Cosmological Newtonian limits on long time scales

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by  Chao Liu

COSMOLOGICAL NEWTONIAN LIMITS ON LONG TIME SCALES
CHAOLIUANDTODDA.OLIYNYK
Abstract. We establish the existence of 1-parameter families of ǫ-dependent solutions to the Einstein-Euler
equationswithapositivecosmologicalconstantΛ>0andalinearequationofstatep=ǫ2Kρ,0<K≤1/3,for
theparametervalues 0<ǫ<ǫ0. Thesesolutionsexistgloballytothefuture,convergeasǫց0tosolutionsof
thecosmologicalPoison-EulerequationsofNewtoniangravity,andareinhomogeneousnon-linearperturbations
7 ofFLRWfluidsolutions.
1
0
2
1. Introduction
n
a
GravitatingrelativisticperfectfluidsaregovernedbytheEinstein-Eulerequations. Thedimensionlessversion
J
of these equations with a cosmological constant Λ are given by
4
1 G˜µν +Λg˜µν =T˜µν, (1.1)

] ˜µT˜µν =0, (1.2)
c ∇
q where G˜µν is the Einstein tensor of the metric
-
r g˜=g˜ dx¯µdx¯ν,
g µν
[ and

1 T˜µν =(ρ¯+p¯)v˜µv˜ν +p¯g˜µν
v
is the perfect fluid stress-energy tensor. Here, ρ¯ and p¯denote the fluid’s proper energy density and pressure,
5
7 respectively, while v˜ν is the fluid four-velocity, which we assume is normalized by
9 v˜µv˜ = 1. (1.3)
3 µ −
0 In this article, we assume a positive cosmological constant Λ > 0 and restrict our attention to barotropic
1. fluids with a linear equation of state of the form
0 1
7 p¯=ǫ2Kρ¯, 0<K .
≤ 3
1
: The dimensionless parameter ǫ can be identified with the ratio
v
v
T
i ǫ= ,
X c
r wherecisthespeedoflightandvT isacharacteristicspeedassociatedtothefluid. Understandingthebehavior
a
of solutions to (1.1)-(1.2) in the limit ǫ 0 is known as the Newtonian limit, which has been the subject of
ց
many investigations. Most work in this subject has been carried out in the setting of isolated systems and has
almost exclusively involved formal calculations, see [1, 2, 8, 11, 12, 13, 21, 34, 35, 36] and references therein,
with a few exceptions [45, 46, 55] where rigorous results were established. Due to questions surrounding the
physicalinterpretationoflargescalecosmologicalsimulationsusingNewtoniangravityandtheroleofNewtonian
gravity in cosmological averaging, interest in the Newtonian limit and the related Post-Newtonian expansions
hasshiftedfromtheisolatedsystemssettingtothecosmologicalone. Heretoo,themajorityofresultsarebased
on formal calculations [6, 9, 14, 22, 23, 28, 26, 27, 31, 32, 42, 43, 44, 53, 67] with the articles [47, 48, 49, 50]
being the only exceptions where rigorous results have been obtained.
From a cosmological perspective, the most important family of solutions to the system (1.1)-(1.2) are the
Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) solutions that represent a homogenous, fluid filled universe
undergoing accelerated expansion. Letting (x¯i), i = 1,2,3, denote the standard periodic coordinates on the
3-torus1 T3 and t=x¯0 a time coordinate on the interval (0,1], the FLRW solutions on the manifold
ǫ
M =(0,1] T3
ǫ × ǫ
1Here,Tn=[0,ǫ]n/∼where∼isequivalence relationthatfollowsfromtheidentification ofthesidesofthebox[0,ǫ]n. When
ǫ
ǫ=1,wewillsimplywriteTn.
1
2 CHAOLIUANDTODDA.OLIYNYK
are defined by
3
h˜(t)= dtdt+a(t)2δ dx¯idx¯j, (1.4)
−Λt2 ij
Λ
v˜ (t)= t ∂ , (1.5)
H t
− 3
r
ρ (1)
H
ρ (t)= , (1.6)
H a(t)3(1+ǫ2K)
where ρ (1) is the initial density (freely specifiable) and a(t) satisfies
H
3 Λ ρ (t)
ta′(t)=a(t) + H , a(1)=1. (1.7)
− Λ 3 3
r r
Remark 1.1. Throughout this article, we take the homogeneous initial density ρ (1) to be independent of ǫ.
H
All of the results establishedin this article remaintrue if ρ (1)is allowedto depend onǫ in a C1 manner, that
H
is the map [0,ǫ ) ǫ ρǫ (1) R is C1 for some ǫ >0.
0 ∋ 7−→ H ∈ >0 0
Remark 1.2. The representation (1.4)-(1.6) of the FLRW solutions is not the standard one due to the choice
of time coordinate that compactifies the time interval from [0, ) in the standard presentation to (0,1] in the
∞
coordinates used here. Letting τ denote the standard time coordinate, the relationship between the two time
coordinates is
t=e−√Λ3τ. (1.8)
Due to our choice of time coordinate, the future lies in the direction of decreasing t and timelike infinity is
located at t=0.
Remark 1.3. As we show in 2.1, FLRW solutions a,ρ depend regularly on ǫ and have well defined
H
§ { }
Newtonian limits. Letting
˚a= lima and ˚ρ = limρ (1.9)
H H
ǫց0 ǫց0
denote the Newtonian limit of a and ρ , respectively, it then follows from (1.6) and (1.7) that ˚a,˚ρ satisfy
H H
{ }
˚ρ (1)
H
˚ρ =
H ˚a(t)3
and
3 Λ ˚ρ (t)
t˚a′(t)=˚a(t) + H , ˚a(1)=1,
− Λ 3 3
r r
which define the Newtonian limit of the FLRW equations.
In the articles [47, 48], the second author established the existence of 1-parameter families of solutions2
g˜µν,ρ¯ ,v˜µ , 0 < ǫ < ǫ , to (1.1)-(1.2), which include the above family of FLRW solutions, on spacetime
{ ǫ ǫ ǫ} 0
regions of the form
(T ,1] T3 M ,
1 × ǫ ⊂ ǫ
for some T (0,1], that converge, in a suitable sense, as ǫ 0 to solutions of the cosmological Poisson-Euler
1
∈ ց
equations of Newtonian gravity. Although this result rigorously established the existence of a wide class of
solutions to the Einstein-Eulerequations thatadmit a (cosmological)Newtonianlimit, the local-in-timenature
of the result left open the question of what happens on long time scales. In particular, the question of the
existence of 1-parameter families of solutions that converge globally to the future as ǫ 0 was not addressed.
ց
In lightof the importance of Newtoniangravityin cosmologicalsimulations [10, 15, 62, 63], this questionneeds
toberesolvedinordertounderstandonwhattimescalesNewtoniancosmologicalsimulationscanbetrustedto
approximaterelativisticcosmologies. Inthisarticle,weresolvethisquestionunderasmallinitialdatacondition.
Informally, we establish the existence of 1-parameter families of ǫ-dependent solutions to (1.1)-(1.2) that: (i)
are defined for ǫ (0,ǫ ) for some fixed constant ǫ >0, (ii) exist globally on M , (iii) converge, in a suitable
0 0 ǫ
∈
sense,asǫ 0tosolutionsofthecosmologicalPoison-EulerequationsofNewtoniangravity,and(iv) aresmall,
ց
non-linear perturbations of the FLRW fluid solutions (1.4)-(1.7). The precise statement of our results can be
found in Theorem 1.6.
2Toconvertthe1-parametersolutionstotheEinstein-Eulerequationsfrom[47,48]tosolutionsof (1.1)-(1.2),themetric,four-
velocity,timecoordinateandspatialcoordinatesmusteachberescaledbyanappropriatepowersofǫ,afterwhichtherescaledtime
coordinatemustbetransformedaccordingtotheformula(1.8).
COSMOLOGICAL NEWTONIAN LIMITS ON LONG TIME SCALES 3
Before proceeding with the statement of Theorem 1.6, we first fix our notation and conventions, and define
a number of new variables and equations.
1.1. Notation.
1.1.1. Indices and coordinates. Unlessstatedotherwise,ourindexingconventionwillbeasfollows: weuselower
case Latin letters, e.g. i,j,k, for spatial indices that run from 1 to n, and lower case Greek letters, e.g. α,β,γ,
forspacetimeindicesthatrunfrom0ton. Whenconsideringthe Einstein-Eulerequations,wewillfocusonthe
physical case where n=3, while all of the PDE results established in this article hold in arbitrary dimensions.
Wewillrefertotheglobalcoordinates(x¯µ)onM definedaboveasrelativistic coordinates. Inordertodiscuss
ǫ
the Newtonian limit and the sense in which solutions converge as ǫ 0, we need to introduce the spatially
ց
rescaled coordinates (xµ) defined by
t=x¯0 =x0 and x¯i =ǫxi, ǫ>0,
which we refer to as Newtonian coordinates. The Newtonian coordinate define a global coordinate system on
M :=M =(0,1] Tn.
1
×
For scalar functions f(t,x¯i) of the relativistic coordinates, we let
f(t,xi):=f(t,ǫxi) (1.10)
denote the representation of f in Newtonian coordinates.
1.1.2. Derivatives. Partial derivatives with respect to the Newtonian coordinates (xµ) = (t,xi) and the rela-
tivistic coordinates (x¯µ) = (t,x¯i) will be denoted by ∂ = ∂/∂xµ and ∂¯ = ∂/∂x¯µ, respectively, and we use
µ µ
Du = (∂ u) and ∂u = (∂ u) to denote the spatial and spacetime gradients, respectively, with respect to the
j µ
Newtonian coordinates, and similarly ∂¯u = (∂¯ u) to denote the spacetime gradient with respect to the rela-
µ
tivistic coordinates. We also use Greek letters to denote multi-indices, e.g. α = (α ,α ,...,α ) Zn , and
1 2 n ∈ ≥0
employ the standard notation Dα =∂α1∂α2 ∂αn for spatial partial derivatives. It will be clear from context
1 2 ··· n
whether a Greek letter stands for a spacetime coordinate index or a multi-index.
Given a vector-valued map f(u), where u is a vector, we use Df and D f interchangeably to denote the
u
derivative with respect to the vector u, and use the standard notation
d
Df(u) δu:= f(u+tδu)
· dt
(cid:12)t=0
(cid:12)
for the action of the linear operator Df on the vector (cid:12)δu. For vector-valued maps f(u,v) of two (or more)
(cid:12)
variables, we use the notation D f and D f interchangeably for the partial derivative with respect to the first
1 u
variable, i.e.
d
D f(u,v) δu:= f(u+tδu,v),
u
· dt
(cid:12)t=0
and a similar notation for the partial derivative with resp(cid:12)ect to the other variable.
(cid:12)
(cid:12)
1.1.3. Function spaces. Givena finite dimensionalvectorspaceV,we letHs(Tn,V), s Z , denote the space
≥0
of maps from Tn to V with s derivatives in L2(Tn). When the vector space V is clear∈from context, we write
Hs(Tn) instead of Hs(T,V). Letting
u,v = (u(x),v(x))dnx,
h i ZTn
where (, ) is a fixed inner product on V, denote the standard L2 inner product, the Hs norm is defined by
· ·
u 2 = Dαu,Dαu .
k kHs h i
0≤X|α|≤s
For any fixed basis e N of V, we follow [47] and define a subspace of Hs(Tn,V) by
{ I}I=1
N
H¯s(Tn,V)= u(x)= uI(x)e Hs(Tn,V) 1,uI =0 for I =1,2, ,N .
I
∈ h i ···
(cid:26) I=1 (cid:12) (cid:27)
X (cid:12)
Specializing to n=3, we define, for fixed ǫ >0 and r >0, the(cid:12)spaces
0 (cid:12)
Xs (T3)=( ǫ ,ǫ ) B (Hs+1(T3,S )) Hs(T3,S ) B (H¯s(T3)) H¯s(T3,R3),
ǫ0,r − 0 0 × r 3 × 3 × r ×
where S denotes the space of symmetric N N matrices.
N
×
4 CHAOLIUANDTODDA.OLIYNYK
To handle to smoothness of coefficients that appear in various equations, we introduce the spaces
Ep((0,ǫ ) (T ,T ) U,V), p Z ,
0 1 2 ≥0
× × ∈
whicharedefinedtobethesetofV-valuedmapsf(ǫ,t,ξ)thataresmoothontheopenset(0,ǫ ) (T ,T ) U,
0 1 2
whereU Tn RN isopen,andforwhichthereexistconstantsC >0,(k,ℓ) 0,1,...,p Z× , such×that
k,ℓ ≥0
⊂ × ∈{ }×
∂ kDℓf(ǫ,t,ξ) C , (ǫ,t,ξ) (0,ǫ ) (T ,T ) U.
| t ξ |≤ k,ℓ ∀ ∈ 0 × 1 2 ×
If V =R or V clearfromcontext,we willdropthe V andsimply write Ep((0,ǫ ) (T ,T ) U). Moreover,we
0 1 2
× ×
willuse the notationEp((T ,T ) U,V)to denotethe subspaceofǫ-independentmaps. Givenf Ep((0,ǫ )
1 2 0
× ∈ ×
(T ,T ) U,V), we note, by uniform continuity, that the limit f (t,ξ):=lim f(ǫ,t,ξ) exists and defines an
1 2 0 ǫց0
×
element of Ep((T ,T ) U,V).
1 2
×
1.1.4. Constants. We employ that standard notation
a.b
for inequalities of the form
a Cb
≤
in situations where the precise value or dependence on other quantities of the constant C is not required. On
the other hand, when the dependence of the constant on other inequalities needs to be specified, for example if
the constant depends on the norms u L∞ and v L∞, we use the notation
k k k k
C =C( u L∞, v L∞).
k k k k
Constants of this type will always be non-negative, non-decreasing, continuous functions of their arguments,
and in general, C will be used to denote constants that may change from line to line. However, when we want
to isolatea particularconstantfor use later on,we willlabel the constantwith a subscript, e.g. C ,C ,C , etc.
1 2 3
1.1.5. Remainderterms. Inordertosimplifythehandlingofremaindertermswhoseexactformisnotimportant,
we will use upper case calligraphic letters, e.g. (ǫ,t,x,ξ), (ǫ,t,x,ξ) and (ǫ,t,x,ξ), to denote vector valued
maps that, for some ǫ ,R>0 and N Z , arSe elements oTf the space E1 (U0,ǫ ) (0,2) Tn B RN .
0 ≥1 0 R
∈ × × ×
1.2. Conformal Einstein-Euler equations. The method we use to es(cid:0)tablish the existence of ǫ-(cid:0)depe(cid:1)n(cid:1)dent
familiesofsolutionstotheEinstein-Eulerequationsthatexistgloballytothefutureisbasedontheonedeveloped
in [51]. Following [51], we introduce the conformal metric
g¯µν =e2Ψg˜µν (1.11)
and the conformal four velocity
v¯µ =eΨv˜µ. (1.12)
Under this change of variables, the Einstein equation (1.1) transforms as
G¯µν =T¯µν :=e4ΨT˜µν e2ΨΛg¯µν +2(¯µ¯νΨ ¯µΨ¯νΨ) (22¯Ψ+ ¯Ψ2)g¯µν, (1.13)
− ∇ ∇ −∇ ∇ − |∇ |g¯
where 2¯ = ¯µ¯ , ¯Ψ2 = g¯µν¯ Ψ¯ Ψ, and here and in the following, unless otherwise specified, we raise
∇ ∇µ |∇ |g¯ ∇µ ∇ν
and lower all coordinate tensor indices using the conformal metric. Contracting the free indices of (1.13) gives
R¯ =4Λ T¯,
−
where T¯ =g¯ T¯µν, which we can use to write (1.13) as
µν
1 ǫ2K
2R¯µν = 4¯µ¯νΨ+4¯µΨ¯νΨ 2 2¯Ψ+2 ¯Ψ2+ − ρ¯+Λ e2Ψ g¯µν
− − ∇ ∇ ∇ ∇ − |∇ | 2
(cid:20) (cid:18) (cid:19) (cid:21)
2e2Ψ(1+ǫ2K)ρ¯v¯µv¯ν. (1.14)
−
We will refer to these equations as the conformal Einstein equations.
Letting Γ˜γ and Γ¯γ denote the Christoffel symbols of the metrics g˜ and g¯ , respectively, the difference
µν µν µν µν
Γ˜γ Γ¯γ is readily calculated to be
µν − µν
Γ˜γ Γ¯γ =g¯γα g¯ ¯ Ψ+g¯ ¯ Ψ g¯ ¯ Ψ .
µν − µν µα∇ν να∇µ − µν∇α
Using this, we can express the Euler equations(cid:0)(1.2) as (cid:1)
¯ T˜µν = 6T˜µν Ψ+g¯ T˜αβg¯µν¯ Ψ, (1.15)
µ µ αβ µ
∇ − ∇ ∇
which we refer to as the conformal Euler equations.
COSMOLOGICAL NEWTONIAN LIMITS ON LONG TIME SCALES 5
Remark 1.4. Due to our choice of time orientation, the conformal fluid four-velocity v¯µ, which we assume is
future oriented, satisfies
v¯0 <0.
We also note that v¯µ is normalized by
v¯µv¯ = 1, (1.16)
µ
−
which is a direct consequence of (1.3), (1.11) and (1.12).
1.3. Conformal factor. Following [51], we choose
Ψ= lnt (1.17)
−
for the conformal factor, and for later use, we introduce the backgroundmetric
3
h¯ = dtdt+E2(t)δ dx¯idx¯j, (1.18)
ij
−Λ
where
E(t)=a(t)t, (1.19)
which is conformally related to the FLRW metric (1.4). Using (1.7), we observe that E(t) satisfies
1
∂ E(t)= E(t)Ω(t), (1.20)
t
t
where Ω(t) is defined by
3 Λ ρ (t)
H
Ω(t)=1 + . (1.21)
− Λ 3 3
r r
A short calculation then shows that the non-vanishing Christoffel symbols of the background metric (1.18) are
given by
Λ 1
γ¯0 = E2Ωδ and γ¯i = Ωδi, (1.22)
ij 3t ij j0 t j
from which we compute
Λ
γ¯σ :=h¯µνγ¯σ = Ωδσ. (1.23)
µν t 0
1.4. Wave gauge. For the hyperbolic reduction of the conformal Einstein equations, we use the wave gauge
from [51] that is defined by
Z¯µ =0, (1.24)
where
Z¯µ =X¯µ+Y¯µ (1.25)
with
1 Λ
X¯µ :=Γ¯µ γ¯µ = ∂¯ g¯µν + g¯µσg¯ ∂¯ g¯αβ Ωδµ Γ¯µ =g¯σνΓ¯µ (1.26)
− − ν 2 αβ σ − t 0 σν
(cid:0) (cid:1)
and
2Λ
Y¯µ := 2¯µΨ+ δµ = 2(g¯µν h¯µν)¯ Ψ. (1.27)
− ∇ 3t 0 − − ∇ν
6 CHAOLIUANDTODDA.OLIYNYK
1.5. Variables. Toobtainvariablesthataresimultaneouslysuitableforestablishingglobalexistenceandtaking
Newtonian limits, we switch to Newtonian coordinates (xµ)=(t,xi) and employ the following rescaled version
of the variables introduced in [51]:
1g¯0µ η¯0µ
u0µ = − , (1.28)
ǫ 2t
1 3(g¯0µ η¯0µ)
u0µ = ∂¯ g¯0µ − , (1.29)
0 ǫ 0 − 2t !
1
u0µ = ∂¯g¯0µ, (1.30)
i ǫ i
1
uij(t,x)= ¯gij η¯ij), (1.31)
ǫ −
1(cid:16)
uij = ∂¯ ¯gij, (1.32)
µ ǫ µ
1
u= ¯q, (1.33)
ǫ
1
u = ∂¯ ¯q, (1.34)
µ µ
ǫ
1
z = v¯, (1.35)
i i
ǫ
1
ζ = ln t−3(1+ǫ2K)ρ¯ , (1.36)
1+ǫ2K
(cid:0) (cid:1)
and
δζ =ζ ζ , (1.37)
H
−
where
¯gij =α−1g¯ij, α:=(detgˇij)−13 =(detg¯kl)13, gˇij =(g¯ij)−1, (1.38)
Λ 2Λ
¯q=g¯00 η¯00 lnα lnE, (1.39)
− − 3 − 3
Λ
η¯µν = δµδν +δµδνδij, (1.40)
−3 0 0 i j
and
1
ζ (t)= ln t−3(1+ǫ2K)ρ (t) . (1.41)
H 1+ǫ2K H
As we show below in 2.1, ζ is given by the explicit fo(cid:0)rmula (cid:1)
H
§
2 C t3(1+ǫ2K)
0
ζ (t)=ζ (1) ln − , (1.42)
H H − 1+ǫ2K C0 1 !
−
where the constants C and ζ (1) are defined by
0 H
Λ+ρ (1)+√Λ
H
C = >1 (1.43)
0
Λ+ρ (1) √Λ
p H
−
and p
1
ζ (1)= lnρ (1), (1.44)
H 1+ǫ2K H
respectively. Letting
˚ζ = limζ (1.45)
H H
ǫց0
denote the Newtonian limit of ζ , it is clear from the formula (1.42) that
H
C t3
˚ζ (t)=lnρ (1) 2ln 0− . (1.46)
H H
− C 1
(cid:18) 0− (cid:19)
COSMOLOGICAL NEWTONIAN LIMITS ON LONG TIME SCALES 7
For later use, we also define
1
zi = v¯i. (1.47)
ǫ
Remark 1.5. Itis importantto emphasize that the abovevariablesaredefined onthe ǫ-independent manifold
M = (0,1] T3. Effectively, we are treating components of the geometric quantities with respect to the
×
relativistic coordinates as scalarsdefined on M and we are pulling them back as scalars to M by transforming
ǫ
to Newtonian coordinates. This process is necessary to obtain variables that have a well defined Newtonian
limit. We stress that for any fixed ǫ>0, the gravitational and matter fields g¯µν,v¯µ,ρ¯ on M are completely
ǫ
{ }
determined by the fields u0µ,uij,u,z ,ζ on M.
i
{ }
1.6. Conformal Poisson-Euler equations. Theǫ 0limitoftheconformalEinstein-EulerequationsonM
ց
are the conformal cosmological Poisson-Euler equations, which are defined by
3 3(1 ˚Ω)
∂ ˚ρ+ ∂ ˚ρ˚zj = − ˚ρ, (1.48)
t j
Λ t
r
Λ (cid:0) (cid:1) Λ1 1 3 δji
˚ρ∂˚zj +K∂j˚ρ+˚ρ˚zi∂˚zj = ˚ρ˚zj ˚ρ∂j˚Φ ∂j := ∂ , (1.49)
r3 t i r3 t − 2Λ E˚2 i
(cid:16) (cid:17)
ΛE˚2
∆˚Φ= Π˚ρ (∆:=δij∂ ∂ ), (1.50)
3 t2 i j
where Π is the projection operator defined by
Πu=u 1,u , (1.51)
−h i
for u L2(T3),
∈
2
C t3 3
E˚(t)= 0− (1.52)
C 1
(cid:18) 0− (cid:19)
and
2t3
˚Ω(t)= , (1.53)
t3 C
0
−
with C given by (1.43).
0
It will be important for our analysis to introduce the modified density variable˚ζ defined by
˚ζ =ln(t−3˚ρ),
whichisthenon-relativisticversionofthevariableζ introducedabove,see(1.36). Ashortcalculationthenshows
that the conformal cosmological Poisson-Euler equations can be expressed in terms of this modified density as
follows:
3 3˚Ω
∂ ˚ζ+ ˚zj∂ ˚ζ+∂ ˚zj = , (1.54)
t j j
Λ − t
r
Λ (cid:0) (cid:1) Λ1 1 3
∂˚zj+˚zi∂˚zj +K∂j˚ζ = ˚zj ∂j˚Φ, (1.55)
t i
3 3 t − 2Λ
r r
∆˚Φ= ΛtE˚2Πe˚ζ. (1.56)
3
1.7. Main Theorem. We are in the position to state the main theorem of the article. The proof is given in
7.
§
Theorem 1.6. Suppose s Z , 0<K 1, Λ>0, ρ (1)>0, r >0 and the free initial data u˘ij,u˘ij,ρ˘ ,ν˘i
∈ ≥3 ≤ 3 H { 0 0 }
is chosen so that u˘ij B (Hs+1(T3,S )), ˘uij Hs(T3,S ), ρ˘ B (H¯s(T3)), ν˘i H¯s(T3,R3). Then for
r > 0 chosen small e∈nougrh, there exis3ts a c0on∈stant ǫ >30 an0d∈marps u˘µν Cω X∈s (T3),Hs+1(T3,S ) ,
0 ∈ ǫ0,r 4
u˘ Cω Xs (T3),Hs+1(T3) , u˘µν Cω Xs (T3),Hs(T3,S ) , u˘ Cω Xs (T3),Hs(T3) , z˘ = (z˘)
∈ ǫ0,r 0 ∈ ǫ0,r 4 0 ∈ ǫ0,r (cid:0) i (cid:1)∈
Cω Xs (T3),Hs(T3,R3 , and δζ˘ Cω Xs (T3),Hs(T3) , such that3
ǫ0,(cid:0)r (cid:1) ∈ (cid:0)ǫ0,r (cid:1) (cid:0) (cid:1)
(cid:0) uµ(cid:1)0(cid:1) :=u˘µ0(ǫ,˘u(cid:0)kl,˘ukl,ρ˘ ,ν˘k)=ǫΛ(cid:1)∆−1ρ˘ δµ+O(ǫ2),
|t=1 0 0 6 0 0
3SeeLemma6.7fordetails.
8 CHAOLIUANDTODDA.OLIYNYK
1
uij :=u˘ij(ǫ,˘ukl,˘ukl,ρ˘ ,ν˘k)=ǫ2 ˘uij u˘pqδ δij +O(ǫ3),
|t=1 0 0 − 3 pq
(cid:18) (cid:19)
2Λ
u :=u˘(ǫ,˘ukl,u˘kl,ρ˘ ,ν˘k)=ǫ2 u˘ijδ +O(ǫ3),
|t=1 0 0 9 ij
ν˘jδ
z :=z˘(ǫ,˘ukl,˘ukl,ρ˘ ,ν˘k)= ij +O(ǫ),
i|t=1 i 0 0 ρ (1)+ρ˘
H 0
ρ˘
δζ :=δζ˘(ǫ,u˘kl,u˘kl,ρ˘ ,ν˘k)=ln 1+ 0 +O(ǫ2),
|t=1 0 0 ρ (1)
(cid:18) H (cid:19)
u :=u˘ (ǫ,u˘ij,˘uij,ρ˘ ,ν˘i)=O(ǫ)
0|t=1 0 0 0
and
uµν :=u˘µν(ǫ,˘ukl,˘ukl,ρ˘ ,ν˘k)=O(ǫ)
0 |t=1 0 0 0
determineviatheformulas (1.28),(1.29),(1.31), (1.33), (1.35),(1.36),and (1.37)asolutionofthegravitational
and gauge constraint equations, see (6.3)-(6.4) and Remark 6.1. Furthermore, there exists a σ >0, such that if
ku˘ijkHs+1 +ku˘i0jkHs +kρ˘0kHs +kν˘ikHs ≤σ,
then there exist maps
uµν C0((0,1],Hs(T3,S )) C1((0,1],Hs−1(T3,S )),
ǫ ∈ 4 ∩ 4
uµν C0((0,1],Hs(T3,S )) C1((0,1],Hs−1(T3,S )),
γ,ǫ ∈ 4 ∩ 4
u C0((0,1],Hs(T3)) C1((0,1],Hs−1((T3)),
ǫ
∈ ∩
u C0((0,1],Hs(T3)) C1((0,1],Hs−1((T3)),
γ,ǫ
∈ ∩
δζ C0((0,1],Hs(T3)) C1((0,1],Hs−1(T3)),
ǫ
∈ ∩
zǫ C0((0,1],Hs(T3),R3)) C1((0,1],Hs−1(T3,R3)),
i ∈ ∩
for ǫ (0,ǫ ), and
0
∈
˚Φ C0((0,1],Hs+2(T3)) C1((0,1],Hs+1(T3)),
∈ ∩
δ˚ζ C0((0,1],Hs(T3)) C1((0,1],Hs−1(T3)),
∈ ∩
˚z C0((0,1],Hs(T3,R3)) C1((0,1],Hs−1(T3,R3)),
i
∈ ∩
such that
(i) uµν(t,x),u (t,x),δζ (t,x),zǫ(t,x) determines, via (1.11), (1.12), (1.16), (1.28), (1.31), (1.33), (1.35),
{ ǫ ǫ ǫ i }
and (1.36)-(1.40), a1-parameter family ofsolutions totheEinstein-Euler equations (1.1)-(1.2)inthewave
gauge (1.24) on M ,
ǫ
(ii) ˚Φ(t,x),˚ζ(t,x) := δ˚ζ +˚ζ ,˚zi(t,x) := E˚(t)−2δij˚z (t,x) , with ˚ζ and E˚ given by (1.46) and (1.52),
H j H
{ }
respectively, solves the conformal cosmological Poisson-Euler equations (1.54)-(1.56) on M with initial
data ˚ζ =ln(ρ (1)+ρ˘ ) and˚zi =ν˘i/(ρ (1)+ρ˘ ),
t=1 H 0 t=1 H 0
| |
(iii) the uniform bounds
kδ˚ζkL∞((0,1],Hs)+k˚ΦkL∞((0,1],Hs+2)+k˚zjkL∞((1,0]×Hs)+kδζǫkL∞((0,1],Hs)+k˚zjǫkL∞((0,1]×Hs) .1
and
kuµǫνkL∞((1,0],Hs)+kuµγ,νǫkL∞((0,1],Hs)+kuǫkL∞((0,1],Hs)+kuγ,ǫkL∞((0,1],Hs) .1,
hold for ǫ (0,ǫ ),
0
∈
(iv) and the uniform error estimates
kδζǫ−δ˚ζkL∞((0,1],Hs−1)+kzjǫ−˚zjkL∞((1,0]×Hs−1) .ǫ,
kuµǫ,ν0kL∞((1,0],Hs−1)+kuµk,νǫ−δ0µδ0ν∂k˚ΦkL∞((0,1],Hs−1)+kuµǫνkL∞((0,1],Hs−1) .ǫ
and
kuγ,ǫkL∞((0,1],Hs−1)+kuǫkL∞((0,1],Hs−1) .ǫ
COSMOLOGICAL NEWTONIAN LIMITS ON LONG TIME SCALES 9
hold for ǫ (0,ǫ ).
0
∈
1.8. Future directions. Although the 1-parameter families of ǫ-dependent solutions to the Einstein-Euler
equations from Theorem 1.6 do provide a positive answer to the question of the existence of non-homogeneous
relativisticcosmologicalsolutionsthataregloballyapproximatedtothefuturebysolutionsofNewtoniangravity,
it does not resolve the question for initial data that is relevant to our Universe. This is because these solutions
haveacharacteristicsize ǫ andshouldbe interpretedascosmologicalversionsofisolatedsystems[23,49,50].
∼
This defect was remedied on shorttime scales in [50]. There the local-in-time existence of 1-parameterfamilies
ofǫ-dependentsolutionsto the Einstein-Eulerequationsthat convergeto solutionsofthe cosmologicalPoisson-
Euler equations on cosmological spatial scales was established.
Inworkthatiscurrentlyinpreparation[38],wecombinethetechniquesdevelopedin[50]withageneralization
of the ones developed in this article to extend the local-in-time existence result from [50] to a global-in-time
result. This resolves the existence question of non-homogeneous relativistic cosmological solutions that are
globallyapproximatedtothe future oncosmologicalscalesby solutionsofNewtoniangravity,atleastforinitial
data that is a small perturbation of FLRW initial data. However, this is far from the end of the story because
there arerelativistic effects thatareimportantfor precisioncosmologythatarenotcapturedby the Newtonian
solutions. To understand these relativistic effects, higher order post-Newtonian (PN) expansions are required
startingwith the 1/2-PNexpansion,whichis, by definition, the ǫ ordercorrectionto the Newtoniangravity. In
particular, it can be shown [52] that the 1-parameter families of solutions must admit a 1/2-PN expansion in
order to view them on large scales as a linear perturbation of FLRW solutions. The importance of this result
is that it shows it is possible to have rigorous solutions that fit within the standard cosmological paradigm of
linearperturbationsofFLRWmetricsonlargescaleswhile,atthesametime,arefullynon-linearonsmallscales
of order ǫ. Thus the natural next step is to extend the results of [38] to include the existence of 1-parameter
families of ǫ-dependent solutions to the Einstein-Euler equations that admit 1/2-PNexpansions globallyto the
future on cosmological scales. This is work that is currently in progress.
1.9. Priorand related work. Thefuturenon-linearstabilityoftheFLRWfluidsolutionsforalinearequation
ofstatep=Kρwasfirstestablishedunderthecondition0<K <1/3andtheassumptionofzerofluidvorticity
byRodnianskiandSpeckin[58]usingageneralizationofawavebasedmethoddevelopedbyRingstr¨omin[56].
Subsequently, it has been shown [19, 24, 40, 61] that this future non-linear stability result remains true for
fluids with non-zero vorticity and also for the equation of state parameter values K = 0 and K = 1/3, which
correspond to dust and pure radiation, respectively. It is worth noting that the stability results established
in [40] and [19] for K = 1/3 and K = 0, respectively, rely on Friedrich’s conformal method [17, 18], which is
completely different from the techniques used in [24, 58, 61] for the parameter values 0 K <1/3.
≤
In the Newtonian setting, the global existence to the future of solutions to the cosmological Poisson-Euler
equations was established in [4] under a small initial data assumptionand for a class of polytropic equations of
state.
Anewmethodwasintroducedin[51]toprovethefuturenon-linearstabilityoftheFLRWfluidsolutionsthat
was based on a wave formulation of a conformal version of the Einstein-Euler equations. The global existence
results in this article are established using this approach. We also note that this method was recently used to
establish the non-linear stability of the FLRW fluid solutions that satisfy the generalized Chaplygin equation
of state [37].
1.10. Overview. In 2,weemploythevariables(1.28)-(1.37)andthewavegauge(1.24)towritetheconformal
§
Einstein-Eulersystem,givenby(1.14)and(1.15),asanon-localsymmetrichyperbolicsystem,see(2.103),that
is jointly singular in ǫ and t.
In 3,westateandprovealocal-in-timeexistenceanduniquenessresultalongwithacontinuationprinciplefor
§
solutionsofthereducedconformalEinstein-Eulerequationsanddiscusshowsolutionstothereducedconformal
Einstein-Euler equations determine solutions to the singular system (2.103). Similarly, in 4, we state and
§
prove a local-in-time existence and uniqueness result and continuation principle for solutions of the conformal
cosmologicalPoisson-Euler equations (1.54)-(1.56).
We establish in 5 uniform a priori estimates for solutions to a class of symmetric hyperbolic equations that
§
arejointlysingularinǫandt,andincludeboththeformulation(2.103)oftheconformalEinstein-Eulerequations
and the ǫ 0 limit of these equations. We also establish error estimates, that is, a priori estimates for the
ց
difference between solutions of the singular hyperbolic equation and the corresponding ǫ 0 limit equation.
ց
10 CHAOLIUANDTODDA.OLIYNYK
In 6,weconstructǫ-dependent1-parameterfamiliesofinitialdataforthe reducedconformalEinstein-Euler
§
equations that satisfy the constraint equations on the initial hypersurface t=1 and limit as ǫ 0 to solutions
ց
of the conformal cosmologicalPoisson-Euler equations.
Using the results from 2 to 6, we complete the proof of Theorem 1.6 in 7.
§ § §
2. A singular symmetric hyperbolic formulation of the conformal Einstein-Euler equations
Inthissection,weemploythevariables(1.28)-(1.37)andthewavegauge(1.24)tocasttheconformalEinstein-
Euler system, given by (1.14) and (1.15), into a form that is suitable for analyzing the limit ǫ 0 globally to
ց
the future. More specifically, we show that the Einstein-Euler system can be written as symmetric hyperbolic
system that is jointly singular in ǫ and t, and for which the singular terms have a specific structure. Crucially,
the ǫ-dependent singular terms are of a form that has been well-studied beginning with the pioneering work of
Browning, Klainerman, Kreiss and Majda [5, 29, 30, 33], while the t-dependent singular terms are of the type
analyzed in [51].
2.1. Analysis of the FLRW solutions. As a first step in the derivation, we find explicit formulas for the
functions Ω(t), ρ (t) and E(t) that will be needed to show that the transformed conformal Einstein-Euler
H
systems is of the form analyzed in 5. We begin by differentiating (1.21) and observe, with the help of (1.6),
§
(1.19) and (1.20), that it satisfies the differential equation
3 3
t∂ (1 Ω)+ (1+ǫ2K)(1 Ω)2 = (1+ǫ2K). (2.1)
t
− − 2 − 2
Integrating gives
2t3(1+ǫ2K)
Ω(t)= , (2.2)
t3(1+ǫ2K) C
0
−
where C is as defined above by (1.43). Then by (1.21), we find that
0
4C Λt3(1+ǫ2K)
0
ρ (t)= , (2.3)
H (C t3(1+ǫ2K))2
0
−
which, in turn, shows that ζ (t), as defined by (1.41), is given by the formula (1.42).
H
It is clear from the above formulas that Ω, ρ and ζ , as functions of (t,ǫ), are in C2([0,1] [0,ǫ ])
H 0
× ∩
W3,∞([0,1] [ ǫ ,ǫ ]) for any fixed ǫ >0. In particular, we can represent t−1Ω and ∂ Ω as
0 0 0 t
× −
1
Ω=E−1∂ E =t2+3ǫ2K (t) and ∂ Ω=t2+3ǫ2K (t),
t 1 t 2
t Q Q
respectively, where we are employing the notation from 1.1.5 for the remainder terms and .
1 2
§ Q Q
Using (2.2), we can integrate (1.20) to obtain
2
t 2s2+3ǫ2K C t3(1+ǫ2K) 3(1+ǫ2K)
0
E(t)=exp ds = − >1 (2.4)
Z1 s3(1+ǫ2K)−C0 ! C0−1 !
for t [0,1]. Fromthis formula,it is clear that E C2([0,1] [ ǫ ,ǫ ]) W3,∞([0,1] [ ǫ ,ǫ ]), and that the
0 0 0 0
∈ ∈ × − ∩ × −
Newtonian limit of E, denoted E˚and defined by
E˚(t)= limE(t),
ǫց0
is given by the formula (1.52). Similarly, we denote the Newtonian limit of Ω by
˚Ω(t)= limΩ(t),
ǫց0
which we see from (2.2) is given by the formula (1.53).
For latter use, we observe that E, Ω, ρ and ζ satisfy
H H
1 1
E−1∂2E+ E−1∂ E = (1+3ǫ2K)ρ , (2.5)
− t t t 2Λt2 H
5 3
E−1∂2E+2E−2(∂ E)2 E−1∂ E = (1 ǫ2K)ρ (2.6)
t t − t t 2Λt2 − H
and
3
∂ ζ = Ω= 3E−1∂ E = γ¯i = γ¯i =t2+3ǫ2K (t) (2.7)
t H −t − t − i0 − 0i Q3