ 2

*R*

^{4}, which follows from the bootstrap assumption (5.14).

*s*

_{0},

*S*). Moreover, if $s\u2208E$, then

*ɛ*is sufficiently small, the set $E\u2282[s0,S)$ is open, and hence equal to [

*s*

_{0},

*S*).

*tp*

^{v})(

*s*). The upper bound for (

*tp*

^{v})(

*s*) then follows from the estimate (5.19), after integrating Eq. (5.18).

*t*(

*s*

_{0}),

*v*

_{0}≥ 1,

*U*

_{1}replaced by

*U*

_{1}/2, provided that

*U*

_{1}≥ 2

*C*.

**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*),

**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

*u*and

*v*be such that

*U*

_{0}≤

*u*≤

*U*

_{1},

*v*≥

*v*

_{0}and

*t*

_{0}≤

*t*(

*u*,

*v*) <

*T*.

*u*and

*v*, using the boundary condition (5.1) and (5.2), and inserting (5.29)–(5.31), it follows that

*ɛ*suitably small,

*u*, that

**The region**

*u*≥*U*_{1}

**:**In order to see that the solution is isometric to the spatially homogeneous FLRW solution (1.7) and (1.8) in the region

*u*≥

*U*

_{1}, recall first (5.23), from which it follows that

*V*

_{1}is such that

*R*(

*U*

_{1},

*V*

_{1}) = 0. Recall the residual gauge freedom of Remark 2.1 and consider the following change of gauge. Define increasing functions $v\u0303(v)$ and $u\u0303(u)$ as solutions of

*u*=

*U*

_{1}, takes the form

*R*=

*R*

_{◦}and $\Omega 2=\Omega \u25e62$ for all

*u*≥

*U*

_{1}. □

### D. The conclusion of the proof of the main theorem

Recall the set $QU1,T\u2282Q$ defined by (5.10). Consider the set $A\u2282[t0,\u221e)$ of times *T* ∈ [*t*_{0}, ∞) such that the solution exists up to time *T* — in the sense that $(u(t,r),v(t,r))\u2208Q$ for all *t*_{0} ≤ *t* < *T*, *r* ≥ 0 — and moreover satisfies the estimates (5.12)–(5.14) for all $(u,v)\u2208QU1,T$ for some *ɛ* > 0 sufficiently small so that the conclusion of Theorem 5.1 holds. The set $A\u2282[t0,\u221e)$ is manifestly connected and open, and is moreover non-empty by local existence and Cauchy stability, provided *ɛ*_{0} is sufficiently small. By Theorem 2.3, the bounds (5.13)–(5.14) imply that $(u(T,r),v(T,r))\u2208Q$ for all *r* ≥ 0 such that *u*(*T*, *r*) ≤ *U*_{1}. By Theorem 5.1, the solution is isometric to the spatially homogeneous FLRW solution (1.7) and (1.8) in the region {*t*_{0} ≤ *t*(*u*, *v*) < *T*}∩{*u* ≥ *U*_{1}} and thus $(u(T,r),v(T,r))\u2208Q$ for all *r* ≥ 0. Theorem 5.1 moreover guarantees that the bounds (5.12)–(5.14) cannot be saturated, and thus the set $A$ is moreover closed, and hence equal to [*t*_{0}, ∞).

## 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.

## REFERENCES

*The Large Scale Structure of Space-Time*

*The Early Universe: Facts and Fiction*

*On the Topology and Future Stability of the Universe*