1. Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008

2. Introduction –Binary black hole problem Some ingredients –3+1 decomposition –Formulation of the equations –Initial data –Gauge –Mesh refinement –Boundary conditions –Excision –Diagnostic tools Applications ConclusionsOverview

3. The big picture Model GR (numerical relativity) PN Perturbation theory Non-GR? External Physics Astrophysics Fundamental physics Cosmology Detectors Physical System describes observe test implications Help detect

4. Numerical relativity -Two 10 solar mass black holes -Frequency ~ 100Hz -Distort the 4km mirror spacing by about 10^-18 m

5. 3+1 decomposition GR: “Space and time exist together as Spacetime’’ Numerical relativity: reverse this process! ADM 3+1 decomposition Arnowitt, Deser, Misner (1962) York (1979) Choquet-Bruhat, York (1980) 3-metric lapse shift lapse, shift Gauge Einstein equations 6 Evolution equations 4 Constraints Constraints preserved under evolution!

6. ADM equations Evolution equations Constraints Evolution Solve constraints initially Evolve data Reconstruct spacetime Extract physics

7. Formulation of the equations ADM: unsuccessful; weakly hyperbolic! BSSN (most popular) Shibata, Nakamura ‘95 Baumgarte, Shapiro ‘99 Balance laws: Bona, Massó (H-code) Many more: Sarbach et.al.‘02; Gundlach, Martin-Garcia Split degrees of freedom (similar to initial data split) Hyperbolicity Generalized harmonic formulation Harmonic gauge well-posed Wave equations for BBH-breakthrough Choquet-Bruhat ‘62 Pretorius ‘05 ADM-like family: Harmonic family: Control of constraints: KST, NOR,… Z4 LSU, Caltech, Gundlach Garfinkle ‘04

8. The BSSN formulation

9. Initial data Two difficulties: Constraints, realistic data York-Lichnerowicz split Conformal transverse traceless Physical transverse traceless Thin sandwich York, Lichnerowicz O’Murchadha, York Wilson, Mathews; York Conformal flatness: Spurious radiation does not seem problematic, but alternatives studied Generalized analytic solutions: Time symmetric, -holes: Spin, linear momenta: Punctures Brill-Lindquist, Misner (1960s) Bowen, York (1980) Brandt, Brügmann (1997) Isotropic Schwarzschild Excision data: Isolated Horizon condition on excision surface Meudon group; Cook, Pfeiffer; Ansorg Quasi-circularity: Effective potential method PN fit helical killing vector

10. Gauge Specific problem in GR: Coordinates constructed during evolutions Highly non-trivial:Prescribe to avoid coordinate singularities Einstein equations say nothing about Maximal slicing, min.distortion shift Smarr, York ‘78 Driver conditions Balakrishna et.al.’96 1+log, -driver AEI Moving punctures UTB, Goddard ‘06 Bona-Massó family Bona, Massó ‘95 Harmonic coords Choquet-Bruhat‘62 Generalized harmonic Garfinkle ‘04 Pretorius ‘05 Study singularity avoidance Alcubierre ‘03 Analytic studies gauge sources relation to Drive to stationarity special case special case

11. Mesh-refinement, boundaries 3 length scales: BH Wave length Wave zone Choptuik ’93 AMR, Critical phenomena Stretch coordinates: Fish-eye Lazarus, AEI, UTB FMR, Moving boxes: Berger-Oliger BAM Brügmann’96 Carpet Schnetter et.al.’03 AMR: Steer resolution via scalar Paramesh: MacNeice et.al.’00, Goddard modified Berger-Oliger: Pretorius, Choptuik ’05 SAMRAI Refinement boundaries: reflections, stability Lehner, Liebling, Reula ‘05

12. Outer boundary conditions Problems: Well-posedness of equations? Constraint violations? BCs that satisfy constraints and/or well-posedness Friedrich, Nagy ‘99 Calabrese, Lehner, Tiglio ‘02 Frittelli, Goméz ‘04 Sarbach, Tiglio ‘04 Kidder et.al.‘05, Lindblom et.al.‘06 Tested with success in BBH simulation: Lindblom et.al.‘06 Conformal, null-formulation: Untested in BBH simulations Compactification in 3+1 Pretorius ‘05 Push boundaries “far out”, use Sommerfeld condition Used successfully by most groups; accuracy limits? Multi-patch approach: Efficiency AEI (Cactus): Thornburg et.al.: excision, Char.Code LSU (below), Austin (below), Cornell-Caltech (below)

13. Black hole excision Cosmic censorship: Causal disconnection of region inside AH Unruh ’84 cited in Thornburg ‘87 Grand Challenge: Causal differencing “Simple Excision” Alcubierre, Brügmann ‘01 Dynamic “moving” excision Pitt-PSU-Texas PSU-Maya Pretorius combined with Dual coordinate frame Caltech-Cornell Mathematical properties: Wealth of literature

14. Diagnostic tools A computer just gives numbers! These are gauge dependent! Convert to physical information… ADM mass, momentum Arnowitt, Deser, Misner ‘62 Bondi mass, News function (Characteristic approach) Gravitational Waves Zerilli-Moncrief formalism Newman-Penrose scalar Black hole quantities: mass, momentum, spin, area,… Apparent Horizon Alcubierre, Gundlach (Cactus) Schnetter ‘03 Thornburg ‘03 (AHFinderDirect) Pretorius Event horizon Diener ‘03 Isolated, Dynamic Hor. Ashtekar, Krishnan ’03 Ashtekar et.al. Dreyer et.al. ’02

15. 2004 2007 How far we are?

16. Spinning holes: The orbital hang-up Spins alligned with inspiral delayed, larger Spins anti-alligned with inspiral fast smaller No extreme Kerr holes produced

17. Gravitational recoil Anisotropic emission of GW carries away linear momentum recoil of remaining system Merger of galaxies Inspiral and merger of black holes Recoil of merged hole Displacement, Ejection? Astrophysical relevance BH inspiral kick possible ejection of BH from host Escape velocities: globular clusters dSph dE large galaxies Merritt et al.’04

18. Non-spinning binaries Emerging picture: Kicks unlikely to exceed a few Numerical relativity allows accurate estimates Campanelli ’05 Herrmann et al.’06 Baker et al.’06 Close limit calculations Sopuerta et al.’06 a,b Upper and lower bounds Including eccentricity increases kick for small eccentricities EOB approximation: account for deviations from Kepler law Damour & Gopakumar ‘06

19. Non-spinning binaries Systematic parameter study Gonzalez et al.’06 Moving puncture method BAM code Nested boxes, resolutions Extract calculate linear momentum Vary mass ratio: 150,000 CPU hours Higher order PN Blanchet et al.’05

20. Non-spinning binaries: Maximal kick Maximal kick: at

21. Recoil of spinning binaries Kidder ’95 : PN study including recoil of spinning holes = “unequal mass” + “spin(-orbit)” Penn State ‘07: Spin-orbit term larger extrapolated: AEI ’07: extrapolated:

22. Recoil of spinning binaries UTB-Rochester maximum predicted: NASA Goddard: Spin effect Unequal-mass effect PN predictions remarkably robust Fitting formulas

23. Discretization error: Trajectories: Getting even larger kicks

24. Dependence on Extraction radius Error fall-off:

25. Reducing eccentricity

26. Data analysis and PN comparisons Thick red line  NR waveforms Dashed black  ‘best matched’ 3.5 PN waveforms Thin green  Hybrid waveforms Since it is expensive to generate an entire physical bank of templates using numerical simulations, it is better to construct a phenomenological bank –unequal mass, non spinning black holes-

27. Eccentricity

28. IMRI’s: Motivation Stellar mass black holes  M~1-10 M sun Intermediate mass bh’s  M~10 2-4 M sun Supermassive bh’s  M~10 6-9 M sun Why IMRIs and EMRIs are interesting? Astrophysics Data Analysis and gravitational waves detection: Gravitational waves emited during the merger of stellar-mass black holes into a IMBHs will lie in the frequencies of Advanced LIGO (Brown et al. 2007) Tests of General Relativity Comparison with PN and perturbation theory

29. Numerical simulations are expensive How many orbits are required? Data analysis 10? 100? How far we need to go in mass ratios?  Compare with PN! 1:100? 1:1000???  Hopefully not!

30. Mass ratio 1:10 –Resolution: [η]= 1/M Problems: M1 = 0.25, M2 = 2.5, M = M1+M2 D = 19.25 = 7M q = M1/M2 = 10, η = q/(1+q) 2 = 0.0826 Parameters: –Gauge:

33. Kick Fitchett (MNRAS 203 1049,1983) Gonzalez et al. (PRL 98 091101, 2007) V~62 km/s

34. Radiated energy ΔE/M=0.580192 η 2 ΔE/M~0.004018 Berti et al. (2007)

35. Final spin Damour and Nagar (2007) a a F /M F ~0.2602

36. E RAD = 0.011001 l=2 75.62% l=3 16.36% l=4 4.96% l=5 1.74% Energy distribution

37. Conclusions After a lot of work and effort….it seems to work! It is over? No way! –It is necessary to improve accuracy –Now it is possible to do physics –the original purpose of everything- –Data analysis –Parameter estimations Matter