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arXiv:2012.14241v1 [math-ph] 28 Dec 2020

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER

Hamed Barzegar,David Fajman

Abstract

It is shown that Milne models (a subclass of FLRW spacetimes with negative spatial curvature) are nonlin-

early stable in the set of solutions to the Einstein-Vlasov-Maxwell system, describing universes with ensembles

of collisionless self-gravitating, charged particles. The system contains various slowly decaying borderline

terms in the mutually coupled equations describing the propagation of particles and Maxwell ﬁelds. The

eﬀects of those terms are controlled using a suitable hierarchy based on the energy density of the matter

ﬁelds.

1. Introduction

Studying the global dynamics of non-vacuum solutions to the Einstein equations is a major eﬀort

in general relativity with the aim to draw conclusions on isolated self-gravitating systems and

cosmology. In recent years several results have contributed substantially to the understanding

of non-vacuum dynamics with self-gravitating matter models, which provide realistic features of

matter in the actual universe such as relativistic ﬂuids of various types and kinetic matter models.

While ﬂuids are known to require expansion of spacetime to avoid shock formation (cf. [34]) Vlasov

matter shows a more regular behavior. When coupled to the Einstein equations, it is expected

to exhibit only those types of singularity formation which are caused by gravity. Indeed, most

nonlinear stability results for the vacuum Einstein equations have recently been generalized to

the Einstein–Vlasov system, most prominently for the de Sitter type spacetimes [33], Minkowski

spacetime [13, 23, 28, 36] and the Milne mo del [1] and lower dimensional analogues [18, 20, 21, 22].

The Einstein-Vlasov system describes spacetimes containing ensembles of self-gravitating collision-

less particles and provides a realistic description of the large scale structure of spacetime. It admits

various steady states modelling isolated self-gravitating matter conﬁgurations such as galaxies and

galaxy clusters [5,6, 31] and similarly a variety of matter dominated cosmological models [8,30]. In

contrast to those mo dels, in the stability analysis of [1, 13, 23, 28, 33, 36] the matter distribution is

considered to be small and disperses in the course of the evolution while the spacetime geometry

asymptotes to the background vacuum geometry.

While the Einstein–Vlasov system models purely gravitative eﬀects there exist generalizations,

which include more detailed physical phenomena such as charged collisionless particles modeled by

the Einstein–Vlasov–Maxwell system on which we focus in the following.

1.1. The Einstein–Vlasov–Maxwell system. The Einstein–Vlasov–Maxwell system (EVMS)

(1.1)

Ric[h]−1

2R[h]·h= 8πT,

Lh,F f= 0 ,

dF= 0 ,

d⋆ F =⋆J ,

describes spacetimes containing Maxwell ﬁelds and ensembles of collisionless charged particles,

which interact via gravity and electromagnetism [14, 17]. Here, hdenotes a Lorentzian metric on a

given 4-manifold M;T=V

T+MTis the total energy-momentum tensor with V

Tand MTdenoting

Date: December 29, 2020.

1

2H. Barzegar,D. Fajman

the energy-momentum tensors of the Vlasov matter and the Maxwell ﬁeld, respectively; Lh,F is

the Liouville–Vlasov operator which consists of the geodesic spray and the Maxwell term; dand ⋆

denote the exterior derivative and the Hodge star operator on M, respectively; Fis the Faraday

tensor, and Jis the matter current 1-form.

When setting the Faraday tensor Fand the charge to zero, (1.1) reduces to the Einstein–Vlasov

system while setting the distribution function fto zero it reduces to the Einstein–Maxwell system.

There exist only few results on the EVMS concerning stationary solutions [4, 37, 38] and evolution

in spherical symmetry [29]. In particular, stability results have not been established yet in the class

of solutions to the EVMS.

1.2. Background spacetimes. We consider a class of cosmological vacuum spacetimes which are

Lorentz cones over a negative closed Einstein space (M, γ) of dimension 3 with Einstein constant

κ=−2

9, i.e., Ric[γ] = −2

9γ, where the value of κis chosen for convenience. Then, the Milne model

(1.2) (0,∞)×M, −dt2+t2

9·γ

is a solution to the vacuum Einstein equations. In contrast to the class of exponentially expand-

ing spacetimes or those with power law inﬂation, the Milne model does not exhibit accelerated

expansion. Its linear scale factor constitutes the threshold between accelerated and deccelerated

expansion, which makes the model particularly interesting from the perspective of the regularizing

eﬀect of its expansion on various matter models [25]. Future stability of (1.2) under the Einstein

ﬂow has been established in the vacuum case in [3] and for diﬀerent types of matter in [1, 15, 26].

1.3. Main result. It is the purpose of the present paper to establish the ﬁrst nonlinear stability

result for the EVMS. In particular, we prove future nonlinear stability for the Milne model in the

set of solutions to the EVMS. The main challenge in contrast to earlier stability results on the

Milne model consists in the direct mutual coupling of the matter ﬁelds, which does induce various

slowly decaying terms in the system. We discuss these aspects further below in detail. The main

theorem reads as follows.

Theorem 1.1. Let (M, γ)be a compact, 3-dimensional negative Einstein manifold without bound-

ary and Einstein constant κ=−2

9, and let ǫ>0. Then, there exists a δ > 0such that for a

rescaled initial data (g0,Σ0, f0, ω0,˙ω0)∈H6(M)×H5(M)×HVl,5,4,c(T M )×H6(M)×H5(M)at

t=t0with compact momentum support of the initial particle distribution and

(1.3) (g0,Σ0, f0, ω0,˙ω0)∈B6,5,5,6,5

δ(γ, 0,0,0,0) ,

the corresponding solution to the rescaled Einstein–Vlasov–Maxwell system is future-global in time

and future complete. Moreover, the rescaled metric and trace-free part of the second fundamental

form converge as

(1.4) (g, Σ) →(γ, 0) for τր0,

with decay rates determined by ǫas in (10.8) below. In particular, the Milne model is future

asymptotically stable for Einstein–Vlasov–Maxwell system in the class of initial data given above.

We will provide precise deﬁnitions of the objects in Theorem 1.1 further below. However, to give an

overview on the theorem, we brieﬂy summarize them. We use (g, Σ, f , ω, ˙ω) to denote the rescaled

variables: the Riemannian metric, the trace-free part of the second fundamental, the distribution

function, the spatial vector potential, and its time-derivative, respectively. τ < 0 denotes the mean

curvature and is related to the time variable in (1.2) by t=−3τ−1. Therefore, τր0 corresponds

to t→ ∞.Hk(M) with k≥0 denotes the L2-based Sobolev norm and HVl,5,4,c(T M ) denotes

the space of distribution functions of compact momentum support on T M corresponding to the

standard L2-Sobolev norms (cf. [20]). We denote by B6,5,5,6,5

δ(·,·,·,·,·) a ball of radius δcentered

at its arguments in the set of H6(M)×H5(M)×HVl,5,4,c(T M )×H6(M)×H5(M).

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 3

Remark 1.2. The main theorem is related to foregoing works and more general settings in the

following sense.

(1) Theorem 1.1 in particular implies Theorem 1 of [1] (the Einstein-Vlasov case) and Theo-

rem 7.1 of [15] (Einstein-Maxwell case).

(2) We consider the EVMS with particles of identical rest mass m > 0 and identical charge q.

The present result however directly generalizes to a collection of ensembles with diﬀerent

masses and charges (also of opposite signs). For simplicity of the presentation we restrict

ourselves to this speciﬁc case.

While there exists a number of stability results for relativistic Vlasov-Maxwell systems [10, 11, 12]

the present result is to our knowledge the ﬁrst stability result for the EVMS.

1.4. Aspects of the proof. The proof of Theorem 1.1 is partially based on the techniques devel-

oped to control the distribution function in [19,20] and [1] and the techniques to control the Maxwell

ﬁelds derived in [15]. However, the present problem poses various new diﬃculties as Maxwell ﬁelds

and distribution function are directly coupled via the Maxwell term in the transport equation and

the particle-current term in the Maxwell equation. In fact, both terms are principal terms in the

sense that they decay a priori at the slowest rate when compared with other perturbative terms

in the respective equation. Moreover, the decay rates of these terms induce a loss in decay for the

corresponding energies of the Maxwell ﬁelds and distribution function, respectively. We refer to

those terms in the following as borderline terms.

In the Einstein equations a borderline term enters in the lapse equation (2.9c) via the rescaled

pressure η. This has already been observed and resolved in [1] in conjunction with the corresponding

borderline term (∗) in the transport equation (2.25). Their mutual coupling was controlled in [1] by

using the continuity equation to obtain a sharp estimate for the energy density of the distribution

function.

In the presence of charges an additional borderline term enters the transport equation. It is caused

by the Maxwell ﬁeld and denoted by (∗∗) in (2.25). It eventually yields a small ε-growth for the

energy of the distribution function similar to the borderline term (∗) in the same equation. To

establish this behavior it is necessary that the energy controlling the Maxwell ﬁeld, which appears

in this term, is uniformly bounded in time. A small loss for that energy would prevent the estimates

from closing.

However, the Maxwell equations themselves have a borderline term marked by (∗) in (2.33). This

term is caused by the presence of charged particles. It can be controlled by the energy of the

distribution function but would in this case pick up its small growth and prevent the bootstrap

argument from closing. At this point it is important to use the fact that the respective term in the

Maxwell equations is determined by the matter current, which, in turn is determined to leading

order by the energy density, which, due to the continuity equation does not have a loss in decay in

comparison with the energy of the distribution function. This observation is the key that enables

us to close the hierarchy of estimates and thereby the bootstrap argument.

1.5. Organization of the paper. In Section 2 notations, geometric setup and energies are intro-

duced. In Section 3, bounds on the energy-momentum tensor in terms of the energies are given.

In Section 4 we provide the estimate for the momentum support. Section 5 provides the energy

estimates for the distribution function and the Maxwell ﬁelds. The energy density is estimated in

Section 6. Lapse and shift vector are estimated in Section 7. The energy estimates for the spatial

geometry are given in Section 8. In Section 9 we provide the energy estimate for the total energy

incorporating all previously derived estimates. Section 10 provides the proof of the main theorem

based on the foregoing sections.

Acknowledgements. This work was supported in part by the Austrian Science Fund (FWF) via

the project Geometric transport equations and the non-vacuum Einstein ﬂow (P 29900-N27).

4H. Barzegar,D. Fajman

2. Preliminaries

In this section we recall various notations and facts on the geometric setup from [1, 15].

2.1. Notation. In this paper, M=R×M, with Mbeing a three-dimensional compact Riemannian

manifold, denotes a four-dimensional Lorentzian manifold equipped with Lorentzian metrics hand

h, and the associated covariant derivatives e

∇and ∇, respectively. Further, Riemannian metrics

on Mwill be denoted by γ, ˜g,g, the associated Christoﬀel symbols by b

Γ[γ], e

Γ[˜g], Γ[g], and the

associated covariant derivatives by b

D,D,D, respectively. We will also denote the determinants of

a generic metric by |·|. The Laplacian of gis then deﬁned as ∆ = trgD2. Moreover, dµgwill stand

for the Riemannian measure induced on Mby g. The Riemannian inner products on a tangent

space TxMat a point xis given by h·,·igand h·,·i˜g, respectively. The Hodge-Laplacian acting on

diﬀerential forms on Mwill be denoted by ∆H=d∗d+dd∗where dand d∗denote the exterior

derivative and the codiﬀernetial with respect to the metric gon M, respectively. LYwill represent

the Lie-derivative in the direction of a vector ﬁeld Y. We occasionally will use the notations

b

N:= N/3−1 and b

X:= X/N. The Greek indices will stand for the spacetime coordinates on M

and the Latin indices will denote the coordinates on M, whereas the coordinates on the tangent

bundle T M of Mwill be denoted by the bold Latin letters a,b,c,... ∈ {1,...,6}. Furthermore,

we denote the standard (L2-based) Sobolev norm with respect to the ﬁxed metric γof order ℓ≥0

by k·kHℓ(M)for all functions and symmetric tensor ﬁelds on M. For brevity we write Hℓ≡Hℓ(M).

Throughout this paper, Cdenotes any positive constant which is uniform in the sense that it does

not depend on the solution of the system once a smallness parameter δfor the initial data and

the initial time T0are chosen. Moreover, if δis decreased or T0is increased, Cwill keep its value.

Nevertheless, the actual value of Cmay change from line to line.

2.2. Background geometry. Throughout the rest of the paper, we consider the Einstein space

(M, γ ) with Ric[γ] = −2

9γ. Then, the Einstein operator ∆E(cf. [9] and [27]) associated with

γacting on symmetric 2-tensors is deﬁned by ∆E≡ −b

∆γ−2˚

Rγ, where b

∆γ:= γij b

Dib

Djand

(˚

Rγu)ij := Riem[γ]ikjℓ ukℓ for an arbitrary symmetric 2-tensor u. The lowest positive eigenvalue of

the Einstein operator λ0obeys 9λ0≥1 in the present setting (cf. [1]), which is relevant for the

energy estimate for metric and second fundamental form. In addition, ker ∆E={0}holds in the

present setting (cf. [3] and [1]), which assures the coercivity of the same energy.

2.3. Spacetime, gauges and rescaled variables. We consider the spacetime (M,h) and write

the unrescaled Lorenztian metric hin ADM formalism as

h=−e

N2dt ⊗dt + ˜gij dxi+e

Xidt⊗dxj+e

Xjdt,

where e

Nand e

Xare the lapse function and the shift vector ﬁeld. Let e

Σ and τbe the trace-free part

and the trace of the second fundamental form of the hypersurfaces {t= const.}which we assume

that all have constant mean curvature τ.

2.3.1. CMCSH gauge. In the spacetime setting introduced above, the constant mean curvature

spatial harmonic (CMCSH) gauge (cf. [2]) is achieved by

(2.1)

t=τ ,

˜gij e

Γ[˜g]ℓ

ij −b

Γ[γ]ℓ

ij = 0 .

2.3.2. Rescaled variables. We further deﬁne rescaled quantities g,N,X, and Σ by

(2.2) gij := τ2˜gij , gij := τ−2˜gij , N := τ2e

N , Xi:= τe

Xi,Σij := τe

Σij ,

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 5

and we introduce a rescaled time Tby τ:= τ0·e−Tfor T∈(−∞,∞) and some ﬁxed τ0<0.

Note that ∂T=−τ∂τand dτ =−τ dT . We occasionally will use the dot notation to denote the

T-derivative.

2.3.3. Rescaled geometry. In the new time coordinate and with the rescaled variables the Lorentzian

metric reads

(2.3) h= (τ0)−2e2T−N2dT 2+gijdxi−XidT ⊗dxj−XjdT =: τ−2h .

Remark 2.1. The Milne solution with the choice of the time coordinate τand τ0=−3 reads

hMilne =e2T−dT 2+1

9γ.

Now, let Π be the second fundamental form of the hypersurfaces {T= const.}with respect to the

metric h. Then, it is readily seen that

(2.4) Π = −Σ + N−11−N

3g .

One therefore ﬁnds that the future-directed timelike unit normal e0of the hypersurfaces {T=

const.}with respect to his

(2.5) e0=N−1(∂T+X).

With this deﬁnition the following lemma holds.

Lemma 2.2 ( [15], Lemma 2.1).For any f∈C∞(M)and ξ∈C∞(R,Ω1(M))

Le0g=−2Π ,

[Le0,∆g]f= 2hΠ, D2fig+hDlog N, D∂e0fig+hS, Df ig

+ 2hΠ, D log N⊗Df ig−trgΠhDlog N, Df ig,

[Le0,divg]ξ= 2hΠ, Dξig+hDlog N, Le0ξig+hS, Df ig

+ 2hΠ, D log N⊗ξig−trgΠhDlog N, ξig,

where S:= 2divgΠ−DtrgΠ.

Throughout this paper, the index ˆ

0 refers to the vector ﬁeld e0whereas 0 refers to the time-function

τ. For the later use we summarize the connection coeﬃcient with respect to the Lorentzian metric

h, i.e. Γ[h]α

βγ , using the Koszul formula

(2.6) Γ[h]ˆ

0

ˆ

0ˆ

0= 0 ,

Γ[h]i

ˆ

0ˆ

0=gij N−1∂jN ,

Γ[h]ˆ

0

iˆ

0= 0 ,

Γ[h]i

jˆ

0=−gikΠkj ,

Γ[h]ˆ

0

ij =−Πij ,

Γ[h]i

jk = Γ[g]i

jk .

2.3.4. Slice-adapted gauge. On a vector potential A∈Ω1(M) the Lorenz gauge d⋆ A = 0 is

usually imposed. As discussed in [15], it turns out to be diﬃcult to work with the Lorenz gauge

in the present context. Instead, we impose the slice-adapted gauge, as introduced in [15], which is

adapted to the hypersurfaces of the foliation. This gauge determines Auniquely by requiring that

the spatial components of Aassociated to the foliation, denoted by the 1-forms ω∈C∞(R,Ω1(M))

with ω(∂i) = A(∂i), are divergence-free and orthogonal to the kernel of the Hodge Laplacian, and

the component of Ain direction of the vector ﬁeld e0, denoted by Ψ := A(e0)∈C∞(M) has

vanishing integral on each spatial slice. This is indeed achievable:

Lemma 2.3 ( [15], Lemma 5.4).Let F∈Ω2(M)be exact. Then, there exists a unique form

A∈Ω1(M)with dA=Fsuch that

(2.7) divgω= 0 , ω ⊥ker(∆H),ZM

Ψdµg= 0 .

6H. Barzegar,D. Fajman

Remark 2.4. We consider ω∈C∞(R,Ω1(M)) as an element in Ω1(M) by requiring ω(e0) = 0.

With this choice one could write A=ω+ Ψe∗

0where e∗

0∈Ω1(M) is dual to e0.

2.4. Rescaled Einstein equations. In this subsection we use the (3 + 1)-dimensional ADM

formalism to establish the rescaled Einstein equations.

2.4.1. Matter quantities. The matter quantities which appear in the ADM formulation of Einstein

equations read (cf., e.g. [32])

(2.8) ˜ρ:= e

N2T00 ,˜

ji:= e

NT0

i,e

Sij := 8πTij −1

2(trhT) ˜gij ,

where ˜ρand ˜

jiare the unrescaled energy density and the matter current, respectively.

2.4.2. Einstein equations. The Einstein ﬂow in CMCSH gauge reads

R− |Σ|2

g+2

3= 4τ·ρ ,(2.9a)

DiΣij =τ2jj,(2.9b) ∆g−1

3N=N|Σ|2

g+τ·η

|{z}

(∗)−1,(2.9c)

∆gXi+RijXj= 2DjNΣij −DiN

3−1+ 2Nτ 2ji

−2NΣjk −DjXkΓi

jk −b

Γi

jk ,(2.9d)

∂Tgij = 2NΣij + 2N

3−1gij −LXgij ,(2.9e)

∂TΣij =−2Σij −NRij +2

9gij +D2

ij N+ 2NΣikΣk

j

−1

3N

3−1gij −N

3−1Σij −LXΣij +Nτ Sij .(2.9f)

Here, we use the notations Rij := Ric[g]ij ,R:= trgRic[g] and the rescaled matter quantities

(2.10) ρ:= 4π˜ρ·τ−3, η := 4π˜ρ+ ˜gij Tij ·τ−3, ji:= 8π˜

ji·τ−5, Sij := 8πe

Sij ·τ−1.

Remark 2.5. The term (∗) is a borderline term, which only yields no loss on the decay for the

gradient of the lapse, when the decomposition for the pressure ηbelow is used, which identiﬁes

the energy density as the leading order term in η. This has already been observed in [1] and is in

particular not due to the coupling between Maxwell and Vlasov part.

We decompose the rescaled energy density as

(2.11) η=ρ+τ2η,

where

(2.12) η= 4π˜gij Tij ·τ−5.

For the later use we also denote Tab := τ−7Tab and we consider the following lemma which is

proved in [2] (cf. also [3]).

Lemma 2.6 ( [3], Lemma 6.2).In the CMCSH gauge the following identity holds

(2.13) Rij +2

9gij =1

2Lg,γ (g−γ)ij +Jij ,

where

Lg,γ (g−γ) := −∆g,γ(g−γ)−2˚

Rγ(g−γ),with ∆g,γ (g−γ)ij := 1

p|g|b

Dkhgkℓp|g|b

Dℓ(g−γ)ij i,

and

kJkHk−1≤Ckg−γk2

Hk.

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 7

An important fact is the uniform positivity of the lapse function, which is used throughout the

remainder of the paper.

Lemma 2.7. The lapse function Nis uniformly positive. In particular, one has

(2.14) 0 < N ≤3.

Proof. The proof follows from applying the maximum principle to the elliptic equation for the lapse

function.

2.5. Vlasov matter. In this subsection we give a quick introduction to the Vlasov matter and

then rescale the momentum and ﬁnally derive the rescaled transport equation.

2.5.1. The mass-shell relation. Throughout this paper we assume that all particles have the same

positive mass m= 1 modeled by a distribution function with the domain

(2.15) P={(x, e

p) : |e

p|2

h=−1,˜p0<0} ⊂ TM,

where e

p:= ˜pµ∂µwith ∂0=∂τand ˜pµbeing the canonical coordinates on the tangent bundle of

M. One can associate an energy-momentum tensor to a distribution function ˜

f:P→[0,∞) by

(2.16) V

Tαβ [˜

f](x) := ZPx

˜

f˜pα˜pβdµPx,

where dµPxis the Riemannian measure induced on Pxby the Lorentzian metric hat a given point

x, and is given by

dµPx:= p|h|

−˜p0

d˜p1∧d˜p2∧d˜p3=e

N

−˜p0

dµ˜p,

where

dµ˜p:= |˜g|1

2d˜p1∧d˜p2∧d˜p3.

We consider the projection map pr : P→TMwhich does (t, xi, p0, pi)7→ (t, xi, pi). Then, instead

of using ˜

fwe deal with the function f:= ˜

f◦pr−1which we refer to as distribution function for the

remainder of the paper.

2.5.2. The rescaled momentum. We rescale the momentum vector ﬁeld as

(2.17) ˜pa=τ2pa.

As a result we have ∂˜pi=τ−2∂pi. The unrescaled mass-shell relation in (2.15) gives (cf., e.g., [35])

(2.18) ˜p0=e

N2− | e

X|2

˜g−1he

X, ˜pi˜g+qhe

X, ˜pi2

˜g+e

N2− | e

X|2

˜g(1 + |˜p|2

˜g).

Rescaling the variables in the previous equation and denoting p0:= τ−2˜p0, we ﬁnd

(2.19) p0=N−11− | b

X|2

g−1τhb

X, pig+bp,

or equivalently,

p0=1 + τ2|p|2

g

Nbp−τhb

X, pig,

where

(2.20) bp:= qτ2hb

X, pi2

g+1− | b

X|2

g(1 + τ2|p|2

g).

Furthermore, we ﬁnd

(2.21) ˜p0=h0α˜pα=−e

Nbp ,

which, in particular, implies that dµPx=bp−1dµ˜p. We further use the notation p:= Np0.

8H. Barzegar,D. Fajman

Remark 2.8. Note that (2.19) reduces to p0=q1 + |p|2

gwhen X= 0 and N= 1. These are the

values that correspond to the background geometry.

2.5.3. The transport equation. The transport equation in the presence of the electromagnetic ﬁeld

reads

(2.22) ˜pµ∂µf−e

Γi

µν ˜pµ˜pν+q˜pαFi

α∂˜pif= 0 ,

where e

Γα

βγ ≡e

Γ[h]α

βγ . We wish to rewrite (2.22) in terms of the rescaled variables. We start with

the unrescaled Christoﬀel symbols of h(cf. [1])

(2.23) e

Γ[h]a

bc = Γ[g]a

bc +N−1Σbc +1

3gbcXa,

e

Γ[h]a

00 =τ−2Γa,

e

Γ[h]a

0b=τ−1(−δa

b+ Γa

b),

where

Γa:= −∂TXa−Xa−2

3(N−3)Xa+XbDbXa−2NΣa

bXb+NDaN

+hN−1∂TN−N−1XbDbN+N−1Σbc +1

3gbcXbXciXa,(2.24a)

Γa

b:= −NΣa

b+1

3δa

b(3 −N) + DbXa−N−1XaDbN+N−1Σbc +1

3gbcXcXa.(2.24b)

We use notations Γ∗and Γ∗

∗when we want to suppress the indices of the above two objects. The

rescaled transport equation ﬁnally takes the following form, using the natural horizontal and vertical

derivatives on T M ,Aa:= ∂a−piΓk

aiBkand Ba:= ∂pa,

∂Tf=τN pa

pAaf−τ−1p

NΓaBaf

|{z }

(∗)

+2paBaf−2pcΓa

cBaf

−τΣbc +1

3gbcXapbpc

pBaf+τqFaBaf

|{z }

(∗∗)

,

(2.25)

where

(2.26) Fi:= hijF0j+pa

p0hij Faj +τ

NFa0b

Xi,

with hij =gij −N−2XiXj, which can be read oﬀ from the metric h.

Remark 2.9. The terms marked by (∗) and (∗∗) are borderline terms. In particular, the term

marked by (∗∗) originates from the Maxwell ﬁeld and the slow decay is caused by the ﬁrst term in

(2.26). In combination with the factor τthe term F0jappears in the energy for the Maxwell ﬁeld

(2.49), which itself is ε-small but does not decay. As a consequence the term (∗∗) in the Vlasov

equation yields a growth for the energy of the distribution function of the type exp(C εT ), however,

only if we obtain sharp estimates for the energy of F. Therefore, we need to avoid that the loss for

the energy of the distribution function couples back into the equation for the Maxwell ﬁeld.

2.6. Maxwell equation. We start with the the Maxwell equation which in Heaviside–Lorentz

units reads

(2.27) hλµ e

∇λFµα =−qZf˜pαdµPx=: Jα.

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 9

Recall that the index 0 refers to the vector ﬁeld ∂τ. Rescaling the result according to (2.2) and

(2.17) yields

Jα=−qZf˜pα

bpdµ˜p=−qτ 3Zf˜pα

bpdµp,

where we used

dµ˜p=τ3|g|1

2dp1∧dp2∧dp3=: τ3dµp.

Thus, by (2.21) we have

(2.28) J0=τqN Zf dµp=: τJ,

and using

(2.29) ˜pa

bp=τ−1Xap0+gabpb

bp=: −NPa

we ﬁnd

(2.30) Ji=qNτ 3ZfPidµp=: τ3Ji.

The deﬁnition of Pais motivated by the relation Bap0=−τ2Pa(cf. Appendix E). Keeping (2.3)

in mind, for the left-hand side of (2.27) we get

hλµ e

∇λFµα =hλµ h∇λFµα −e

Γβ

µλ −Γβ

µλFβα −e

Γβ

αλ −Γβ

αλFµβ i=hλµ∇λFµα =τ2hλµ∇λFµα ,

where we used the facts from Appendix A. Hence,

(2.31) hλµ∇λFµα =τ−2˜

Jα.

We compute the following for Proposition below, which gives the rescaled Maxwell equations

(2.32)

F0i=−τ−1∂Tωi+∂iNΨ−Xjωj,

Fˆ

0i=N−1∂Tωi+Xj∂jωi−∂iΨ,

Fij =∂iωj−∂jωi,

and

τ−2˜

Jˆ

0=τ−2˜

J(e0) = τ−2N−1˜

JT+b

Xj˜

Jj=τ−2−τN −1˜

J0+b

Xj˜

Jj=−N−1J+τb

XjJj,

where we used ˜

JT=˜

J(∂T) = ˜

J(−τ∂τ) = −τ˜

J0.

Proposition 2.10. Let F∈Ω2(M)be exact which solves (2.31) and let A∈Ω1(M)be a vector

potential for Fwhich satisﬁes the slice-adapted gauge conditions in Lemma 2.3 with the same Ψ

and ωgiven there. Then, we have

∆gΨ = −divg(Ψ ·Dlog(N)) −[Le0,divg]ω−N−1J

|{z}

(∗)

+τb

XjJj,(2.33a)

(Le0(Le0ω))k+ ∆Hωk=∂k(∂e0Ψ) + ∂e0Ψ·∂kN

N+ Ψ ·∂k(∂e0log(N))

+gij ∂iN

NFjk + trgΠ·Fˆ

0k+ 2gij ΠikFjˆ

0−τJk.(2.33b)

Remark 2.11. The term (∗) is borderline in the following sense. As deﬁned in (2.28) it is given up

to a factor by the integral of the distribution function. A straightforward estimate by the energy

of the distribution function would induce a small growth for the ﬁeld Ψ, which would prevent the

estimates from closing. To obtain a uniform bound on Ψ it is crucial to observe that the leading

order term in (∗) is in fact the energy density, which is established in Lemma 5.3.

10 H. Barzegar,D. Fajman

Proof of Proposition 2.10. From (2.27) and using the slice-adapted gauge, we get (cf. Eqs. (5.16)–

(5.18) in [15])

gij DiFjˆ

0= ∆gΨ + divg(Ψ ·Dlog(N)) + [Le0,divg]ω ,(2.34)

gij DiFjk =−∆Hωk+ trgΠ·Fˆ

0k+gij ΠikFjˆ

0,(2.35)

Dˆ

0Fˆ

0k= (Le0(Le0ω))k−∂k(∂e0Ψ) −∂e0Ψ·∂kN

N−Ψ·∂k(∂e0log(N))

−gij ∂iN

NFjk +gij Πik Fˆ

0j.(2.36)

Inserting the results into (2.31) and using (2.28) and (2.30) ﬁnishes the proof.

2.7. Local existence. A local-existence-theory for the system (2.9), (2.25) and (2.33) can be

derived based on the ideas of [15,19].

Proposition 2.12. Consider CMC-initial data

(2.37) (g0,Σ0, f0, ω0,˙ω0)∈B6,5,5,6,5

δ(γ, 0,0,0,0)

with δ > 0suﬃciently small. Let I⊂Rbe a compact interval with T0∈I. Then, there exists

aT > 0and a unique solution (g, k, f , ω, ˙ω)to the Einstein–Vlasov–Maxwell system with J=

(T0−T, T0+T)launched by this initial data. Tdepends continuously on the H6(M)-, H5(M)-,

HVl,5,4,c(T M )-, H6(M)-, and H5(M)-norm of g0,Σ0,f0,ω0, and ˙ω0, respectively. The following

regularity properties hold

(g, k),(ω, ˙ω)∈Cb(J, H 6×H5)∩C1

b(J, H5×H4),

f∈Cb(J, HVl,5,4,c(T M )) .

Moreover, the solution is either global in time, i.e. T=∞or

lim sup

tրT

(kg−γkH6+kΣkH5+kωkH6+k˙ωkH5+k|fk|5,2)≥2δ.

Proof. The local existence theory can be established similar to the corresponding theorems for the

Vlasov case in [1] and the Maxwell case in [15]. When considering the Maxwell equations in Lorenz

gauge we obtain wave-type equation for the 4-potential in the form

H,hAα=τ−2Jα,

where H,h denotes the Hodge wave operator of the metric h.

Complementing the Einstein-Vlasov system with Maxwell terms by this equations does not change

the structure of the elliptic-hyperbolic system with respect to the Einstein-Vlasov case considered

in [19]. The additional wave-type equation can be treated at the same order of regularity as the

evolution equations for the spatial metric g (which is decomposed into ﬁrst order equations). A

local-existence theorem analogue to Theorem 4.2 of [19] follows by a similar proof, where the vector

potential is controlled in the same regularity class as the metric. In consequence we obtain a local-

existence theory for the EVMS in CMCSH-Lorenz gauge.

Then, as shown in [15], we perform a gauge-transformation for the vector potential of the Maxwell

ﬁeld to obtain a potential obeying the slice-adapted gauge while conserving the regularity of the

potential and thereby of the solution. We conclude at this point that we have existence and

uniqueness of a local-in-time solution of the EVMS in CMCSH-slice-adapted gauge in the respective

regularity class.

It remains to prove the continuation criterion. For this purpose we need to ensure that a solution,

which is small in CMCSH-slice-adapted gauge remains small when its vector potential is transformed

to the Lorenz gauge. We therefore consider the gauge transformation of the vector potential

(2.38) Aµ=A′

µ+∂µΛ,

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 11

such that A′

µfulﬁls the Lorenz gauge. Then Λ solves the wave equation

(2.39) hΛ = −∂e0Ψ−N−1(1 −hij hij) (∂e0Λ−Ψ) −b

Xib

XjDiωj+hijΠij Ψ + N−1gij ∂jωi,

with trivial initial data. Here, we used (2.6) and relations in Appendix A. From this follows that

smallness of the solution in slice-adapted gauge in the considered regularity class implies smallness

of the solution in Lorenz gauge in the corresponding regularity class on short time intervals, as

Λ is determined by the right-hand side of the wave equation. Note therefore that ∂e0Ψ fulﬁls the

elliptic equation (2.33a), which improves its regularity by ellipticity of the equation. This implies

the continuation criterion in slice-adapted gauge.

2.8. Coupling of the equations and consequences for the long-time behavior. Having

derived all equations we give a brief summary on the coupling issues in the remark below.

Remark 2.13. The rescaled EVMS consists of systems of equations (2.9), (2.25), and (2.33). In this

rescaled system we have particular matter-electromagnetic coupling terms, i.e., τ qFiBifand N−1J,

which appear as source terms in the Maxwell equations and in the Vlasov equations, respectively.

Those terms are not present in the respective individual cases, i.e., pure electromagnetism in [15]

or pure uncharged matter in [1]. In addition, they constitute principal terms in the sense of their

decay properties. Both terms decay slower or at the same rate as the slowest decaying neighbouring

source terms and thereby constitute potential obstacles when analyzing the rescaled equations. The

major observation of the following bootstrap analysis is that those terms can indeed be handled

in the existing bootstrap hierarchy of the previous works and in turn yield the same asymptotic

behaviour as the respective cases.

2.9. Energy-momentum tensor. We compute and collect all relevant terms from the energy-

momentum tensor in the following. The unrescaled energy-momentum tensor is given by

T=V

T+MT.

Accordingly, we have for the matter variables deﬁned in (2.8),

˜ρ≡˜ρV+ ˜ρM,˜

ja≡˜

ja

V+˜

ja

M, η ≡ηV+ηM,

where scripts V and M stand for Vlasov and Maxwell matter ﬁelds, respectively.

Then, the rescaled Vlasov matter quantities take the following form

ρV= 4πN 2Zf(p0)2

bpdµp,(2.40a)

ja

V= 8πN Zfp0pa

bpdµp,(2.40b)

ηV= 4πZf|p+τ−1p0X|2

g

bpdµp,(2.40c)

V

Tab = 8πZfpapb

bpdµp.(2.40d)

For the Maxwell ﬁeld we start with the unrescaled energy-momentum tensor (again in the Heaviside–

Lorentz units)

(2.41) MTµν =Fα

µFνα −1

4hµν Fαβ Fαβ .

We ﬁrst compute Fαβ Fαβ in terms of the rescaled variables

(2.42) FαβFαβ =−2τ6N−2gij F0iF0j+ 2τ5N−2XigjkF0kFij +τ4hikhj ℓFij Fkℓ .

12 H. Barzegar,D. Fajman

Thus, in terms of rescaled variables MT00 takes the form

(2.43) MT00 =τ2hijF0iF0j+τ−4

41− | b

X|2

gN2Fαβ Fαβ .

For the oﬀ-diagonal components of the energy-momentum tensor we have

(2.44) MT0i=τ2hjkF0kFij −1

4τ−3XiFαβ Fαβ .

Finally, the spatial part reads

(2.45) MTij =−τ4N−2F0iF0j−τ3N−2Xk(F0iFjk +F0jFik) + τ2hkℓFik Fj ℓ −1

4τ−2Fαβ Fαβ .

Then, the rescaled Maxwell quantities can be expressed by the components of the energy-momentum

tensor of Maxwell ﬁeld computed above as

ρM= 4πτ N−2MT00 −2τ−1Xi(MT0i) + τ−2XiXj(MTij),(2.46a)

ji

M= 8πN −1hN−2Xi(MT00 ) + τ−1b

Xib

Xj+hij MT0j+τ−2hikXj(MTij )i,(2.46b)

ηM= 4πτ −3gij (MTij).(2.46c)

2.10. Norms for matter ﬁelds. We recall in the following brieﬂy the norms used to control

distribution function and Faraday tensor as introduced in [20] and [15], respectively.

2.10.1. L2-Sobolev energy of the distribution function. L2-Sobolev energies of the distribution func-

tion can be deﬁned based on the Sasaki metrics with respect to γand g. For γ,γ≡γijdxi⊗dxj+

γij b

Dpi⊗b

Dpj, where b

Dpi:= dpi+b

Γi

jk dpjdpk, deﬁnes a metric on T M . We denote the associated co-

variant derivative by b

D. The volume form induced by γis then given by dµγ:= −|γ|Q3

i=1 dxi∧dpi.

A weighted metric on T M is deﬁned by γ:= γijdxi⊗dxj+hpi−2

γγij b

Dpi⊗b

Dpj,where hpiγ:=

q1 + |p|2

γ. We then deﬁne the L2-Sobolev energy of the distribution function with respect to the

associated Sasaki metric of γby

(2.47) |||f|||ℓ,µ := sX

k≤ℓZT M hpi2µ+4(ℓ−k)

γ|b

Dkf|2

γdµγ

The function space associated with the above norm is denoted by HVl,ℓ,µ(T M ). Moreover, pointwise

estimates are taken with respect to the following L∞

xL2

p-norm

|||f|||∞,ℓ,µ := sup

x∈M

sX

k≤ℓZTxMhpi2µ+4(ℓ−k)

γ|b

Dkf|2

γdbµp

,

where dbµp:= |γ|1

2dp1∧dp2∧dp3. Then, the following lemma holds.

Lemma 2.14. For fsuﬃciently regular, there exists a constant Csuch that

|||f|||∞,ℓ,µ ≤C|||f|||ℓ+2,µ

holds.

In addition, we consider the L2-Sobolev energy associated with the dynamical metric g. The Sasaki

metric with respect to gis g≡gij dxi⊗dxj+gij Dpi⊗Dpj,where Dpi:= dpi+Γi

jk dpjdpk. The asso-

ciated covariant derivative and the connection coeﬃcient are denoted by Dand Γ(cf. Appendix C),

respectively, and the volume form induced by gon T M is given by dµg:= −|g|Q3

i=1 dxi∧dpi.

Then, we deﬁne analogously a weighted version of the Sasaki metric associated with gby g:=

gij dxi⊗dxj+hpi−2gij Dpi⊗Dpj,where hpi:= q1 + |p|2

g. We ﬁnally deﬁne the L2-Sobolev energy

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 13

of the distribution function with respect to the Sasaki metric associated with the dynamical metric

gby

(2.48) Eℓ,µ(f) := sX

k≤ℓZT M hpi2µ+4(ℓ−k)|Dkf|2

gdµg

Under suitable smallness assumptions, E5,4(f) and |||f|||5,4are equivalent.

2.10.2. L2-Sobolev norm of the Faraday tensor. In the following we deﬁne the L2-Sobolev norm of

the Faraday tensor which measures the perturbation of the Maxwell ﬁeld. It is deﬁned as follows

(cf. [15])

|||F|||2

Hℓ:= X

k≤ℓZMτ2gij Di1···DikF0iDi1···DikF0j

+gij gabDi1···DikFia Di1···DikFjbdµg.(2.49)

The factor τ2in the ﬁrst term compensates the the growth of the τ-component coming from

Aτ=dT

dτ AT=−τ−1AT, relative to the spatial components of F.

2.11. Smallness. To determine the ﬁnal estimates we use a standard bootstrap argument. We

deﬁne a set of smallness conditions for the dynamical quantities which serves as the bootstrap

assumptions. We deﬁne

B6,5,5,6,5

δ,τ (γ, 0,0,0,0) := n(g, Σ, f, ω, ˙ω)∈H6×H5×HVl,5,4×H6×H5

|τ|−1

2(kg−γkH6+kΣkH5) + kρkH4+|τ|1

2· |||f|||5,4+|||F|||H5< δo.(2.50)

We say that (g(τ),Σ(τ), f(τ), ω(τ),˙ω(τ)) is δ-small if (g, Σ, f , ω, ˙ω)∈B6,5,5,6,5

δ,τ (γ, 0,0,0,0). To

refer to δ-small data we also use the term smallness assumptions. A direct consequence of the

smallness assumption is stated in the following lemma.

Lemma 2.15. For δ-small data (2.50) with δsuﬃciently small, the energies E5,4(f)and |||f|||5,4

are equivalent.

Proof. The proof follows straightforwardly from the deﬁnitions and the smallness assumption.

The smallness assumptions imply the smallness of the perturbation for the lapse function and shift

vector. The following result is a corollary of Proposition 7.1.

Corollary 2.16. For any δ > 0there exists a ¯

δ > 0such that (g, Σ, f , ω, ˙ω)∈B6,5,5,6,5

δ,τ (γ, 0,0,0,0)

implies

(2.51) |τ|−1(kN−3kH6+kXkH6)<¯

δ .

3. Estimating the energy-momentum tensor

In this section we will estimate the components of the energy-momentum tensor of the Maxwell

ﬁeld and other matter quantities by the norms deﬁned in the previous section.

Lemma 3.1. For k > 3/2we have the following estimate

(3.1) kMT00kHk+|τ|−1· kMT0ikHk+|τ|−2· kMTkHk≤C|||F|||2

Hk,

where MTdenotes the spatial part of MTand C=CkNkHk,kNkL∞,kN−1kL∞,kXkHk.

Proof. The statement of the lemma follows directly from the expressions (2.43)–(2.45) (cf. also

Lemma 4.2 in [15]).

We summarize the estimates of the rescaled matter quantities in the following proposition.

14 H. Barzegar,D. Fajman

Proposition 3.2. For k > 3/2the following estimates hold

(3.2)

kρkHk≤Ch|τ| · |||F|||2

Hk+Ek,3(f)i,

kjkHk≤Ch|||F|||2

Hk+Ek,3(f)i,

kηkHk≤Ch|τ|−1· |||F|||2

Hk+Ek,4(f)i,

kSkHk≤Ch|τ| · |||F|||2

Hk+τ2Ek,4(f) + kρkHki,

where C=CkNkHk,kNkL∞,kN−1kL∞,kXkHk.

Proof. Combining Lemma 13 in [1] with the Lemma 3.1 result in the estimates of the proposition.

3.1. Pointwise estimates on the momentum variables. In this subsection we provide some

useful auxiliary estimates used further below.

Lemma 3.3. The following estimates hold for large T, using the smallness of the lapse function

and shift vector, and provided that the momenta have compact support

|p|g

bp.|τ|−1,or |p|g

bp.|p|g,(3.3)

p0

bp.1

N1− | b

X|2

g,(3.4)

|P|g.|τ|−1· | b

X|g+N−1|p|g.(3.5)

Proof. The proof follows straightforwardly from the explicit expression for the respective quantities.

4. Control of the momentum support

Based on the characteristic system associated to the transport equation, we derive an estimate on

the radius of the momentum support of the distribution function.

The characteristic system associated with the rescaled transport equation reads

dxa

dT =−τpa

p0,(4.1)

dpa

dT =τ−1Γap0−2pa+ 2Γa

bpb+τΓa

bc +N−1Σbc +1

3gbcXapbpc

p0−τqFa.(4.2)

With the deﬁnition of the auxiliary quantity (cf. [1])

(4.3) G(T, x, p) := |p|2

g,

one ﬁnds its derivative along a characteristic

(4.4) dG

dT =|p|2

˙g+ 2τ−1hΓ∗, pigp0+ 4gikΓi

jpjpk+2τ

NΣbc +1

3gbcpbpc

p0hX, pig−4τ qhF, pig,

which implies the following result.

Lemma 4.1. For δ-small data (2.50) with δsuﬃciently small, the following estimate holds for any

characteristic

(4.5)

dG

dT ≤C[|˙g|g+|Γ∗

∗|g+ (1 + |Σ|g)|X|g]G+C|τ|−1· |Γ∗|g+|qτ | · |F|g√G.

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 15

Deﬁning the supremum of the values of Gin the support of fat a ﬁxed time Tanalogous to [1] as

(4.6) G[T] := sup npG(T, x, p)|(x, p)∈supp f(T,.,.)o,

we derive an estimate to control the momentum support in the following proposition.

Proposition 4.2. For δ-small data (2.50) with δsuﬃciently small, the following estimate holds

G|T≤G|T0+CZT

T0

(eskΓ∗kH2+|||F|||H2)ds

×exp CZT

T0kΣkH2+kN−3kH2+kXkH2+kΓ∗

∗kH2+|q|e−s|||F|||H2ds.(4.7)

Proof. From Lemma 4.1 and the deﬁnition of Fi, we get

d

dT √G≤C(kΣkH2+kN−3kH2+kXkH2+kΓ∗

∗kH2+|qτ | · |||F|||H2)√G

+C|τ|−1kΓ∗kH2+|||F|||H2.

Applying Gr¨onwall’s lemma completes the proof.

5. Energy estimates

5.1. Energy estimate of the distribution function. We ﬁrst give the energy estimate for the

distribution function.

Before we proceed, we prove the following lemma:

Lemma 5.1. Assume that δ-small data (2.50) with δsuﬃciently small holds. Then, for a generic

suﬃciently regular function φon T M , the following inequality holds

(5.1) X

m≤kZ|DmZφ dµp|2

gdµg.X

m≤kZZ¯p|Dmφ|gdµp2

dµg.

Proof. To obtain this inequality, we use (cf. (4.26) in [1])

(5.2) DaZφ dµp=ZAaφ dµp.

For higher covariant derivatives one gets terms of the form

(5.3) pi•Riem[g]•Dℓφ ,

where ℓ≤m−1 and •denotes various abstract contractions of tensor indices. Then, the smallness

assumption on the metric yields

X

m≤kZDmZφdµp

2

dµg.X

m≤kZZ|Dmφ|gdµp+Z|p|g|Dm−1φ|gdµp2

dµg

.X

m≤kZZ(1 + |p|g)|Dmφ|gdµp2

dµg

.X

m≤kZZ¯p|Dmφ|gdµp2

dµg.(5.4)

16 H. Barzegar,D. Fajman

Proposition 5.2. Let fbe a solution of the transport equation (2.25). For δ > 0suﬃciently small

and (g, Σ, f, ω, ˙ω)∈B6,5,5,6,5

δ,τ (γ, 0,0,0,0) the estimate

∂TE2

k,µ(f)≤ChkN−3kHk+kΣkHk+kXkHk+1 +|τ|·kN−1∂TXkHk+|τ|−1· kN−1Γ∗kHk

+kΓ∗

∗kHk+k(Σ + g)XkHk+|τ|G+|q| · |||F|||HkiE2

k,µ(f),(5.5)

holds for k > 5/2and µ≥3, provided that |τ|Gis bounded by a constant.

Proof. Recall the following notation. Using the introduced bold Latin indices we deﬁne the frame

{θa}a≤6:= {A1,A2,A3,B1,B2,B3}. That is, the indices a∈ {1,2,3}correspond to the horizontal

directions whereas the indices a∈ {4,5,6}correspond to the vertical directions. The proof is

identical to the proof of Proposition 12 in [1] except for the last term. However, for the last term

which comes from the Maxwell part of the transport equation (2.25) one can proceed similarly.

Indeed; we consider

(5.6) Da1···Dak∂Tf ,

which arises from ∂T|Dkf|2

gwhen applying ∂Tto the L2-energy (2.48). For all the other terms

involving ∂Twe refer to Proposition 12 in [1]. Therefore, we will focus on the Maxwell part of the

transport equation in (5.6)

Da1···Dak(τqFaBaf) = τ qθa1Da2···Dak−X

2≤j≤k

Γc

aja1Da2···Dac···Dak(FaBaf)

=τqθa1···θak(FaBaf) + ...+τ q(−1)k−1Γc1

aka1Γc2

c1a2···Γck−1

ck−2ak−1(FaBaf),

where we suppressed the mixed terms. After commuting the operator FaBato the front by using

the relations in Appendix D one obtains

(5.7) τqFaBa(Da1···Dakf),

and other mixed terms which are of the form (cf. (5.3))

(5.8) τqDk1FapiDk2Riemk3Dk4f ,

with Piki=k. From (5.7) and after integration by parts when evaluating

τq X

k≤ℓZT M hpi2µ+4(ℓ−k)ga1b1···gakbkDb1···Dbkf·Da1···Dak∂Tf dµg,

one ﬁnds the corresponding term |q| · |||F|||Hkin (5.5). All the mixed terms are of lower order and

can be absorbed in the latter term.

5.2. Estimate of current density. As argued previously, the speciﬁc form of the eﬀective terms

arising from the current, Jand Jiis essential to obtain suﬃcient bounds on the Maxwell ﬁelds.

The following lemma establishes these bounds.

Lemma 5.3. The following estimates

kJkHk≤C|q| · kρVkHk,(5.9)

kJkHk≤C|q||τ|−1kXkHkkρVkHk+kjVkHk,(5.10)

hold, provided that |τ|Gis bounded by a constant and δ-small data (2.50) with δsuﬃciently small

holds.

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 17

Proof. To obtain the estimates above we need following inequalities

sup

(x,p)∈supp f(T,.,.)bp≤C(1 + |τ|G),sup

(x,p)∈supp f(T,.,.)bp

p2≤C(1 + |τ|G)3,

and

sup

(x,p)∈supp f(T,.,.)

1

p≤C(1 + |τ|G),

where we applied the smallness conditions. Then,

kJkHk=

qN Zf dµp

Hk

≤C|q|(1 + |τ|G)3kN−1ρVkHk,(5.11)

from which the ﬁrst estimate follows. Note here that the expression bpand p0appear via Rf µp=

Rf(p0)2/bp·bp/(p0)2µpto generate ρV. When the factor bp/(p0)2is hit by derivatives of the norm,

the generated terms can be estimated by a uniform constant invoking the smallness assumptions.

All those terms are absorbed into the constant C.

For the second estimate we have

kJkHk=

qτ −1XiZfp0

bpdµp+qZfpi

bpdµp

Hk

≤C|q|(1 + |τ|G)|τ|−1kXkHkkρVkHk+kjVkHk,(5.12)

where ρVand jVare deﬁned in (2.40). Then, the assumption on the boundedness of |τ|Gﬁnishes

the proof.

Remark 5.4. Estimate (5.9) enables us to estimate Jby the energy density of the distribution

function and eventually by the full energy density, which, as shown below, is uniformly bounded.

This prevents this term to pick up the loss from the Vlasov energy.

5.3. Energy estimate for the vector potential. As explained in Section 5.1 of [15], the struc-

ture of the system (2.33a)–(2.33b) suggests the following energy

(5.13)

Ek(ω) :=

k−1

X

ℓ=0 ZMh(∆H)ℓLe0ω, Le0ωig+h(∆H)ℓ+1ω, ωigdµg

≃ kLe0ωk2

Hk−1(M[g]) +kωk2

Hk(M[g])

with k≥1. In the following we omit the argument ωand write Ek≡Ek(ω) for simplicity.

Remark 5.5. The gauge condition ω⊥ker(∆H) allows us to control the L2-norm of ωby Ek.

Remark 5.6. Under smallness assumptions the L2-Sobolev norm of the Faraday tensor (2.49) is

equivalent to the energy (5.13) and the norm of Ψ in the following sense

(5.14) |||F|||Hk≃ kΨkHk+1 +pEk+1 .

Remark 5.7. Both terms on the right-hand side, as shown below, are uniformly bounded.

Lemma 5.8. Let Fbe a solution of (2.31) and let A∈Ω1(M)be a gauged vector potential for F,

Ψ, and ωas in Lemma 2.3. Then, for k > 5/2and assuming kNkHkis bounded by some constant,

we have

∂TEk≤C(kDlog NkHk−1+kΠkHk−1+kSkHk−1)Ek

+C[k∂e0ΨkHk+ (k∂e0log NkHk+kΠkHk−1)kΨkHk+|τ| · kJ kHk−1]pEk.(5.15)

18 H. Barzegar,D. Fajman

Proof. The lemma is proved along the same lines as the proof of Lemma 5.7 in [15], and by using

(2.33b) and (cf. Eq. (5.37) in [15])

(5.16) kie0FkHk−1≤CkDΨkHk−1+kΨkHk−1kDlog NkHk−1+pEk,

where iY: Ωℓ(M)−→ Ωℓ−1(M) is the interior product for any vector ﬁeld Y∈X(M). Here, by

abuse of notation, we consider the Sobolev norm of ie0F∈C∞(R,Ω1(M)).

Lemma 5.9. As long as kDlog NkHkis small enough the following estimate holds

kΨkHk+1 ≤ChkΠkHk−1+ (1 + kΠkHk−1)kDlog NkHk−1+kSkHk−1ipEk

+CkJkHk−1

|{z }

(∗)

+|τ| · kXkHk−1kJkHk−1.(5.17)

Remark 5.10. The term (∗) is borderline, as it would yield a growth for the norm of Ψ. Uniform

boundedness is achieved due to the estimated using the energy density as outlined above.

Proof of Lemma 5.9. By elliptic regularity and the fact that (2.33a) has a unique solution, we ﬁnd

kΨkHk+1 ≤CkΨkHkkDlog NkHk+k[Le0,divg]ωkHk−1

+kN−1kHk−1kJkHk−1+|τ|k b

XkHk−1kJkHk−1.

And from Lemma 2.2 it follows

k[Le0,divg]ωkHk−1≤C[kΠkHk−1+ (1 + kΠkHk−1)kDlog NkHk−1+kSkHk−1]pEk.

Combining the results and using Lemma 5.3 completes the proof.

We need to estimate the term k∂e0ΨkHkappearing in (5.15). This is done in Lemma 5.14 which

need further preparations:

Lemma 5.11. Let the integer k > 7/2and assume that kNkHk−2is bounded. Then, we have

ki∂τFkHk−2≤C|τ|−1kNkHk−2kΨkHk−1+kΨkHk−2kDlog NkHk−2+pEk−1

+C|τ|−1kXkHk−2pEk−1.(5.18)

Proof. We recall that i∂τF∈C∞(R,Ω1(M)). Hence,

(5.19) i∂τF(∂a) = Fa0=−τ−1N·ie0F(∂a) + τ−1XiFai .

The ﬁrst term can be estimated by (5.16) and for the second term we use (5.14). Then, invoking

the assumption on kNkHk−2ﬁnishes the proof.

Lemma 5.12. Let k, µ ∈Nsuch that k > 7/2and µ≥3. Further, assume that kNkHk−2is

bounded. Then, the following estimate holds

(5.20) k∂e0JkHk−2≤Ck∂TNkHk−2+kb

NkHk−2+kXkHk−1kJkHk−2+C|q|Ek−1,µ(f),

provided that kΓ∗kHk−2is bounded.

Proof. We have

(5.21) ∂e0J=N−2∂TNJ+qZ∂Tfdµp+qZf ∂T(dµp) + b

Xk∂kJ.

The third term can be rewritten as

qZf∂T(dµp) = N−13b

N−DiXi+ 6J,

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 19

where we used the evolution equation for |g|1

2and the rescaling of the momentum according to

(2.17) to get

(5.22) ∂T(dµp) = 3b

N−DiXi+ 6dµp.

For the second term in (5.21), using the transport equation, we ﬁnd

Z∂Tf dµp=ZτN pi

pAif−τ−1p

NΓaBaf+ 2paBaf

−2Γa

cpcBaf−τΣbc +1

3gbcXapbpc

pBaf−τqFiBifdµp.(5.23)

The last term after integration by parts gives

−τq ZFiBif dµp=τ q ZBiFif dµp

=τq Zfhgij −b

Xib

XjFaj +τ

NFa0b

XiiBipa

p0dµp.(5.24)

The third term of the (5.23) cancels the third term of (5.22), after integration by parts. Hence,

qZ∂Tf dµp+qZf∂T(dµp) = qZτ N pi

pAif−τ−1p

NΓaBaf−2Γa

cpcBaf

−τΣbc +1

3gbcXapbpc

pBaf+τqBiFifdµp

=: q(I1+I2+I3+I4+I5).(5.25)

Invoking Lemmas 3.3 and 5.1 and the relations in Appendix E, integrating by parts whenever

possible, and imposing the smallness assumptions, leads to

(5.26)

kI1kHk−2≤C· |τ| · Ek−1,µ(f),

kI2kHk−2≤C· |τ| · kΓ∗kHk−2·Ek−2,µ(f),

kI3kHk−2≤C· kΓ∗

∗kHk−2·Ek−2,µ(f),

kI4kHk−2≤C· |τ| · k(Σ + g)XkHk−2·Ek−2,µ+1(f)

kI5kHk−2≤C· |τq| · |||F|||Hk−2·Ek−2,µ(f).

Note that Ek−2,µ+1(f)≤Ek−1,µ(f). Hence,

k∂e0JkHk−2≤CkN−2kHk−2· k∂TNkHk−2+kb

NkHk−2+kXkHk−1kJkHk−2

+|q|

5

X

i=1 kIikHk−2.

Assuming that kNkHk−2and kΓ∗kHk−2are bounded, we arrive at the estimate of the lemma.

Lemma 5.13. Let the integers k > 7/2and let µ≥3and assume that kNkHk−2is bounded. Then,

k∂e0JkHk−2≤Ck∂TNkHk−2+kb

NkHk−2+kXkHk−1kJkHk−2

+C|q|h1 + |τ|+|τ|−1· kΓ∗kHk−2+kΓ∗

∗kHk−2+kΣkHk−2+kb

NkHk−2+kXkHk−1

+|τ|−1(kXkHk−2+k∂TXkHk−2) + k∂TNkHk−2+|q| · |||F|||Hk−2iEk−1,µ(f).(5.27)

20 H. Barzegar,D. Fajman

Proof. We have

(5.28) ∂e0Jk=N−2∂TNJk+qZ∂TfPkdµp+qZf∂TPkdµp+qZfPk∂T(dµp) + b

Xi∂iJk.

Again, using (5.22), the fourth term can be rewritten as

(5.29) qZfPk∂T(dµp) = N−13b

N−DiXi+ 6Jk.

For the second and the ﬁfth terms of (5.28) we repeat a similar calculation to the one in Lemma

5.12. The corresponding term which after integrating by parts cancels the ﬁfth term of (5.28) is

2RPkpaBaf dµp. Thus,

(5.30) 2qZPkpaBaf dµp+ 6N−1Jk=−2qZBaPkpaf dµp.

Using (E.1g), under smallness assumptions, we ﬁnd

(5.31)

qZBaPkpaf dµp

Hk−2≤C|q|Ek−2,µ(f).

For the Maxwell part of the second term of (5.28) one gets, after integration by parts,

(5.32)

τq ZFiBifPkdµp

Hk−2≤C|q|(1 + |τ|)|||F|||Hk−2·Ek−2,µ(f).

Therefore,

qZ∂TfPkdµp

Hk−2≤C|q|h|τ|+|τ|−11 + τ2kΓ∗kHk−2+kΓ∗

∗kHk−2

+|τ| · k(Σ + g)XkHk−2+|q| · |||F|||Hk−2iEk−1,µ(f).(5.33)

Finally, putting (F.3) into the third term of (5.28) and making use of the relations in Appendix F,

one ﬁnds

qZf∂TPkdµp

Hk−2≤C|q|h|τ|−1(kXkHk−2+k∂TXkHk−2) + k∂TNkHk−2

+kΣkHk−2+kb

NkHk−2iEk−2,µ(f).(5.34)

Inserting the results above and invoking the smallness assumptions yields the proof.

Lemma 5.14. Let integers k > 7/2and and µ≥3. Then, as long as kDlog NkHkis small enough

and kΠkHk−2is bounded, the following holds

k∂e0ΨkHk≤CkΠkHk−2+kSkHk−2+kDlog NkHk−2+kD∂e0log NkHk−2

+kLe0SkHk−2+kLe0ΠkHk−2pEk+kΨkHk

+Ck∂TNkHk−2+kb

NkHk−2+kXkHk−1kJkHk−2

+C1 + k∂TNkHk−2+k∂TXkHk−2+kb

NkHk−2+kXkHk−1|τ| · kXkHk−2· kJ kHk−2

+C|q|n|τ| · h1 + kΓ∗kHk−2+k(Σ + g)XkHk−2+kXkHk−2· k b

NkHk−2

+k∂TXkHk−2+k∂TNkHk−2+|q| · |||F|||Hk−2i+kΓ∗

∗kHk−2oEk−1,µ(f).(5.35)

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 21

Proof. Diﬀerentiating (2.33a) in the direction of ∂e0and using the elliptic regularity, we arrive at

(cf. (5.39) in [15])

k∂e0ΨkHk≤Ck[Le0,∆g]ΨkHk−2+k∂e0divg(ΨDlog N)kHk−2

+k∂e0[Le0,divg]ωkHk−2+k∂e0(N−1J+τb

XjJj)kHk−2.(5.36)

The ﬁrst three terms are estimated in Lemma 5.9 in [15] by assuming that kDlog NkHkis small

enough and kΠkHk−2is bounded:

k[Le0,∆g]ΨkHk−2+k∂e0divg(ΨDlog N)kHk−2+k∂e0[Le0,divg]ωkHk−2

≤CkΠkHk−2+kSkHk−2+kDlog NkHk−2+kD∂e0log NkHk−2

+kLe0SkHk−2+kLe0ΠkHk−2pEk+kΨkHk+kDlog NkHk−1k∂e0ΨkHk−1.(5.37)

We estimate the last term of (5.36). To this end, we use the following relations:

∂e0N−1=−N−3∂TN+Xk∂kN,

∂e0τ=−τN −1,

∂e0Xk=N−1∂TXk+Xj∂jXk,

∂e0(τb

XjJj) = −τXkJk+τ ∂e0b

XkJk+τb

Xk∂e0Jk,

k∂e0N−1JkHk−2≤CkN−3kHk−2(k∂TNkHk−2+kXkHk−2kNkHk−1)kJkHk−2

+CkN−1kHk−2k∂e0JkHk−2,

k∂e0b

XkHk−2≤CkN−3kHk−2(k∂TNkHk−2+kNkHk−1kXkHk−2)kXkHk−2

+kNkHk−2(k∂TXkHk−2+kXkHk−1kXkHk−2),

k∂e0(τhb

X, Jig)kHk−2≤C|τ|kN−1kHk−2· k b

XkHk−2+k∂e0b

XkHk−2kJkHk−2

+C|τ|k b

XkHk−2k∂e0JkHk−2.

Then, by the smallness assumptions and boundedness of kNkHk−2, one ﬁnds

(5.38)

k∂e0(N−1J+τb

XjJj)kHk−2

≤Ck∂TNkHk−2+kb

NkHk−2+kXkHk−1kJkHk−2

+C1 + k∂TNkHk−2+k∂TXkHk−2+kb

NkHk−2+kXkHk−1|τ| · kXkHk−2· kJ kHk−2

+C|q|n|τ| · h1 + kΓ∗kHk−2+k(Σ + g)XkHk−2+|q| · |||F|||Hk−2+kXkHk−2· k b

NkHk−2

+k∂TXkHk−2+k∂TNkHk−2+|q| · |||F|||Hk−2i+kΓ∗

∗kHk−2oEk−1,µ(f).

Inserting (5.37) and (5.38) into (5.36) and assuming that kDlog NkHk−2is small enough, one

obtains the claim of the lemma.

22 H. Barzegar,D. Fajman

Proposition 5.15. For the energy deﬁned in (5.13) we have the estimate

(5.39)

∂TEk≤CkΣkHk−1+kdivgΣkHk−1+k∂e0NkHk+kN−3kHk

+kLe0ΣkHk−2+kLe0divgΣkHk−2Ek

+Ck∂TNkHk−2+kN−3kHk+k∂e0NkHk+kXkHk−1+kΣkHk−1

+kdivgΣkHk−1+kLe0ΣkHk−2+kLe0divgΣkHk−2kJkHk−2pEk

+C|τ| · kJ kHk−2pEk+C|q||τ|+kΓ∗

∗kHk−2Ek−1,µ(f)pEk,

as long as the norms in the brackets are bounded.

Proof. The boundedness of kΠkHk−2and kDlog NkHk−2together with Lemma 5.9 implies

(5.40) pEk+kΨkHk≤CpEk+kJkHk−2+|τ| · kXkHk−2kJkHk−2.

Then, combining Lemmas 5.8 and 5.14, applying the smallness assumptions, and ﬁnally assuming

the boundedness of the norms appearing in the estimate yield the proof.

Remark 5.16. Note that all terms appearing in the right-hand side of the inequality in Proposi-

tion 5.15 have good decay behaviour, i.e., they do not cause problems in the ﬁnal estimate.

6. Energy estimates from divergence identity

We deﬁne the L2-Sobolev energy of the rescaled energy density by

(6.1) ̺k:= sX

ℓ≤kZM|Dℓρ|2

gdµg.

We introduce

(6.2) b

∂T:= ∂T+LX.

Then, the divergence identity reads (cf. (G.2))

(6.3) b

∂Tρ= (3 −N)ρ+1

2τN −1Da(N2ja)−1

6τ2NgabTab −1

2τ2NΣabTab .

Moreover, we need the following identities (cf. [16] and [1])

(6.4) b

∂T, DiDj1. . . Djmu=−X

1≤a≤m

Dj1. . . Dja−1DbDja+1 . . . Djmu·b

∂TΓb

jai.

and

(6.5) ∂TZM

u dµg=−ZM

(3 −N)u dµg+ZMb

∂Tu dµg

for any function uon M. Then, one ﬁnds the following estimate for ̺k.

Proposition 6.1. For integers k > 7/2the following estimate holds

(6.6)

∂T̺k≤C(kD(Nk)kHk−2+kN−3kHk)̺k

+Cτ2

N(Σab +1

3gab)Tab

Hk+C|τ| ·

N−1divg(N2j)

Hk,

provided that |τ|Gis bounded by a constant.

Proof. The proof is analogous to the proof of Proposition 16 in [1].

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER 23

7. Elliptic Estimates for the lapse function and the shift vector

In this section we obtain elliptic estimates for the lapse function and the shift vector and their

respective time derivatives.

Proposition 7.1. The following estimates hold for the lapse function and the shift vector

kN−3kHk≤CkΣk2

Hk−2+|τ| · kρkHk−2+|τ|3· kηkHk−2,(7.1)

kXkHk≤CkΣk2

Hk−2+kg−γk2

Hk−1+|τ|·kρkHk−3+|τ|3· kηkHk−3+τ2kNjkHk−2.(7.2)

Proof. The estimates follows directly by the elliptic regularity applied to the elliptic equations for

the lapse function and the shift vector.

7.1. Estimate of the time derivatives. Further, we can estimate the time derivatives of the

lapse function and shift vector by using the elliptic estimate.

Proposition 7.2. Let k, µ ∈Nwith k > 9/2and µ≥3. For δ-small data (2.50) with δsuﬃciently

small the following estimates hold

k∂TNkHk≤Chkb

NkHk+kXkHk+kΣk2

Hk−1+kg−γk2

Hk+|τ| · kSkHk−2+|τ| · kρkHk−1

+|τ|3· kηkHk−2+τ2kjkHk−1+|τ|3· kTkHk−1+|τ|3Ek−1,µ+1(f) + τ2|||F|||2

Hk−1i,(7.3)

k∂TXkHk≤Chkb

NkHk−1+kXkHk+kΣk2

Hk−1+kg−γk2

Hk−1+|τ| · kSkHk−3+|τ| · kρkHk−2

+|τ|3· kηkHk−3+τ2kjkHk−1+|τ|3· kTkHk−1+|τ|3Ek−2,µ+1(f) + τ2|||F|||2

Hk−2i,(7.4)

where Tdenotes the spatial part of the rescaled energy-momentum tensor Tand integers µ≥3and

k > 7/2.

Proof. Diﬀerentiating the elliptic system for (N , X) and using the relations in Appendix B yields

(7.5)

∆−1

3∂TN= 2NhD2N, Σig+ 2 b

N∆N− hD2N, LXgig

+h2Dj(NΣij )−Dib

N−∆Xi+RijXjiDiN

+ 2N−3N−1

3|Σ|2

g+ 2h∇X, Σ,Σig−NhΣ,1

2Lg,γ (g−γ) + Jig

+hΣ, D2Nig− hΣ,LXΣig+Nτ hΣ, Sig

+∂TN|Σ|2

g+τρ +τ3η+N∂T(τ ρ) + ∂T(τ3η),

where h∇X, Σ,Σig≡DiXjΣi

kΣk

j, and

∆(∂TXi) + Rij∂TXj=−[∂T,∆] Xi−∂TRijXj

+ 2Dj(∂TN)Σij + 2DjN∂TΣij −∂Tgij Djb

N−1

3Di(∂TN)

+ 2∂TNτ 2ji+ 2N∂T(τ2ji)−2∂TNΣjk +N ∂TΣjk −∂TgjℓDℓXk

−Dj∂TXk−gjℓ ∂TΓk

mℓXmΓi

jk −b

Γi

jk −2NΣjk −DjXk∂TΓi

jk .(7.6)

For the lapse function the elliptic regularity by using the smallness assumptions yields

(7.7) k∂TNkHk≤Ckb

NkHk+kXkHk+kΣk2

Hk−1+kg−γk2

Hk+|τ| · kSkHk−2

+|τ| · kρkHk−2+|τ|3· kηkHk−2+|τ| · k∂TρkHk−2+|τ|3· k∂TηkHk−2

+kΣk2

Hk−2+|τ| · kρkHk−2+|τ|3· kηkHk−2k∂TNkHk−2.

Using elliptic regularity iteratively in conjunction with the smallness assumptions for both equations

yields the estimates on the time-derivatives. The procedure is straighforward and follows the

analogue case in [1].

24 H. Barzegar,D. Fajman

8. Energy Estimate For Geometric Objects

In this section we mainly adapt the results on the energy for the geometric perturbation from [3]

and [1] to the present case.

8.1. Decomposing the evolution equations. We use the following form of the evolution equa-

tions.

Lemma 8.1. The evolution equations for gand Σare equivalent to the system

∂T(g−γ) = 2NΣ + Fg−γ,(8.1a)

∂T(6Σ) = −12Σ −3NLg,γ (g−γ) + 6N τ S −Xib

Di(6Σ) + FΣ,(8.1b)

where

kFg−γkHk≤CkΣk2

Hk−1+kg−γk2

Hk+|τ| · kρkHk−2+|τ|3kηkHk−2+τ2kNjkHk−1,

kFΣkHk−1≤CkΣk2

Hk−1+kg−γk2

Hk+|τ| · kρkHk−1+|τ|3kηkHk−1+τ2kNjkHk−2.

Proof. The proof is formally identical to [1].

8.2. Energy. We deﬁne an energy for the trace-free part of the second fundamental form and the

metric perturbation. This deﬁnition depends on the lowest eigenvalue λ0of the Einstein operator

of γ. We deﬁne the constant α=α(λ0, δα) by

(8.2) α:= 1 −δα,

where

(8.3) δα:= (0, λ0>1/9,

p1−9(λ0−ε), λ0= 1/9,

with 0 < ε ≪1. Next, we deﬁne the correction constant accordingly by

(8.4) cE:= 1 −δ2

α.

Once εis ﬁxed, in case λ0= 1/9, δαcan be made suitably small, independent of the other constants

which play role in the ﬁnal estimate.

We are now able to deﬁne the energy for the geometric perturbation by

(8.5) Ek(g−γ, Σ) := X

1≤m≤k

E(m)(g−γ, Σ) := X

1≤m≤kE(m)(g−γ, Σ) + cEΓ(m)(g−γ, Σ),

where

(8.6) E(m)(g−γ, Σ) := 18 ZMhΣ,Lm−1

g,γ Σidµg+9

2ZMh(g−γ),Lm

g,γ (g−γ)idµg,

Γ(m)(g−γ, Σ) := 6 ZMhΣ,Lm−1

g,γ (g−γ)idµg,

for integers m≥1. Note that here h·,·i is deﬁned for the symmetric covariant 2-tensors uand vby

hu, vi:= uij vkℓ γik γjℓ .

Lemma 8.2. For the integers k > 5/2, there exists a constant C > 0such that for δ-small data

(2.50) with δsuﬃciently small the inequality

(8.7) kg−γk2

Hk+kΣk2

Hk−1≤CEk(g−γ, Σ)

holds.

Proof. For the proof we refer to Lemma 19 in [1].

STABLE COSMOLOGIES WITH COLLISIONLESS CHARGED MATTER