The Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant

The Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant Jared Speck & Igor Rodnianski jspeck@math.princeton.edu University of Cambridge & Princeton University October 26, 2009

Indices Greek indices {0, 1, 2, 3} Latin indices {1, 2, 3} t = 0 = ( t, 1, 2, 3 ) = ( 1, 2, 3 )

Einstein s equations G µν + Λg µν = T µν We fix Λ > 0 Bianchi identities imply D µ T µν = 0 Our spacetimes will have topology [0, T ] T 3

The notion of stability" Need a background solution (Ours will be FLRW type) Initial value problem formulation of Einstein equations We will use wave coordinates (de Donder 1921, Choquet-Bruhat, 1952) Goal: show that if we slightly perturb the background data, then the resulting solution exists for all t 0 and that the spacetime is future causally geodesically complete We show convergence as t Our proofs are based on energy estimates for quasilinear wave equations

Previous stability results For Λ = 0 : Vacuum Einstein using maximal foliation (Christodoulou & Klainerman, 1993) Einstein-Maxwell (Zipser, 2000) Vacuum Einstein using double-null foliation (Klainerman & Nicolò, 2003) Einstein-scalar field using wave coordinates (Lindblad & Rodnianski, 2004-2005) Einstein-Maxwell-scalar field in 1 + n dimensions (Loizelet, 2008) Vacuum Einstein for more general data (Bieri, 2009)

Previous stability results cont. For Λ > 0 using the conformal method Vacuum Einstein (Friedrich, 1986) Einstein-Maxwell & Einstein-Yang-Mills (Friedrich, 1991) Vacuum Einstein in (1 + n) dimensions, n odd (Anderson, 2005) For Λ = 0 using wave coordinates Einstein-scalar field with a potential V (Φ) in (1 + n) dimensions (Ringström, 2008) Φ = V (Φ) V (Φ) emulates Λ > 0 : V (0) > 0, V (0) = 0, V (0) > 0

Previous stability results cont. Euler-Poisson with a cosmological constant (Brauer, Rendall, & Reula, 1994)

Why Λ > 0? Answer # 1: Einstein was the first to envision Λ : he was looking for static solutions. Answer # 2: In 1929, Hubble formulated an empirical law," which was based on observations of the redshift effect, and which suggested that the universe is expanding. Hubble s law:" galaxies are receding from Earth, and their velocities are proportional to their distances from it. In the 1990 s: data from type Ia supernovae and the Cosmic Microwave Background suggested accelerated expansion. Λ > 0 = solutions with accelerated expansion. e.g. g = dt 2 + e 2( Λ/3)t 3 a=1 (dx a ) 2

The modified irrotational Euler-Einstein system def h jk = e 2Ω g jk, W = def 3/(1 + 2s), H 2 = def Λ, def ˆ 3 g = g αβ α β ˆ g (g 00 + 1) = 5H t g 00 + 6H 2 (g 00 + 1) + 00 ˆ g g 0j = 3H t g 0j + 2H 2 g 0j 2Hg ab Γ ajb + 0j ˆ g h jk = 3H t h jk + jk t 2 Φ + m jk j k Φ + 2m 0j t j Φ = W ω t Φ + ( Φ) m µν is the reciprocal acoustic metric The terms are error terms

The appearance of z j in the fluid equation 2 t Φ + m jk j k Φ + 2m 0j t j Φ = wω t Φ + ( Φ) m jk = g jk jk (m) (1 + 2s) + (m) m 0j = 2sg00 e Ω g ja z a + (1 + 2s) + (m) (m) = (1 + 2s)(g 00 + 1)(g 00 1) + 2(1 + 2s)e Ω g 00 g 0a z a + jk (m) = 2eΩ (g jk g 0a + 2sg 0k g aj )z a + z j def = e Ω j Φ t Φ

The global existence theorem Theorem (I.R. & J.S. 2009) Assume that N 3 and 0 < c 2 s < 1/3. Then there exist ɛ 0 > 0 and C > 1 such that for all ɛ ɛ 0, if Q N (0) C 1 ɛ, then there exists a global future causal geodesically complete solution to the reduced irrotational Euler-Einstein system. Furthermore, holds for all t 0. Q N (t) ɛ

Asymptotics of the solution Theorem (I.R. & J.S. 2009) Under the assumptions of the global existence theorem, with the additional assumption N 5, there exist q > 0, a smooth Riemann metric g ( ) jk corresponding Christoffel symbols Γ ( ) ijk with and inverse g jk ( ) on T 3, and (time independent) functions Φ ( ), Ψ ( ) on T 3 such that e 2Ω g jk (t, ) g ( ) jk H N Cɛe qht e 2Ω g jk (t, ) g jk ( ) H N Cɛe qht e 2Ω t g jk (t, ) 2ωg ( ) jk H N Cɛe qht g 0j (t, ) H 1 g ab ( ) Γ( ) ajb HN 3 Cɛe qht t g 0j (t, ) H N 3 Cɛe qht

Asymptotics cont. Theorem (Continued) g 00 + 1 H N Cɛe qht t g 00 H N Cɛe qht e 2Ω K jk (t, ) ωg ( ) jk H N Cɛe qht In the above inequality, K jk is the second fundamental form of the hypersurface t = const. e wω t Φ(t, ) Ψ ( ) H N 1 Cɛe qht Φ(t, ) Φ ( ) H N 1 Cɛe qht.

Comparison with flat space Christodoulou s monograph The Formation of Shocks in 3-Dimensional Fluids" shows that on the Minkowski space background, shock singularities can form in solutions to the irrotational fluid equation arising from data that are arbitrarily close to that of a uniform quiet state. Conclusion: Exponentially expanding spacetimes can stabilize irrotational fluids.

Topologies beyond that of T 3 The study of wave equations arising from metrics featuring accelerated expansion is a very local" problem A patching argument can very likely" be used to allow for many of the topologies considered by Ringström: Unimodular Lie Groups different from SU2; H 3 ; H 2 R;

Future directions Other equations of state Sub-exponential expansion rates Non-zero vorticity : forthcoming article

Thank you

The norms S g00 +1;N S g0 ;N S h ;N def = e qω t g 00 H N + e (q 1)Ω g 00 H N + e qω g 00 + 1 H N 3 ( ) def = e (q 1)Ω t g 0j H N + e (q 2)Ω g 0j H N + e (q 1)Ω g 0j H N def = j=1 3 j,k=1 ( ) e qω t h jk H N + e (q 1)Ω h jk H N + h jk H N 1 def S N = e wω t Φ Ψ 0 H N + e (w 1)Ω Φ H N def Q N = S g00 +1;N + S g0 ;N + S h ;N + S N

g µν building block energies If α > 0, β 0, and v is a solution to ˆ g v = αh t v + βh 2 v + F, then we control solutions to this equation using an energy of the form E(γ,δ)[v, 2 v] = def + 1 { g 00 ( t v) 2 + g ab ( a v)( b v) 2γHg 00 v t v + δh 2 v 2 } d 3 x 2 T 3

Energies for g µν E 2 g 00 +1;N E 2 g 0 ;N E 2 h ;N def = α N def = α N j=1 def = α N e 2qΩ E 2 (γ 00,δ 00 )[ α (g 00 + 1), ( α (g 00 + 1))] 3 e 2(q 1)Ω E(γ 2 0,δ 0 )[ α g 0j, ( α g 0j )] { 3 e 2qΩ E(0,0)[0, 2 ( α h jk )] + 1 } c α H 2 ( α h jk ) d 3 x 2 T 3 j,k=1

Energies for Φ E 2 0 E 2 N def = 1 (e wω t Φ 2 Ψ 0 ) 2 + e 2wΩ m jk ( j Φ)( k Φ) d 3 x T 3 def = E0 2 + 1 α N 1 2 T 3 e 2W Ω ( t α Φ) 2 + e 2wΩ m jk ( j α Φ)( k α Φ) d 3 x.

Equivalence of norms and energies Lemma If Q N (t) is small enough, then the norms and energies are equivalent.

The Euler-Einstein system With R = g αβ R αβ, Einstein s field equations are R µν 1 2 Rg µν + Λg µν = T (fluid) µν T (fluid) µν = (ρ + p)u µ u ν + pg µν ρ = proper energy density, p = pressure, u = four-velocity u is future-directed, and u α u α = 1

The relativistic Euler equations Consequence of the Bianchi identities D µ ( R µν 1 2 Rgµν ) = 0: D µ T µν (fluid) = 0 (1) Conservation of particle number: D µ (nu µ ) = 0 (2) n = proper number density (1) and (2) are the relativistic Euler equations.

Equation of state To close the Euler system, we need more equations. Fundamental thermodynamic relation: Our equation of state: ρ + p = n ρ n c s is the speed of sound p = c 2 s ρ 0 < c s < 1 3

Fluid vorticity σ def = ρ + p n, where σ 0. σ is known as the enthalpy per particle. The fluid vorticity is defined by where def v µν = µ β ν ν β µ, def β µ = σu µ.

Irrotational fluids An irrotational fluid is one for which Local consequence: def v µν = µ β ν ν β µ = 0 def β µ = σu µ = µ Φ Φ is the fluid potential. We think of Φ existing globally; Φ itself may only exist locally (Poincaré s lemma). Only Φ enters into the equations.

Scalar formulation of irrotational fluid mechanics Consequence: for irrotational fluids, the Euler equations D µ T µν (fluid) = 0, D µ(nu µ ) = 0 are equivalent to a scalar quasilinear wave equation that is the Euler-Lagrange equation corresponding to a Lagrangian L(σ) : [ ] L D α σ Dα Φ = 0. L = p u µ = σ 1/2 µ Φ σ = g αβ ( α Φ)( β Φ)

Energy-momentum tensor Corresponding to L, we have the energy-momentum tensor: T (scalar) µν D µ T µν (scalar) def = 2 L For an irrotational fluid, σ ( µφ)( ν Φ) + g µν L = 0 for solutions to the E-L equation T µν (scalar) = T µν (fluid)

The case p = c 2 sρ For p = cs 2 ρ, s = 1 c2 s ; c 2 2cs 2 s = 1, it follows that 2s+1 L = p = (1 + s) 1 σ s+1 T (scalar) µν = 2σ s ( µ Φ)( ν Φ) + g µν (1 + s) 1 σ s+1 The fluid equation: D α (σ s D α Φ) = 0

Background solution ansatz To construct a background solution, g, Φ, we make the ansatz g = dt 2 + a(t) 2 3 (dx k ) 2, k=1 from which it follows that the only corresponding non-zero Christoffel symbols are 0 Γ j k = Γ k 0 j = aȧδ jk k Γ j 0 = Γ 0 k j = ȧ a δk j.

Background solution Plugging this ansatz into the Euler-Einstein system, we deduce that: ρa 3(1+A2) ρå 3(1+A2 ) = def κ 0 Λ ȧ = a 3 + ρ Λ 3 = a 3 + κ 0 3a 3(1+A2 ) ρ > 0, å = a(0) Ω(t) def e = a(t) e Ht, H = def Λ 3 Φ = ( t Φ, 0, 0, 0) ( s + 1 ) 1/(2s+2)e t Φ = W κ Ω 0, W = 3 2s + 1 1 + 2s

Wave coordinates To hyperbolize the Einstein equations, we use a variant of the wave coordinates: Γ ν = Γ ν = 3ωδ 0ν ω = def t Ω H = Λ 3 g φ = def g αβ D α D β φ ˆ g φ def = g αβ α β φ In wave coordinates, g φ = 0 ˆ g φ = 3ω t Φ }{{} dissipative term

Energy vs. norm comparison Lemma If S g00 +1;N + E g0 ;N + E h ;N + S N is sufficiently small, then C 1 E g00 +1;N S g00 +1;N CE g00 +1;N C 1 E g0 ;N S g0 ;N CE g0 ;N C 1 E h ;N S h ;N CE h ;N C 1 E N S N CE N

Integral inequalities t E N (t) E N (t 1 ) + C e qhτ Q N (τ) dτ τ=t 1 E g00 +1;N(t) E g00 +1;N(t 1 ) + E g0 ;N(t) E g0 ;N(t 1 ) + E h ;N(t) E h ;N(t 1 ) + t t τ=t 1 2qHE g00 +1;N(τ) + Ce qhτ Q N (τ) dτ τ=t 1 2qHE g0 ;N(τ) + CE h ;N(τ) + Ce qhτ Q N (τ) dτ t τ=t 1 1 2 He qhτ E h ;N(τ) + Ce qhτ Q N (τ) dτ.