On the cosmic no-hair conjecture in the Einstein-Vlasov setting
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1 On the cosmic no-hair conjecture in the Einstein-Vlasov setting KTH, Royal Institute of Technology, Stockholm Recent Advances in Mathematical General Relativity Institut Henri Poincaré, Paris September 23-25, 215
2 Introduction Causality Cosmic no-hair conjecture, formulation and results Introduction Observations of supernovae of type Ia, 1998: the universe is expanding at an accelerated rate. One possible mechanism: positive cosmological constant Λ.
3 Introduction Causality Cosmic no-hair conjecture, formulation and results de Sitter space Model solution: de Sitter space g ds = dt 2 +e 2Ht ḡ on R T 3, where H = Λ/3. Solves G +Λg =, where G is the Einstein tensor G = Ric 1 2 Sg.
4 Introduction Causality Cosmic no-hair conjecture, formulation and results Cosmic no-hair, rough formulation Cosmic no-hair conjecture: In a spacetime solving Einstein s equations with a positive cosmological constant, the geometry appears de Sitter like to late time observers. In particular: solutions are expected to homogenise and isotropise.
5 Introduction Causality Cosmic no-hair conjecture, formulation and results Minkowski space Let γ(t) = (t,,,). Then γ is an observer in Minkowski space. How much of the t = hypersurface does γ see? x.2 x.4 x x2.5.5 x x x x x Figure : The causal past of γ(t) intersected with the causal future of the t = hypersurface for t = 1/2, t = 1 and t = 2.
6 Introduction Causality Cosmic no-hair conjecture, formulation and results de Sitter space Consider the metric g ds = dt 2 +e 2t ḡ x.4 x x2.5.5 x x x Figure : The causal past of γ(t) intersected with the causal future of the t = hypersurface for t = 1/2 and for all t.
7 Introduction Causality Cosmic no-hair conjecture, formulation and results Regions of interest Relevant spacetime regions: where H = Λ/3. C Λ,K,T = {(t, x) : t > T, x < KH 1 e Ht }, Role of T: to specify what is meant by late times. Role of K 1: to provide a margin in the general case.
8 Introduction Causality Cosmic no-hair conjecture, formulation and results Cosmic no-hair, formal definition Let (M,g) be a time oriented, globally hyperbolic Lorentz mfd which is future causally geodesically complete. Assume, moreover, that (M,g) is a solution to Einstein s equations with a positive cosmological constant Λ. Then (M,g) is said to be future asymptotically de Sitter like if there is a Cauchy hypersurface Σ in (M, g) such that for every future oriented and inextendible causal curve γ in (M,g), the following holds: there is an open set D in (M,g), such that J (γ) J + (Σ) D, and D is diffeomorphic to C Λ,K,T for a suitable choice of K 1 and T >,
9 Introduction Causality Cosmic no-hair conjecture, formulation and results...continued using ψ : C Λ,K,T D to denote the diffeomorphism; letting R(t) = KH 1 e Ht ; using ḡ ds (t, ) and k ds (t, ) to denote the metric and second fundamental form induced on S t = {t} B R(t) () by g ds ; using ḡ(t, ) and k(t, ) to denote the metric and second fundamental form induced on S t by ψ g; and letting N N, the following holds: lim t ( ḡ ds (t, ) ḡ(t, ) C N ds (S t) + k ds (t, ) k(t, ) ) C N ds (S t) =.
10 Introduction Causality Cosmic no-hair conjecture, formulation and results Norms Here: h C N ds (S t) = (sup S t N l= ḡ ds,i1 j 1 ḡ ds,il j l ḡ im dsḡjn ds i 1 ds i l dsh ij j 1 ds j l dsh mn ) 1/2 for a covariant 2-tensor field h on S t, where ds denotes the Levi-Civita connection associated with ḡ ds (t, ).
11 Introduction Causality Cosmic no-hair conjecture, formulation and results Cosmic no-hair conjecture Cosmic no-hair conjecture: Let A denote the class of initial data such that the corresponding maximal Cauchy developments are future causally geodesically complete solutions to Einstein s equations with a positive cosmological constant Λ (for some fixed matter model). Then generic elements of A yield maximal Cauchy developments that are future asymptotically de Sitter like.
12 Introduction Causality Cosmic no-hair conjecture, formulation and results Some results in the Einstein-Vlasov setting Spatially homogeneous setting; H. Lee 4. General perturbations thereof; H. R. 13. The surface symmetric setting; S. B. Tchapnda, A. D. Rendall 3. General perturbations thereof; E. Nungesser 14. The T 3 -Gowdy symmetric setting and general perturbations thereof; H. Andréasson and H. R. 13.
13 Heuristics Formal definition The Einstein-Vlasov system : collection of particles, where the particles all have unit mass, collisions are neglected, the particles follow geodesics, collection described statistically by a distribution function.
14 Heuristics Formal definition The Einstein-Vlasov system Stress energy tensor, In the Vlasov setting, the relevant mathematical structures are the mass shell P; the future directed unit timelike vectors in (M,g), the distribution function f : P [, ), the stress energy tensor T αβ ξ = fp α p β µ Pξ, P ξ the Vlasov equation Lf =.
15 Heuristics Formal definition The Einstein-Vlasov system The Einstein-Vlasov system The Einstein-Vlasov system consists of the equations G +Λg = T, Lf = for g and f. Note that the second equation corresponds to the requirement that f be constant along timelike geodesics.
16 T 2 -symmetry and equations Rough sketch of arguments Results T 2 -symmetry Metric: g =t 1/2 e λ/2 ( dt 2 +α 1 dθ 2 )+te P [dx +Qdy +(G +QH)dθ] 2 +te P (dy +Hdθ) 2, Here α >, λ, P, Q, G and H only depend on t and θ. T 3 -Gowdy: H and G time-independent. Frame: e =t 1/4 e λ/4 t, e 1 = t 1/4 e λ/4 α 1/2 ( θ G x H y ), e 2 =t 1/2 e P/2 x, e 3 = t 1/2 e P/2 ( y Q x ).
17 T 2 -symmetry and equations Rough sketch of arguments Results Matter components and equations Matter components ρ = T(e,e ), J i = T(e,e i ), P i = T(e i,e i ), S ij = T(e i,e j ), Main equations t(tα 1/2 P t) = θ (tα 1/2 P θ ) + tα 1/2 e 2P (Q 2 t αq2 θ ) + t1/2 e λ/2 α 1/2 (P 2 P 3 ), t(tα 1/2 e 2P Q t) = θ (tα 1/2 e 2P Q θ ) + 2t 1/2 α 1/2 e λ/2+p S 23, = t [P 2 t + αp2 θ + e2p (Q 2 ] t + αq2 θ ) + 4t 1/2 e λ/2 (ρ + Λ), λ t 2 αt α λ t = t [P 2 t + αp2 θ + e2p (Q 2 t + αq2 θ ) ] + 4t 1/2 e λ/2 (P 1 Λ), λ θ = 2t(P tp θ + e 2P Q tq θ ) 4t 1/2 e λ/2 α 1/2 J 1.
18 T 2 -symmetry and equations Rough sketch of arguments Results Vlasov equation Vlasov equation: f t + α [ 1/2 v 1 f 1 v θ 4 α1/2 λ θ v + 1 ( λ t 2αt 4 α 1 )v 1 α 1/2 e P v 2 v 3 Q θ t v + 1 (v 3 ) 2 (v 2 ) 2 ] [ ( f 1 2 α1/2 P θ v v 1 P t + 1 )v v 1 v 2 ] f 2 t α1/2 P θ v v 2 [ ( 1 1 )v 2 t Pt 3 12 v 1 v 3 ( α1/2 P θ v + e P v 2 Q t + α 1/2 v 1 )] f Q θ v v 3 =, where f : I S 1 R 3 [, ) is the distribution function, v = [1 + (v 1 ) 2 + (v 2 ) 2 + (v 3 ) 2 ] 1/2. Matter quantities: k (v ) 2 ρ = R 3 v f dv, P k = R 3 v f dv, J k = j v v k R 3 vk f dv, S jk = R 3 v f dv,
19 T 2 -symmetry and equations Rough sketch of arguments Results An estimate for α Note that λ t α t α = t [ P 2 t +αp 2 θ +e2p (Q 2 t +αq 2 θ ) ]+2t 1/2 e λ/2 (ρ+p 1 ). In particular, there is a < c R such that α 1/2 e λ/2 c. Since α t α 4t1/2 e λ/2 Λ, we obtain t α 1/2 2t 1/2 α 1/2 e λ/2 Λ c 1 t 1/2, whence α(t,θ) Ct 3 for t t 1.
20 T 2 -symmetry and equations Rough sketch of arguments Results Asymptotics for λ Letting it can be estimated that ( ) 3 ˆλ = λ+3lnt 2ln, 4Λ tˆλ 3 t (1 eˆλ/2 ). For every ǫ >, there is thus a T such that for all t T. λ(t,θ) 3lnt +2ln 3 4Λ ǫ
21 T 2 -symmetry and equations Rough sketch of arguments Results Energy ( ) E = t 2 α 1/2 S 1 Pt 2 +αpθ 2 +e 2P (Qt 2 +αqθ)+4t 2 1/2 e λ/2 ρ dθ. Lower bound on λ upper bound on E of the form E(t) C a t a, a > 1/2. This bound on E ˆλ. In the end: 3 λ(t, )+3lnt 2ln Ct 1/2, 4Λ Moreover P and Q are bounded in C. C E(t) Ct 1/2.
22 T 2 -symmetry and equations Rough sketch of arguments Results Asymptotics There are smooth functions α >, P, Q, G and H on S, and, for every N Z, a constant C N > such that t P t(t, ) C N + t Q t(t, ) C N + P(t, ) P C N + Q(t, ) Q C N C N t 1, (1) α t α + 3 t + λ t + 3 C N t 2, (2) C N t C N t 3 C α(t, ) α N + λ(t, ) + 3lnt 2ln 3 C N t 1, (3) 4Λ C N for all t t 1.
23 T 2 -symmetry and equations Rough sketch of arguments Results Asymptotics, continued Define f sc via f sc (t,θ,v) = f(t,θ,t 1/2 v). Then there is a smooth, non-negative function with compact support, say f sc,, on S R 3, such that t t f sc (t, ) C N (S R 3 ) + f sc (t, ) f sc, C N (S R 3 ) C N t 1 for all t t 1.
24 T 2 -symmetry and equations Rough sketch of arguments Results Cosmic no-hair conjecture, T 3 -Gowdy symmetric setting Theorem T 3 -Gowdy symmetric solutions to the Einstein-Vlasov system with a positive cosmological constant are future asymptotically de Sitter like.
25 T 2 -symmetry and equations Rough sketch of arguments Results Stability Let ḡ bg and k bg denote the metric and second fundamental form induced by a T 3 -Gowdy symmetric solution (to the Einstein-Vlasov system with a positive cosmological constant Λ) on a constant t hypersurface. Let, moreover, f bg denote the induced initial datum for the distribution function. Then there is an ǫ > such that if (T 3,ḡ, k, f) are initial data satisfying ḡ ḡ bg H 5 + k k bg H 4 + f f bg H 4 Vl,µ ǫ, then the associated maximal Cauchy development is future causally geodesically complete, future asymptotically de Sitter like.
26 T 2 -symmetry and equations Rough sketch of arguments Results Thank you!
27 T 2 -symmetry and equations Rough sketch of arguments Results Induced initial data Let (M,g,f) be a solution and Σ be a spacelike hypersurface in (M,g). Then the initial data induced on Σ consist of the Riemannian metric induced on Σ by g, say ḡ, the second fundamental form induced on Σ by g, say k, the induced distribution function f : TΣ [, ). Here f = f proj 1 Σ, where proj Σ : P Σ TΣ represents projection orthogonal to the normal.
28 T 2 -symmetry and equations Rough sketch of arguments Results Function spaces If Σ is a compact manifold, D µ (TΣ) denotes the space of smooth functions f : TΣ R such that f H l Vl,µ j = i=1 α + β l 2µ+2 β χ i ( ξ)( ᾱ x i (U i ) R n ξ β f xi ) 2 ( ξ, )d ξd is finite for every l, where = (1+ 2 ) 1/2. 1/2
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