Self-similarity breaking of cosmological solutions with collisionless matter

0.2 MB·

by  Ho Lee

Self-similarity breaking of cosmological solutions with
collisionless matter
Ho Lee∗1 and Ernesto Nungesser†2
1Department of Mathematics and Research Institute for Basic Science, Kyung Hee
University, Seoul, 02447, Republic of Korea
7 2Instituto de Ciencias Matema´ticas (CSIC-UAM-UC3M-UCM), 28049 Madrid,
1
Spain
0
2

n
January 30, 2017
a
J

6
2
Abstract

]
c In this paper we consider the Einstein-Vlasov system with Bianchi VII0 symmetry. Un-
q der the assumption of small data we show that self-similarity breaking occurs for reflection
- symmetric solutions. This generalizes the previous work concerning the non-tilted fluid case
r
g [13] to theVlasov case, and we obtain detailed information about the late-time behaviour of
[ metric and matter terms.

1
v 1 Introduction
0
0
9 Due to the observations of the microwave background we know that there is electromagnetic
7 radiationallovertheplacewhichisalmostisotropic. Onlysmalltemperaturevariationsof10−5K
0 are present, which makes them slightly anisotropic. The question is whether this implies that
.
1 the spacetime has to be almost isotropic as well. If the matter distribution is exactly isotropic,
0 there is a known result, called the EGS-theorem [2], which tells you that the spacetime will also
7 be isotropic if the collision term vanishes, i.e. in the Vlasov case or if there is a so-called detailed
1
balance. These results were extended to the Boltzmann case in [12]. There are also “Almost”
:
v EGS-theorems, however a critical question is what almost means. We refer for an overview of
Xi these results to [6] and Section 4.1 of [11].
On the other hand there is a very interesting result by Wainwright, Hancock and Uggla [13]
r
a which points in a negative direction at least if there is no cosmological constant present and in
a homogeneous setting. There it is shown for the case of a perfect fluid as a matter model that
a solution of Bianchi type VII can be arbitrary close to a FLRW solution, but still can become
0
very anisotropic as regards the Weyl curvature. Here we show a similar result for collisionless
matterwithasmallmomentadispersion. Inourcaseitisasmalldataresult,whereonecurvature
variableN isverybiginitially,whichmeansthatitsinverseM,isverysmall. Theothercurvature
+
variableN− andthe shearvariablesΣ+ andΣ− aresmallinitially as well. Whatis obtainedis an
analogue to the result of [13] using the methods of [7].
Note that the Weyl curvature becomes unbounded, but there are no upper bounds to the
Weyl curvature from observations. The Weyl curvature might be of importance from theoretical
considerations concerning the initial singularity or even in a broader cosmological context [10].
Another aspect which makes this Bianchi model interesting is that there are oscillations in both
∗[email protected]
†[email protected]
1
N− andΣ−. Inorderto do the analysiswe follow the analysisof [13]which consistsin separating
oscillating and non-oscillating part.
Finally let us note that in the perfect fluid analysis, the dust case is particularly important
because a Weyl curvature bifurcation occurs for the value γ = 1 which corresponds to dust. If
the value of γ is smaller then one, then the Weyl curvature will tend to zero. For dust the Weyl
curvaturetends toaconstant,andifitisbiggerthanonethe Weylcurvaturebecomesunbounded
cf. Theorem 2.4 of [13].
2 The Einstein-Vlasov system
In this section we introduce the Einstein-Vlasov system with Bianchisymmetry. Consider a four-
dimensional oriented and time oriented Lorentzian manifold ( ,4g) and a distribution function
M
f, then the Einstein-Vlasov system is written as
G =T ,
αβ αβ
f =0,
L
where G is the Einstein tensor and T is the energy-momentum tensor defined by
αβ αβ
T = χp p .
αβ α β
ZH
Here, the integration is over the mass-shell at a given space-time point which is defined by
H
p p gαβ = 1
α β
−
formassiveparticles,andχisthedistributionfunctionmultipliedbytheLorentzinvariantmeasure
and the Liouville operator.
L
The basic equations we will use can be found in Sections 7.3–7.4 and Chapter 25 of [11]. We
also refer to this book for an introduction to the Einstein-Vlasov system. Let Σ be a spacelike
hypersurfacein withnitsfuturedirectedunitnormal. Wedefinethesecondfundamentalform
M
as k(X,Y)= n,Y for vectors X and Y tangent to Σ, where is the Levi-Civita connection
X
h∇ i ∇
of 4g. The Hamiltonian and momentum constraint equations are as follows:
R k kij +k2 =2ρ,
ij
−
j
k k = J ,
ji i i
∇ −∇ −
whereg istheinducedmetriconΣ,k =k gab thetraceofthesecondfundamentalform,Rand
ab
∇
the scalar curvature and the Levi-Civita connection of g respectively, and matter terms are given
by ρ=T nαnβ and J Xi = T nαXβ for X tangent to Σ. Here and throughout the paper we
αβ i αβ
−
assumethatGreeklettersrunfrom0to 3,while Latinletters from1to 3,andalsofollowthe sign
conventions of [11].
2.1 The Einstein-Vlasov system with Bianchi symmetry
A Bianchi spacetime is defined to be a spatially homogeneous spacetime whose isometry group
possessesathree-dimensionalsubgroupthatactssimplytransitivelyonspacelikeorbits. ABianchi
spacetimeadmitsaLiealgebraofKillingvectorfields. Thesevectorfieldsaretangenttothegroup
orbits, which are the surfaces of homogeneity. Using a left-invariantframe, the metric induced on
thespacelikehypersurfacesdependsonlyonthetimevariable. LetGbethethree-dimensionalLie
group, e a basis of the Lie algebra,and ξi the dual of e . The metric of the Bianchi spacetime in
i i
the left-invariant frame is written as
4g = dt dt+g ξi ξj
ij
− ⊗ ⊗
2
on = I G with e future oriented. We will need equations (25.17)–(25.18) of [11] (without
0
M ×
scalar field) with the notation T =S :
ab ab
g˙ =2k , (1)
ab ab
1
k˙ = R +2kik kk +S + (ρ S)g , (2)
ab − ab a bi− ab ab 2 − ab
where the dot means the derivative with respect to time t and S = gabS and R is the Ricci
ab ab
tensor associated to the induced 3-metric. Since k does not depend on spatial variables, the
constraint equations are as follows:
R k kij +k2 =2ρ, (3)
ij
−
jk = J , (4)
ji i
∇ −
where we have dropped the bar on the covariantderivative by a slight abuse of notation.
Below, we collect and derive severaluseful equations. The mixed versionof the second funda-
mental form is given by
1
k˙a = Ra kka+Sa+ (ρ S)δa, (5)
b − b − b b 2 − b
and by taking the trace of (5) we have
1 3
k˙ = R k2 S+ ρ. (6)
− − − 2 2
By the constraint equation (3) one can eliminate the energy density such that (6) reads:
1 1
k˙ = (k2+R+3k kab) S, (7)
ab
−4 − 2
and if we substitute for the Ricci scalar with (3), we obtain
1
k˙ = k kab (S+ρ). (8)
ab
− − 2
It is convenient to express the second fundamental form as
k =σ +Hg ,
ab ab ab
where σ is the trace free part and H is the Hubble parameter:
ab
1
H = k,
3
and we obtain
k kab =σ σab+3H2.
ab ab
By a simple calculation we obtain from (7)
3 R ΣbΣa S
∂ (H−1)= + + a b + , (9)
t 2 12H2 4 6H2
where we have defined
σb
Σb = a.
a H
3
Moreover,let us introduce the following quantities:
ρ H˙
Ω= , q = 1 .
3H2 − − H2
We can see that the constraint equation (3) is written as
1 R
Ω+ ΣiΣj =1. (10)
6(cid:18) j i − H2(cid:19)
Except for Bianchi IX, the scalar curvature is always non-positive (see the expression (E.12) in
Appendix E.2 on page 699 of [11]) so that
Ω 1.
≤
We also derive an evolution equation of σ by combining the above results:
ab
1 1
σ˙b =k˙b H˙δb = 3Hσb +Sb Sδb Rb + Rδb.
a a− a − a a− 3 a− a 3 a
We now define the shear variables:
1 1 1
Σ+ = 2(Σ22+Σ33)=−2Σ11, Σ− = 2√3(Σ22−Σ33),
and the dimensionless time variable τ:
dt
=H−1,
dτ
and combine (5), (9), and (10) to obtain the evolution equations for Σ− and Σ+:
2R 3(R2+R3)
Σ′ =(q 2)Σ + − 2 3 +S , (11)
+ − + 6H2 +
R3 R2
Σ′− =(q 2)Σ−+ 3− 2 +S−, (12)
− 2√3H2
where we have used the notation
1
S = (3S2+3S3 2S),
+ 6H2 2 3 −
S2 S3
S− = 2 − 3.
2√3H2
2.2 Vlasov equation with Bianchi symmetry
Since we use a left-invariant frame, f will not depend on xa. Moreover,since g =g00 = 1 and
00
−
g0a = 0, we have p0 = p = 1+p p gab, ρ = T , and J = T . The frame components of
0 a b 00 a 0a
− −
the energy-momentum tensor apre thus
ρ=(detg)−21 f(t,p∗) 1+papbgabdp∗,
Z
p
−1
Ji =(detg) 2 f(t,p∗)pidp∗,
Z
−1 pipj
Sij =(detg) 2 f(t,p∗) dp∗,
Z 1+p p gab
a b
p
where the distribution function is understood as f =f(t,p∗) with p∗ =(p1,p2,p3).
We define P as the supremum of the square of momenta at a given time:
P(t)=sup gabpapb f(t,p∗)=0 . (13)
{ | 6 }
4
Proposition 1. Consider the Vlasov equation in a Bianchi spacetime. Then, the support in
momentum space is bounded as follows:
τ
P(τ)≤P(τ0)exp(cid:16)2Zτ0(−1+(ΣabΣba)12)ds(cid:17).
Proof. The Vlasov equation is written as
∂f ∂f
p0 +Cd pbp =0, (14)
∂t ba d∂p
a
whereCd arethestructureconstantsoftheLiealgebra(see[4]fordetails). Acharacteristiccurve
ba
is defined for each V (t)=p by
a a
dV
a =(V0)−1Cd VbV . (15)
dt ba d
For the rest of the paper the capital V will indicate that p is parametrised by the coordinate
a a
time t (or τ if we express all the variables in these terms). Note that if we define
P˜ =gabV V , (16)
a b
due to the antisymmetry of the structure constants we have from (15)
P˜˙ =g˙abV V . (17)
a b
In the sense of quadratic forms we have
σab (σcσd)12gab,
≤ d c
and as a consequence
g˙ab 2H( 1+(ΣaΣb)21)gab. (18)
≤ − b a
We combine (17) and (18) with the dimensionless time variable τ to obtain
P˜′ 2( 1+(ΣaΣb)12)P˜.
≤ − b a
Integrating this,
τ
P˜(τ)≤P˜(τ0)exp(cid:16)2Zτ0(−1+(ΣabΣba)12)ds(cid:17), (19)
and taking the supremum, we obtain the desired result.
As a consequence we have a nice result in the case of massive particles in a Bianchi spacetime
which is not Bianchi IX since the quotient of the trace of the energy-momentum tensor and the
energy density is in fact bounded by the support of the momenta.
Corollary 1. Consider massive solutions to the Einstein-Vlasov system in a Bianchi spacetime
which is notBianchi IX.Iftheshearis boundedlike (ΣaΣb)21 <1 forall times, we have adust-like
b a
behaviour in the sense that
S
0.
ρ →
5
Proof. Denoting by a hat that we use an orthonormal frame we have
S =S gab =Sˆ +Sˆ +Sˆ ,
ab 11 22 33
so that the following holds for the trace
pˆ2
S = f(t,pˆ)| | dpˆ P f(t,pˆ)pˆdpˆ Pρ,
Z p0 ≤ Z | | ≤
where we used pˆ2 =pˆ2+pˆ2+pˆ2 =gabp p and p0 1. Applying the previous result we obtain
| | 1 2 3 a b ≥
Sρ ≤P(τ0)exp(cid:16)2Zτ0τ(−1+(ΣabΣba)12)ds(cid:17),
which tends to zero as τ .
→∞
For later use we remark that the quantity S/(3H2) has the same bound with S/ρ as follows:
S ρ
P =ΩP P. (20)
3H2 ≤ 3H2 ≤
Thus we have that
3HS2 ≤P(τ0)exp(cid:16)2Zτ0τ(−1+(ΣabΣba)12)ds(cid:17). (21)
2.3 Reflection and Bianchi VII symmetries
0
We will assume an additional symmetry namely the reflection symmetry:
f(t,p ,p ,p )=f(t, p , p ,p )=f(t,p , p , p ).
1 2 3 1 2 3 1 2 3
− − − −
If we suppose that the distribution function is initially reflection symmetric and the metric and
thesecondfundamentalformareinitiallydiagonal,thenweseethattheenergy-momentumtensor
is diagonal as well. One can see from the equations that the metric and the second fundamental
form will remain diagonal. This symmetry implies in particular that there is no matter current.
In the diagonal case there is a simple formula for the Ricci tensor. Let (ijk) denote a cyclic
permutation of (123) and let us suspend the Einstein summation convention for the next three
formulas. Introduce ν as the signs depending on the Bianchi type (see Table 1 of [1]), and define
i
g
ii
n =ν ,
i i
rgjjgkk
which implies n =ν g /(g g ), etc, and the Ricci tensor (cf. (11a) of [1]) is given by
2 2 22 33 11
p
1
Ri = n2 (n n )2 .
i 2(cid:16) i − j − k (cid:17)
We also have in the diagonal case that
1 1
Σ2 +Σ2 = (Σ1)2+(Σ2)2+(Σ3)2 = ΣaΣb.
+ − 6 1 2 3 6 b a
(cid:16) (cid:17)
We will study the Bianchi VII case. For Bianchi VII , since ν = 0 and ν = ν = 1, we
0 0 1 2 3
have that n and n are positive definite and the only non-vanishing structure constants are (cf.
2 3
Appendix E, page 695 of [11]):
C2 =1= C2 , C3 =1= C3 . (22)
31 − 13 12 − 21
6
The curvature expressions are
1
R1 =R= (n n )2,
1 −2 2− 3
1
R2 = R3 = (n2 n2).
2 − 3 2 2− 3
Let N denote n /H, then straightforwardcomputations lead to
ii i
N2′2 =N22(q+2Σ++2√3Σ−),
N3′3 =N33(q+2Σ+−2√3Σ−),
(see [8] for more details in the Bianchi II case). In order to compare this with the equations of
[13] we use the following quantities
N +N
22 33
N = >0,
+
2
N N
22 33
N− = − .
2√3
Theevolutionequationsfortheshearandcurvaturevariablesusingthedimensionlesstimevariable
are as follows:
Σ′ =(q 2)Σ 2N2 +S ,
+ − +− − +
′
Σ− =(q 2)Σ− 2N+N−+S−,
− −
′
N+ =(q+2Σ+)N++6Σ−N−,
′
N− =(q+2Σ+)N−+2Σ−N+,
which one can compare with (3.10) of [13]. Since we expect a similar behaviour as in the fluid
case, we introduce as (3.15)–(3.16)in [13] the following variables (we use X instead of R to avoid
confusion with the scalar curvature):
1
M = >0,
N
+
N− =Xsinψ, X >0,
Σ− =Xcosψ, X >0.
We also split q in what we expect to be the non-oscillatory part Q and the oscillatory part
H˙ 1 R ΣiΣj S
j i
q = 1 = + + +
− − H2 2 12H2 4 6H2
1 1 3 3 S
= N2 + Σ2 + Σ2 +
2 − 2 − 2 + 2 − 6H2
1 3 1 S
= + Σ2 + X2+X2cos2ψ+ =Q+X2cos2ψ,
2 2 + 2 6H2
where
1 3 1 S
Q= + Σ2 + X2+ .
2 2 + 2 6H2
The differential equations for Σ+, Σ−, N+, and N− are now written as follows:
Σ′ =Σ (Q 2) X2+(1+Σ )X2cos2ψ+S , (23)
+ + − − + +
M′ = M[Q+2Σ +X2(cos2ψ+3Msin2ψ)], (24)
+
−
X′ =[Q+Σ+ 1+(X2 1 Σ+)cos2ψ]X +cosψS−, (25)
− − −
1
ψ′ = 2+M[(1+Σ+)sin2ψ X−1S−sinψ] , (26)
M(cid:16) − (cid:17)
7
which can be compared to (3.17)–(3-20) of [13]. An important quantity in the analysis of [13] is
what they call the Weyl parameter. It is a dimensionless measure of the Weyl curvature tensor.
We have defined shear and curvature variables in such a way that we have the same expressions
as in [13]. The square of it is defined as (cf. (3.37)–(3.38)of [13])
2 = +2+ −2+ −2+ −2,
W E E E H
where
1
=Σ (1+Σ )+ X2(1 3cos2ψ),
+ + +
E 2 −
2X 1
− = sinψ+ M(1 2Σ+)cosψ ,
E M (cid:16) 2 − (cid:17)
3
= X2sin2ψ,
+
H −2
2X 3
− = cosψ MΣ+sinψ .
H M (cid:16)− − 2 (cid:17)
For details we refer to [13] and references therein.
3 The bootstrap argument
3.1 Bootstrap assumptions
We will use a bootstrap argument. Let us assume that there exists an interval [τ ,τ ) where the
0 1
following estimates hold:
M(τ) εe−52(τ−τ0),
≤
X(τ) εe−25(τ−τ0),
≤
Σ+(τ) εe−35(τ−τ0),
| |≤
where the epsilon is a small positive constant.
3.2 Matter terms
Due to the bootstrap assumptions the shear is bounded. Since ΣaΣb =6(Σ2 +Σ2), we have
b a + −
ΣaΣb 6(ε2+X2cos2ψ) Cε2.
b a ≤ ≤
As a consequence using (21) we have
S τ
P(τ )exp 2 ( 1+Cε)ds P(τ )e(−2+Cε)(τ−τ0).
3H2 ≤ 0 (cid:16) Zτ0 − (cid:17)≤ 0
The variables S+ and S− are bounded by that up to an irrelevant constant. Note that
1 1
S+ = 6H2(S−3S11), S− = 2√3H2(S22−S33),
hence we have
1 1
(S+)2+(S−)2 = 36H4(S2−6SS11+9(S11)2)+ 12H4((S22)2+(S33)2−2S22S33)
1
= (S2+3((S1)2+(S2)2+(S3)2) 6SS1+6(S1)2 6S2S3)
36H4 1 2 3 − 1 1 − 2 3
1
= (S2+6((S1)2+(S2)2+(S3)2) 3(S1+S2+S3)2)
36H4 1 2 3 − 1 2 3
1 1
= SaSb S2.
6H4 b a− 18H4
8
Since SbaSab ≤S2,we have(S+)2+(S−)2 ≤S2/(9H4), andconclude thatS+ andS− arebounded
as follows:
S± P(τ0)e(−2+Cε)(τ−τ0).
| |≤
3.3 Estimate of M
In orderto obtaina better estimate for M it will be necessaryto estimate the term in bracketsof
(24) which can be expressed using the definition of Q as follows:
1 d 3 1 S
logM =2Σ + Σ2 +X2 +cos2ψ+3Msin2ψ + . (27)
−2 − dτ + 2 + (cid:16)2 (cid:17) 6H2
Using the bootstrap assumptions and the bound for the matter term we obtain for the evolution
equation of M
1 d
+ logM C(ε+P(τ )),
0
(cid:12)2 dτ (cid:12)≤
(cid:12) (cid:12)
which implies that (cid:12) (cid:12)
M(τ0)e(−21−C(ε+P(τ0)))(τ−τ0) M(τ) M(τ0)e(−12+C(ε+P(τ0)))(τ−τ0). (28)
≤ ≤
3.4 Estimate of X
The evolution equations for M, X, and ψ are written as
′
M = dM,
−
′
X =(a+bcos2ψ)X+c,
ψ′ =2M−1+e,
where a, b, c, d, and e are functions of τ given by
a=Q+Σ 1, (29)
+
−
b=X2 1 Σ , (30)
+
− −
c=S−cosψ, (31)
d=Q+2Σ +X2(cos2ψ+3Msin2ψ), (32)
+
e=(1+Σ+)sin2ψ X−1S−sinψ. (33)
−
An oscillation term appears in the evolution equation of X with the factor b which is not small.
In order to obtain an estimate of X we have to get rid of the b-factor. We introduce the variable
1
X¯ =fX with f = .
1+ 1Mbsin2ψ
4
Note that f >0 as long as M, X, and Σ are small. By direct calculations we have
+
bd b′ be
f′ =f2M − sin2ψ cos2ψ f2bcos2ψ,
(cid:18) 4 − 2 (cid:19)−
and the evolution equation of X¯ is given by
ab+bd b′ be b2
X¯′ = a+M − sin2ψ cos2ψ+ sin4ψ fX¯ +cf. (34)
(cid:18) (cid:18) 4 − 2 8 (cid:19)(cid:19)
We need the following estimates. By the definition of Q we have
1 3 1 S
a= + Σ2 + X2+ +Σ 1,
2 2 + 2 6H2 +−
9
which implies that a+ 1 C(ε+P(τ )) by the bootstrap assumption. Similarly we have
| 2|≤ 0
1
b+1 Cε, d C(ε+P(τ )).
| |≤ (cid:12) − 2(cid:12)≤ 0
(cid:12) (cid:12)
(cid:12) (cid:12)
For b′ we write (cid:12) (cid:12)
b′ =2cX+X2(2a+2bcos2ψ 1+cos2ψ)
−
3 3 3 1 S
+ Σ Σ + Σ2 + X2 X2cos2ψ+ +S ,
2 +− +(cid:18)− 2 2 + 2 − 6H2(cid:19) +
and this shows that b′ C(ε+P(τ )) by the bootstrap assumptions and the estimates above.
0
| | ≤
Consequently, we obtain
ab+bd b′ be b2
X¯′ = a+M − sin2ψ cos2ψ+ sin4ψ fX¯ +cf
(cid:18) (cid:18) 4 − 2 8 (cid:19)(cid:19)
ab+bd b′ b(1+Σ )sin2ψ b2
= a+M − sin2ψ + cos2ψ+ sin4ψ fX¯
(cid:18) (cid:18) 4 − 2 8 (cid:19)(cid:19)
Mb
+ S−sinψcos2ψf2+cf.
2
Applyingtheestimatesofa,b,d,andb′togetherwiththebootstrapassumptionsandtheestimates
of matter terms, we obtain
1
X¯′ +C(ε+P(τ0)) fX¯ +(Cεf2+f)S−
≤(cid:18)− 2 (cid:19)
1
+C(ε+P(τ )) X¯ +(1+Cε)P(τ )e(−2+Cε)(τ−τ0),
0 0
≤(cid:18)− 2 (cid:19)
where we used f =1+O(ε) by the bootstrap assumption, and this can be written as
d e21(τ−τ0)X¯ C(ε+P(τ0))e12(τ−τ0)X¯ +(1+Cε)P(τ0)e(−23+Cε)(τ−τ0).
dτh i≤
Integrating the above inequality we obtain
τ
e12(τ−τ0)X¯(τ) X¯(τ0)+C(ε+P(τ0)) e21(s−τ0)X¯(s)ds
≤ Z
τ0
τ
+(1+Cε)P(τ0) e(−32+Cε)(s−τ0)ds
Z
τ0
≤X¯(τ0)+C(ε+P(τ0))Z τ e21(s−τ0)X¯(s)ds+ (1+3Cε)CPε(τ0),
τ0 2 −
which can be written for small ε as follows:
τ
e21(τ−τ0)X¯(τ) X¯(τ0)+P(τ0)+C(ε+P(τ0)) e21(s−τ0)X¯(s)ds.
≤ Z
τ0
By Gronwall’s inequality we obtain
e21(τ−τ0)X¯(τ) (X¯(τ0)+P(τ0))eC(ε+P(τ0))(τ−τ0),
≤
which shows that
X¯(τ) (X¯(τ0)+P(τ0))e(−12+C(ε+P(τ0)))(τ−τ0).
≤
Since X¯ =fX with f =1+O(ε), we conclude that
X(τ) (1+Cε) X(τ0)+P(τ0) e(−21+C(ε+P(τ0)))(τ−τ0). (35)
≤ (cid:16) (cid:17)
10