Numerical Refinement of a Finite Mass Shock-Wave Cosmology

Transcription

1 Numerical Refinement of a Finite Mass Shock-Wave Cosmology Blake Temple, UC-Davis Collaborators: J. Smoller, Z. Vogler, B. Wissman

2 References Exact solution incorporating a shock-wave into the standard FRW metric for cosmology... Smoller-Temple, Shock-Wave Cosmology Inside a Black Hole, PNAS Sept Smoller-Temple, Cosmology, Black Holes, and Shock Waves Beyond the Hubble Length, Meth. Appl. Anal., 2004.

3 References: Connecting the shock wave cosmology model with Guth s theory of inflation... How inflationary spacetimes might evolve into spacetimes of finite total mass, with J. Smoller, Meth. and Appl. of Anal., Vol. 12, No. 4, pp (2005). How inflation is used to solve the flatness problem, with J. Smoller, Jour. of Hyp. Diff. Eqns. (JHDE), Vol. 3, no. 2, (2006).

4 References: The locally inertial Glimm Scheme... A shock-wave formulation of the Einstein equations, with J. Groah, Meth. and Appl. of Anal., 7, No. 4,(2000), pp Shock-wave solutions of the Einstein equations: Existence and consistency by a locally inertial Glimm Scheme, with J. Groah, Memoirs of the AMS, Vol. 172, No. 813, November Shock Wave Interactions in General Relativity: A Locally Inertial Glimm Scheme for Spherically Symmetric Spacetimes, with J. Groah and J. Smoller, Springer Monographs in Mathematics, 2007.

5 Our Shock Wave Cosmology Solution Expanding Universe Shock Wave Matter Filled Black Hole t = 0 Big Bang t < t crit Schwarzschild Metric t = t crit Shock emerges from White Hole r = 2M = H c

6 The solution can be viewed as a natural generalization of a k=0 Oppenheimer- Snyder solution to the case of non-zero pressure, inside the Black Hole---- 2M/r>1

7 The Shock Wave Cosmology Solution Shock-Wave k=0 ds 2 = dt 2 + R(t) 2 { dr 2 + r 2 dω 2} FRW TOV: ds 2 = B( r)d t M( r) r d r 2 + r 2 dω 2 2M r > 1 r is timelike

8 1 In [Smoller-Temple, PNAS] we constructed an exact shock wave solution of the Einstein equations by matching a (k=0)- FRW metric to a TOV metric inside the Black Hole across a subluminal, entropysatisfying shock-wave, out beyond one Hubble length To obtain a large region of uniform expansion at the center consistent with observations we needed 2M r 2M > 1 r = 1 r = Hubble Length H c

9 The TOV metric inside the Black Hole is the simplest metric that cuts off the FRW at a finite total mass. Approximately like a classical explosion of finite mass with a shock-wave at the leading edge of the expansion. The solution decays time asymptotically to Oppenheimer-Snyder----a finite ball of mass expanding into empty space outside the black hole, something like a gigantic supernova.

10 Limitation of the model: TOV density and pressure are determined by the equations that describe the matching of the metrics p = c2 can only be imposed on the FRW side 3 ρ Imposing p = c2 on the TOV side introduces secondary 3 ρ waves which can not be modeled in an exact solution Question: How to model the secondary waves?

11 OUR QUESTION: How to refine the model to incorporate the correct TOV equation of state, and thereby model the secondary waves in the problem? OUR PROPOSAL: Get the initial condition at the end of inflation Use the Locally Inertial Glimm Scheme to simulate the region of interaction between the FRW and TOV metrics

12 DETAILS OF THE EXACT SOLUTION

13 The Final Equations p = σρ σ = const. Autonomous System S = 2S(1 + 3u) {(σ u) + (1 + u)s} F (S, u) u = (1 + u) { (1 3u)(σ u) + 6u(1 + u)s} G(S, u) Big Bang 0 S 1 Emerges From White Hole Entropy Condition: p < p, ρ < ρ, p < ρ S < ( ) ( ) 1 u σ u 1 + u σ + u E(u)

14 u = p ρ Entropy Inequalities 1/3 σ p < ρ p < p ρ < ρ u = E(S) S = 0 S = 1 Big Bang Solution emerges from White Hole 1 S The entropy conditions for an outgoing shock hold when u σ (S) < E(S)

15 u = p ρ Phase Portrait p = σρ 1/3 Rest Point σ < 1/3 Isocline u = E(S) σ Rest Point S = 0 Big Bang Q σ (S) u σ (S) S = 1 Solution emerges from White Hole 1 S s σ (S) shock speed < c for 0 < S < 1 s σ (S) 0 as S 0

16 u = p ρ Phase Portrait p = σρ σ = 1/3 Double Rest Point u = E(S) σ = 1/3 Isocline S = 0 Big Bang Q 1/3 (S) u σ (S) S = 1 Solution emerges from White Hole 1 S s σ (S) c as S 0

17 r = r R Shock Position= r(t) r r = c H 1 R =Hubble Length t = 0 S = 0 t = t S = 1 The Hubble length catches up to the shock-wave at S=1, the time when the entire solution emerges from the White Hole

18 Kruskall Picture 2M = 1 r t = + Black Hole Singularity FRW FRW Shock Surface FRW Universe TOV t = t Inside Black Hole Schwarzschild Spacetime Outside Black Hole p = 0 ρ = 0 p 0 ρ 0 White Hole Singularity t = 2M = 1 r

19 We are interested in the case σ = 1/3 correct for t = Big Bang to t = 10 5 yr ρ and p on the FRW and TOV side tend to the same values as t 0 It is as though the solution is emerging from a spacetime of constant density and pressure at the Big Bang Inflation

20 u = p ρ Phase Portrait p = σρ σ = 1/3 Double Rest Point u = E(S) σ = 1/3 Isocline S = 0 Big Bang Q 1/3 (S) u σ (S) S = 1 Solution emerges from White Hole 1 S s σ (S) c as S 0

21 Conclude: A solution like this would emerge at the end of inflation if the fluid at the end of inflation became co-moving wrt a (k = 0) FRW metric for r < r 0, and co-moving wrt the simplest spacetime of finite total mass for r > r 0. The inflationary disitter spacetime has all of the symmetries of a vaccuum, and so there is no preferred frame at the end of inflation

22 disitter spacetime in (k=0)-frw coordinates Finite-Mass time-slice at the end of Inflation disitter spacetime in TOV-coordinates 2M r > 1 and r timelike t = const. u co-moving wrt t = const. No preferred coordinates Inflation End of Inflation Shock T ij = ρ g ij T ij = (ρ + p)u i u j + pg ij M, r = const. u co-moving wrt M = const. (k = 0)-FRW ds 2 = dt 2 + R(t) 2 { dr 2 + r 2 dω 2} ds 2 = B( r)d t 2 + TOV 1 1 2M( r) r d r 2 + r 2 dω 2

23 time Proposed Numerical Simulation ds 2 = B( r, t)d t M( r, t) r d r 2 + r 2 dω 2 k=0 FRW metic Standard Schwarschild coordinates TOV metric inside the black hole ds 2 = dt 2 + R(t) 2 { dr 2 + r 2 dω 2} light cone ds 2 = B( r)d t M( r) r d r 2 + r 2 dω 2 space t = t 0 r = r 0 M = M( r 0 ) 2M( r 0 ) r 0 > 1 t = the end of inflation s = t 0

24 A Locally Inertial Method for Computing Shocks

25 Einstein equations-spherical Symmetry ds 2 = A(r, t)dt 2 + B(r, t)dr 2 + r 2 ( dθ 2 + sin 2 θ dφ 2) A r 2 B } {r B B + B 1 = κa 2 T 00 (1) G = 8πT 1 r 2 {r A A B t rb } = κabt 01 (2) (B 1) = κb 2 T 11 (3) 1 rab 2 {B tt A + Φ} = 2κr B T 22, (4) B = 1 1 2M r Φ = BA tb t 2AB B 2 ( Bt ) 2 A B r + AB rb + A ( ) A 2 + A A 2 A 2 A B B. (1)+(2)+(3)+(4) (1)+(3)+div T=0 (weakly)

26 References: The locally inertial Glimm Scheme... A shock-wave formulation of the Einstein equations, with J. Groah, Meth. and Appl. of Anal., 7, No. 4,(2000), pp Shock-wave solutions of the Einstein equations: Existence and consistency by a locally inertial Glimm Scheme, with J. Groah, Memoirs of the AMS, Vol. 172, No. 813, November Shock Wave Interactions in General Relativity: A Locally Inertial Glimm Scheme for Spherically Symmetric Spacetimes, with J. Groah and J. Smoller, Springer Monographs in Mathematics, 2007.

27 t Locally Inertial Glimm/Godunov Method u t + f(a, u) x = g(a, u, x) A = h(a, u, x) A 0 (t) R ij t = t j x = r 0 A ij = const. Fractional Step Method: = x i 1 u L = u i 1,j x i 1 2 R ij x i u R = u ij x i+ 1 2 x i+1 : A = A ij A = (A, B) Locally Flat in each Grid Cell Solve RP for 1/2 timestep Solve ODE for 1/2 time step Sample or Average u = (u 1, u 2 ) A = h(a, u, x) x A(r 0, t j+1 ) = A 0 (t j+1 ) p = c2 3 ρ = Nishida System = Global Exact Soln of RP, [Smol,Te]

28 Remarkable Change of Variables Equations close under change to Local Minkowski variables: I.e., Div T=0 reads: T u = T M 0 = T 00,0 + T 01, = T 01,0 + T 11, ( 2At A + B t B ) T ( At A + 3B ) t T B 2 ( 3A ( A A + B B + 4 r A + 2B B + 4 r ) ) + B t 2A T 11 T 11 + A 2B T 00 2 r B T 22 Time derivatives A t and B t cancel out under change T u Good choice because o.w. there is no A t equation to close Div T = 0! A r 2 B 1 r 2 } {r B B + B 1 {r A A = κa 2 T 00 (1) B t rb } = κabt 01 (2) (B 1) = κb 2 T 11 (3) 1 rab 2 {B tt A + Φ} = 2κr B T 22, (4)

29 Locally Inertial Formulation u = (T 00 M, T 01 M ) A = (A, B) { } { } T 00 A M +,0 B T M 01 { } { } T 01 A M +,0 B T M 11,1,1 B B A A = 2 x = 1 2 A A B = (B 1) x = (B 1) x B T 01 M, (1) { 4 x T 11 M + (B 1) (TM 00 TM 11 ) (2) x +2κxB(T 00 M T 11 M (T 01 M ) 2 ) 4xT 22}, + κxbt 00 M, (3) + κxbt 11 M. (4) TM 00 = c4 + σ 2 v 2 c 2 v 2 ρ T 01 M = c2 + σ 2 c 2 v 2 cvρ T 11 M = v2 + σ 2 c 2 v 2 ρc2 Flat Space Relativistic Euler { T 00 M },0 + { TM 10 },1 = 0 { T 01 M },0 + { TM 11 },1 = 0 u t + f(a, u) x = g(a, u, x) A = h(a, u, x)

30 Grid Rectangle R ij A = A ij u ij u i 1,j (i 12 ) (i, j (i, j) + 12 ), j Solve RP for 1 2 -timestep u t + f(a ij, u) x = 0 u = { u i 1,j u ij x x i x > x i u RP ij Solve ODE for 1 2 -timestep u t = g(a ij, u, x) A f A u(0) = u RP ij Sample/Average then update A to time t j+1 A = h(a, u, x) A(r 0, t j+1 ) = A 0 (t j+1 )

31 time Proposed Numerical Simulation ds 2 = B( r, t)d t M( r, t) r d r 2 + r 2 dω 2 k=0 FRW metic Standard Schwarschild coordinates TOV metric inside the black hole ds 2 = dt 2 + R(t) 2 { dr 2 + r 2 dω 2} light cone ds 2 = B( r)d t M( r) r d r 2 + r 2 dω 2 space t = t 0 r = r 0 M = M( r 0 ) 2M( r 0 ) r 0 > 1 t = the end of inflation s = t 0

32 Speculative Question: Could the anomolous acceleration of the Galaxies and Dark Energy be explained within Classical GR as the effect of looking out into a wave? This model represents the simplest simulation of such a wave

33 Standard Model for Dark Energy Assume Einstein equations with a cosmological constant: G ij = 8πT ij + Λg ij Assume k = 0 FRW: Leads to: ds 2 = dt 2 + R(t) 2 { dr 2 + r 2 dω 2} H 2 = κ 3 ρ + κ 3 Λ Divide by H 2 = ρ crit : 1 = Ω M + Ω Λ Best data fit leads to Ω Λ.73 and Ω M.27 Implies: The universe is 73 percent dark energy

34 Could the Anomalous acceleration be accounted for by an expansion behind the Shock Wave? Shock-Wave k=0 ds 2 = dt 2 + R(t) 2 { dr 2 + r 2 dω 2} FRW TOV: ds 2 = B( r)d t M( r) r d r 2 + r 2 dω 2 2M r > 1 r is timelike

35 Conclusion We think this numerical proposal represents a natural mathematical starting point for numerically resolving the secondary waves neglected in the exact solution. Also a possible starting point for investigating whether the anomalous acceleration/ Dark Energy could be accounted for within classical GR with classical sources?