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Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros

Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros. Jose Antonio Gonzalez IFM-UMSNH 25-Abril-2008 ENOAN 2008. Overview. Introduction Binary black hole problem Some ingredients 3+1 decomposition Formulation of the equations Initial data Gauge Mesh refinement

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Solucion numerica de las ecuaciones de Einstein: Choques de agujeros negros

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

  2. Overview • 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 • Conclusions

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

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

  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: Isotropic Schwarzschild Time symmetric, -holes: Brill-Lindquist, Misner (1960s) Spin, linear momenta: Bowen, York (1980) Punctures Brandt, Brügmann (1997) • Excision data: Isolated Horizon condition on excision surface Meudon group; Cook, Pfeiffer; Ansorg Effective potential method PN fit helical killing vector • Quasi-circularity:

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

  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 • CarpetSchnetter 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 ’84cited in Thornburg ‘87 • Grand Challenge: Causal differencing • “Simple Excision” • Alcubierre, Brügmann ‘01 • Dynamic “moving” excision Pitt-PSU-Texas PSU-Maya Pretorius • combined withDual 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, momentumArnowitt, Deser, Misner ‘62 • Bondi mass, News function (Characteristic approach) • Gravitational Waves Zerilli-Moncriefformalism 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 How far we are? 2007

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

  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 • Higher order PN • Blanchet et al.’05 • Systematic parameter study Gonzalez et al.’06 Moving puncture method BAMcode Nested boxes, resolutions Extract calculate linear momentum Vary mass ratio: 150,000 CPU hours

  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. Getting even larger kicks • Trajectories: • Discretization error:

  24. Dependence on Extraction radius • Error fall-off:

  25. Reducing eccentricity

  26. Data analysis and PN comparisons 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- Thick red line  NR waveforms Dashed black  ‘best matched’ 3.5 PN waveforms Thin green  Hybrid waveforms

  27. Eccentricity

  28. IMRI’s: Motivation • Stellar mass black holes  M~1-10 Msun • Intermediate mass bh’s  M~102-4 Msun • Supermassive bh’s  M~106-9 Msun 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?  Compare with PN! • How far we need to go in mass ratios? 1:100? 1:1000???  Hopefully not!

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

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

  32. Radiated energy ΔE/M=0.580192 η2 Berti et al. (2007) ΔE/M~0.004018

  33. Final spin aF/MF~0.2602 Damour and Nagar (2007)

  34. Energy distribution ERAD = 0.011001 l=2 75.62% l=3 16.36% l=4 4.96% l=5 1.74%

  35. 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

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