Binary black holes and gravita4onal waves: Introduc4on to SpEC

Transcription

1 Binary black holes and gravita4onal waves: Introduc4on to SpEC Abdul Mroue Canadian Ins4tute for Theore4cal Astrophysics 2011 Interna4onal School on Numercical Rela4vity and Gravita4onal waves July 27- August 3, APCTP 1

2 Mo4va4on Explore fully non- linear general rela4vity solve the two body- problem Test general rela4vity and understand gravity Gravita4onal wave astronomy LIGO/Virgo, LISA, ET Astrophysics Solve challenging systems of PDEs numerically Finite difference Pseudo- spectral methods 2

3 Gravita4onal wave detectors Possible sources: Binary black holes Black hole/neutron star No direct detec4on yet of Gravita4onal waves 3

4 How to measure gravita4onal waves? Data analysis: Challenge: detection + parameters estimation Accurate template banks (NR waveforms to tune analytical models i.e. PN, EOB ) Small numerical phase error 4 δl L ~ h ~

5 How to measure gravita4onal waves? Data analysis: Challenge: detection + parameters estimation Accurate template banks (NR waveforms to tune analytical models i.e. PN, EOB ) Small numerical phase error 5 δl L ~ h ~

6 Evolu4on of binary black holes Need accurate model of GWs 6

7 Evolu4on of binary black holes GW? Need accurate model of GWs Knowledge of of the full waveform allows detec4on Astrophysics 7

8 No analy4c solu4on of the two- body problem Analy4cally, orbital dynamics of binaries can be analyzed using several approxima4ons: Post- Newtonian expansions Black hole perturba4on theory Effec4ve- one- body formalism Fully nonlinear numerical rela4vity 8

9 From Blanchet et al. arxiv:

10 Using the NR benchmark: Crucial cross- valida4on tests Domains of validity of each method Develop a universal semi- analy4cal model of binary dynamics 10

11 Numerical Rela4vity Basic idea : ADM formula4on 1962 Split space- 4me ds 2 = α 2 dt 2 + g ij (dx i + β i dt)(dx j + β j dt) Evolu4on equa4ons t g ij =... t K ij =... K ij 1 2 I n g ij Constraints R[g ij ] + K 2 K ij K ij = 0 j (K ij g ij K) = 0 Maxwell equa4ons t E = B t B = E E = 0 B = 0 11

12 Difficul4es with binary black holes simula4ons Solving for the space- 4me when the coordinate system evolves Constraint viola4ons grow exponen4ally Singulari4es inside BH: excision/moving puncture method Numerical methods: accuracy, supercomputers, spectral methods, mul4- domain decomposi4on, length scale 12

13 First binary black holes simula4ons: 2005 first binary back holes simula4ons (Pretorius 05: generalized harmonic system/ Campanelli ea 06, Baker ea 06: BSSN formula4on) Exploring the parameter space (mass ra4os, spin configura4ons) and compare to analy4cal approxima4ons Two approaches to evolve binary black holes: BSSN: finite difference with AMR Mul4- domain spectral methods: SpEC (Cornell- Caltech- CITA- Wash.) using GH system, difficult but very accurate, lower computa4onal cost 13

14 Spectral Einstein Code The spectral Einstein Code (SpEC) is a flexible infrastrucutre for solving par4al differen4al equa4ons using mul4- domain spectral methods. Originally, developed by Lawrence Kidder (Cornell), Harald Pfeiffer (CITA), and Mark Scheel (Caltech). Maj Duez, Francois foucart developed hydrodynamics module. Bela Szilagyi made numerous valuable addi4ons throughout the code. Further contribu4on by: Mike Boyle, Jeandrew Brink, Luisa Buchman, Tony Chu, Gregogry Cook, Dan Hemberger, Jeff Kaplan, Lee Lindblom, Geoffrey Lovelace, Keith Majhews, Abdul Mroue, Curran Muhlberger, Rob Owen, Nick Taylor, Saul Teukolsky, Anil Zenginoglu, and Fan Zhang Spectral Einstein Code SpEC (Caltech- Cornell- CITA) hjp:// holes.org/spec.html 14

15 Spectral Einstein Code Ini4al data Solve the constraints Write as ellip4c equa4ons Use spectral methods to solve to numerically Time evolu4on Generalized Harmonic system First order formula4on Domain decomposi4on Evolu4on using spectral methods and method of lines 15

16 Finite difference method: equa4ons on a grid, local method Approximate the par4al differen4al equa4ons (e.g. 1D wave equa4on) 2 u t 2 = 2 u x 2 u j n +1 2u j n + u j n 1 Δt 2 = u j +1 n 2u n n j + u j 1 Δx 2 16

17 Finite difference method: equa4ons on a grid, local method Approximate the par4al differen4al equa4ons (e.g. 1D wave equa4on) u j n +1 2u j n + u j n 1 Δt 2 = u j +1 n 2u n n j + u j 1 Δx 2 Approximate deriva4ves (e.g. spa4al deriva4ve) u(x,t) x u n j +1 n u j 1 h + O(h 2 ) 17

18 Spectral methods Global method that requires smooth solu4ons Approximate the solu4on Exact deriva4ves Method of choice for high spa4al resolu4on: error drops exponen4al Codes: (high accuracy, fast, memory efficient) u(x) u (N ) (x) du N dx = N 1 u k k =0 N 1 k =0 ε ~ e N u k φ k (x) dφ k (x) dx 18

19 Ini4al value problem Ini4al data must sa4sfy the constraints R[g ij ] + K 2 K ij K ij = 0 j (K ij g ij K) = 0 Unknown mathema4cal type Reformulate as an ellip4c problem For example, rewrite the Hamiltonian constraint equa4on using Choose free data g ij =ψ 4 g ij Solve numerically the ellip4c equa4on in ψ 19

20 Example: BBH ini4al data 20

21 Evolu4on equa4ons: GH system Using the metric, one can define Christoffel symbols and the Ricci tensor : γ Γ µν = 1 2 gγδ (g µδ,ν + g νδ,µ g µν,δ ) γ R µν = Γ µν,γ γ Γ γν,µ + Γ δ µν Γ γ δγ Γ δ γ γν Γ δµ Rewrite Einstein s equa4on as G µν R µν 1 2 Rg µν = 8πT µν R µν = 8π(T µν 1 2 g µν T) 21

22 Evolu4on equa4ons: GH system In vacuum, Einstein s equa4ons 0 = R µν [g µν ] = 1 2 gγδ γ δ g µν + (µ Γ ν ) +... where Γ µ = g µν γ γ x ν Harmonic coordinates Get a wave- like system γ γ x ν = 0 γ γ g µν =... 22

23 Evolu4on equa4ons: GH system Generalized harmonic coordinates: g µν γ γ x ν H µ New Constraints: C µ H µ g µν γ γ x ν = 0 Constraints damping (Gundlach et al., Pretorius 2005): 0 = 1 2 gγδ γ δ g µν + ( µ C ν ) + λ{t ( µ C ν ) 1 2 g µν t γ C γ } +... t C µ ~ λc a 23

24 First order formula4on Theory of hyperbolicity well know for first order systems Rewrite as first order symmetric hyperbolic system (Lindblom et al ) t u α + A(u) kα β k u β = F β (u) Approximate solu4on by truncated series u(x,t) N 1 k =0 u k (t)φ k (x) Evolve by using method of lines (use e.g. Runge- Kuja): t u(x i ) = [F A(u) k k u] x =xi 24

25 Boundary condi4ons and singularity excision Compute the Characteris4c fields: e α ˆ αn k A kα β = v ( α ˆ ) e α ˆ β Impose BCs on incoming fields v ( α ˆ ) > 0 Near the black holes, all characteris4c fields are incoming, no need for boundary condi4ons Sketches courtesy of Harald Pfeiffer 25

26 Domain- decomposi4on Spectral Einstein Code SpEC (Caltech- Cornell- CITA) hjp:// holes.org/spec.html 26

27 Computer simula4on (movie by H. Pfeiffer et al.) (Caltech/Cornell/CITA) 27

28 References B. Fornberg (1996), A Prac4cal Guide to Pseudospectral Methods, Cambridge University Press New York J. P. Boyd (2001), Chebyshev and Fourier Spectral Methods, 2 nd ed., Dover Publica4ons, Mineola, New York., hjp://www- personal.engin.umich.edu/~jpboyd M. Bjorhus (1995), The ODE formula4on of Hyperbolic PDEs discre4zed by the Spectral Colloca4on Method, SIAM J. Sci. Comput. 16, 542. H. Pfeiffer, L. Kidder, M. Scheel and S. Teukolsky (2003), A mul4domain spectral method for solving ellip4c equa4ons, Comput. Phys. Commun. 152, 253 G. Cook, H. Pfeiffer, Excision boundary condi4ons for black hole ini4al data, PRD 70, 2004, L. Kidder et al, Black hole evolu4on by spectral methods, PRD 62, F. Pretorius, Evolu4on of Binary Black Hole Space4mes, PRL 95: , 2005 Lindblom et al., A New Generalized Harmonic Evolu4on System, CQG.23: S447- S462, 2006 M. Scheel et al., Solving Einstein s equa4ons with dual coordinate frames, PRD 74, 2006, M. Scheel et al., High- accuracy waveforms for binary black hole inspiral, merger and ringdown, PRD79:024003,