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Binary Black Hole Simulations Binary Black Hole Simulations Harald Pfeiffer California Institute of Technology Mike Boyle, Duncan Brown, Lee Lindblom, Geoffrey Lovelace, Larry Kidder, Mark Scheel, Saul Teukolsky APS April Meeting, Jacksonville, Apr 17, 2007 Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 1 / 1 Gravitational wave detectors LISA (201x) LIGO (2 sites) VIRGO GEO 600 Among prime targets: Binary black hole systems Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 2 / 1 Stages of binary black hole evolution Knowledge of waveform allows to I Enhance detector sensitivity I Test general relativity I Extract information about source (→ astrophysics) Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 3 / 1 Tasks for Numerical relativity: – simulate “late” inspiral and merger. – determine what “late” means. Tools for computing the waveform 2 Merger Inspiral – v c: perturbative expansion in v/c 1 (post-Newtonian expansion) Inspiral Ringdown – v/c large: Numerical relativity 0 Merger – Numerical relativity -1 Ringdown – BH perturbation theory -300 -200 -100 0 100 t/m – Numerical relativity Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 4 / 1 Tools for computing the waveform 2 Merger Inspiral – v c: perturbative expansion in v/c 1 (post-Newtonian expansion) Inspiral Ringdown – v/c large: Numerical relativity 0 Merger – Numerical relativity -1 Ringdown – BH perturbation theory -300 -200 -100 0 100 t/m – Numerical relativity Tasks for Numerical relativity: – simulate “late” inspiral and merger. – determine what “late” means. Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 4 / 1 Since late 1990s: Groundwork and foundations I New evolution systems I AMR and spectral infrastructures I Initial data I Boundary conditions, gauge conditions 2005: The last pieces! a) Pretorius – constraint damping for generalized harmonic b) Goddard and Brownsville – Gauge conditions for moving punctures Since 2005: The Golden Age of Numerical Relativity History 1964 Hahn & Lindquist: Collisions of Wormholes 1970’s Smarr & Eppley: Head on collisions 1994-99 NSF Binary black hole grand challange Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 5 / 1 2005: The last pieces! a) Pretorius – constraint damping for generalized harmonic b) Goddard and Brownsville – Gauge conditions for moving punctures Since 2005: The Golden Age of Numerical Relativity History 1964 Hahn & Lindquist: Collisions of Wormholes 1970’s Smarr & Eppley: Head on collisions 1994-99 NSF Binary black hole grand challange Since late 1990s: Groundwork and foundations I New evolution systems I AMR and spectral infrastructures I Initial data I Boundary conditions, gauge conditions Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 5 / 1 Since 2005: The Golden Age of Numerical Relativity History 1964 Hahn & Lindquist: Collisions of Wormholes 1970’s Smarr & Eppley: Head on collisions 1994-99 NSF Binary black hole grand challange Since late 1990s: Groundwork and foundations I New evolution systems I AMR and spectral infrastructures I Initial data I Boundary conditions, gauge conditions 2005: The last pieces! a) Pretorius – constraint damping for generalized harmonic b) Goddard and Brownsville – Gauge conditions for moving punctures Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 5 / 1 History 1964 Hahn & Lindquist: Collisions of Wormholes 1970’s Smarr & Eppley: Head on collisions 1994-99 NSF Binary black hole grand challange Since late 1990s: Groundwork and foundations I New evolution systems I AMR and spectral infrastructures I Initial data I Boundary conditions, gauge conditions 2005: The last pieces! a) Pretorius – constraint damping for generalized harmonic b) Goddard and Brownsville – Gauge conditions for moving punctures Since 2005: The Golden Age of Numerical Relativity Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 5 / 1 Computational approaches in the Golden Age Finite difference AMR I Albert-Einstein Institut (Germany), Goddard, Jena (Germany), LSU, PSU, Gravitational wave flux small Princeton, Rochester −5 I Impressive short inspirals with I E˙ /E ∼ 10 mergers I E˙ drives inspiral I Accurate long inspirals difficult High accuracy required Multi-domain spectral methods I Absolute phase error δφ 1 I Cornell/Caltech Solutions are smooth I Impressive long inspiral simulations I Merger difficult Overview Problem characteristics Multiple length scales I Size of BH’s ∼ 1 I Separation ∼ 10 I Wavelength λ ∼ 100 I Wave extraction at several λ Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Computational approaches in the Golden Age Finite difference AMR I Albert-Einstein Institut (Germany), Goddard, Jena (Germany), LSU, PSU, Princeton, Rochester I Impressive short inspirals with mergers I Accurate long inspirals difficult High accuracy required Multi-domain spectral methods I Absolute phase error δφ 1 I Cornell/Caltech Solutions are smooth I Impressive long inspiral simulations I Merger difficult Overview Problem characteristics Multiple length scales I Size of BH’s ∼ 1 I Separation ∼ 10 I Wavelength λ ∼ 100 I Wave extraction at several λ Gravitational wave flux small −5 I E˙ /E ∼ 10 I E˙ drives inspiral Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Computational approaches in the Golden Age Finite difference AMR I Albert-Einstein Institut (Germany), Goddard, Jena (Germany), LSU, PSU, Princeton, Rochester I Impressive short inspirals with mergers I Accurate long inspirals difficult Multi-domain spectral methods I Cornell/Caltech Solutions are smooth I Impressive long inspiral simulations I Merger difficult Overview Problem characteristics Multiple length scales I Size of BH’s ∼ 1 I Separation ∼ 10 I Wavelength λ ∼ 100 I Wave extraction at several λ Gravitational wave flux small −5 I E˙ /E ∼ 10 I E˙ drives inspiral High accuracy required I Absolute phase error δφ 1 Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Computational approaches in the Golden Age Finite difference AMR I Albert-Einstein Institut (Germany), Goddard, Jena (Germany), LSU, PSU, Princeton, Rochester I Impressive short inspirals with mergers I Accurate long inspirals difficult Multi-domain spectral methods I Cornell/Caltech I Impressive long inspiral simulations I Merger difficult Overview Problem characteristics Multiple length scales I Size of BH’s ∼ 1 I Separation ∼ 10 I Wavelength λ ∼ 100 I Wave extraction at several λ Gravitational wave flux small −5 I E˙ /E ∼ 10 I E˙ drives inspiral High accuracy required I Absolute phase error δφ 1 Solutions are smooth Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Finite difference AMR I Albert-Einstein Institut (Germany), Goddard, Jena (Germany), LSU, PSU, Princeton, Rochester I Impressive short inspirals with mergers I Accurate long inspirals difficult Multi-domain spectral methods I Cornell/Caltech I Impressive long inspiral simulations I Merger difficult Overview Problem characteristics Computational approaches Multiple length scales in the Golden Age I Size of BH’s ∼ 1 I Separation ∼ 10 I Wavelength λ ∼ 100 I Wave extraction at several λ Gravitational wave flux small −5 I E˙ /E ∼ 10 I E˙ drives inspiral High accuracy required I Absolute phase error δφ 1 Solutions are smooth Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Multi-domain spectral methods I Cornell/Caltech I Impressive long inspiral simulations I Merger difficult Overview Problem characteristics Computational approaches Multiple length scales in the Golden Age I Size of BH’s ∼ 1 Finite difference AMR I Separation ∼ 10 I Albert-Einstein Institut I Wavelength λ ∼ 100 (Germany), Goddard, Jena I Wave extraction at several λ (Germany), LSU, PSU, Gravitational wave flux small Princeton, Rochester −5 I Impressive short inspirals with I E˙ /E ∼ 10 mergers I E˙ drives inspiral I Accurate long inspirals difficult High accuracy required I Absolute phase error δφ 1 Solutions are smooth Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Overview Problem characteristics Computational approaches Multiple length scales in the Golden Age I Size of BH’s ∼ 1 Finite difference AMR I Separation ∼ 10 I Albert-Einstein Institut I Wavelength λ ∼ 100 (Germany), Goddard, Jena I Wave extraction at several λ (Germany), LSU, PSU, Gravitational wave flux small Princeton, Rochester −5 I Impressive short inspirals with I E˙ /E ∼ 10 mergers I E˙ drives inspiral I Accurate long inspirals difficult High accuracy required Multi-domain spectral methods I Absolute phase error δφ 1 I Cornell/Caltech Solutions are smooth I Impressive long inspiral simulations I Merger difficult Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 6 / 1 Split space-time into space and time t+dt t Evolution equations ∂t gij = ... cf. Maxwell equations ∂t Kij = ... ~ ~ ∂t E = ∇ × B ~ ~ Constraints ∂t B = −∇ × E R[g ] + K 2 − K K ij = 0 ij ij ∇ · E~ = 0 ∇ K ij − gij K = 0 j ∇ · B~ = 0 Solving Einstein’s equations – basic idea Task: Find space-time metric gab such that Rab[gab] = 0 Harald Pfeiffer (Caltech) Binary black hole simulations APS April Meeting, 2007 7 / 1 t+dt t Evolution equations ∂t gij = ... cf. Maxwell equations ∂t Kij = ... ~ ~ ∂t E = ∇ × B ~ ~ Constraints ∂t B = −∇ × E
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