Numerical relativity simulations for GW Astrophysics Harald Pfeiffer AEI Program Advances in Computational Relativity ICERM, Oct 7, 2020 Image: Nils Fischer (AEI)
“GW150914” Abbott+ PRL 12016 Waveform knowledge essential for GW astronomy Parameter estimation Detection by matched filtering LIGO+Virgo, PRX 2016 (1606.04856) Validation Testing GR LIGO+Virgo, PRL 2017 (1706.01812) 2 H. Pfeiffer LIGO & Virgo: CQG 2017 (1611.07531)
In future, need higher accuracy for more diverse systems LISA 3G & LISA: expected SNRs q = 1…10 − 6 among sources: BBH needed accuracy ~ 1/SNR among science targets: eccentricity measurement to δ e < 0.001 LISA proposal 2017 GWIC, https://gwic.ligo.org/3Gsubcomm/documents/science-case.pdf 3 H. Pfeiffer
Methods for modeling BBH Inspiral Merger Ringdown frequency 4 H. Pfeiffer
Methods for modeling BBH Inspiral Merger Ringdown 0 mass-ratio q 1 post-Newtonian theory BH perturbation (and PM & EOB) theory , perturbation theory in 1/q y t i c … i r t frequency n , n e i c p c s e 4 H. Pfeiffer
Methods for modeling BBH Inspiral Merger Ringdown 0 mass-ratio q 1 post-Newtonian theory BH perturbation (and PM & EOB) theory , perturbation theory in 1/q y t i c … i r t frequency n , n e i c p c s e 4 H. Pfeiffer
Methods for modeling BBH Inspiral Merger Ringdown 0 mass-ratio q 1 post-Newtonian LISA theory BH perturbation (and PM & EOB) x theory , perturbation theory in 1/q y t i c … i r t frequency n , n e i c p c s e 4 H. Pfeiffer
Role of NR • Solution of GR for late inspiral + merger • Provide error estimates • Determine regions of validity of perturbative methods - all available perturbation orders needed for science - No extra order for error estimate • Validate GW data-analysis • Black holes, neutron stars … and exotic objects, alternative theories 5 H. Pfeiffer
Role of NR • Solution of GR for late inspiral + merger • Provide error estimates • Determine regions of validity of perturbative methods - all available perturbation orders needed for science - No extra order for error estimate • Validate GW data-analysis • Black holes, neutron stars … and exotic objects, alternative theories 5 H. Pfeiffer
Solving Einstein Equations - Basic idea • Goal: Space-time metric g ab satisfying • Split spacetime into space and time • Evolution equations cf. Maxwell’s equations • Constraints 6 H. Pfeiffer
Why is this hard? • ADM equations ill-posed ; rewrite as hyperbolic system • Singularities inside black holes • Constraints difficult to preserve • Coordinate freedom - How to choose coordinates for a space-time one does not know yet? • Many common numerical challenges - 20-50 variables - 10,000 FLOP / grid-point / time-step - Different length scales, high accuracy requirements 7 H. Pfeiffer
The very beginning 8 H. Pfeiffer
The first 50 Years of numerical relativity for BBH 2007- 2005 Pretorius 2000-04 1962 ADM 1992,3 Ajith, AEI, Jena inspiral-merger- AEI / UTB-NASA 3+1 formulation Choptuik; 1999-00 2011 ringdown (IMR) phenom GW models revive crashing Abrahams+Evans w/ harmonic AEI/PSU Lousto ea codes ( Lazarus) critical phenomena 2009- q=100 grazing collisions 2005-06 1964 UMD, SXS 2014- Campanelli+; Baker+ 1997 ~2000 Choptuik; EOB GW models Hahn-Lindquist IMR w/ BSSN & precessing Brandt- Schnetter;Brügmann moving punctures 2 wormholes 2011 GW models Brügmann mesh refinement Schmidt ea; 2015 2006-08 1984 puncture data 2005 Boyle ea Szilagyi ea Scheel..HP+ SXS Unruh 1994-98 Gundlach ea Radiation aligned 175 orbits IMR w/ spectral excision BBH Grand Challenge frame constraint damping 2015 1964 ~1999 ~2005 1975-77 2006,07 1999 2000-02 2011 2008 1994 Cook Smarr-Eppley BSSN Baker ea; Alcubierre Lovelace ea all of NR Bowen-York head-on evolution Gonzalez ea S/M 2 =0.97 gauge conditions NINJA initial data collision system 2004 non-spinning BBH 2011- 1994-95 1979 York Brügmann ea kicks 1999 Y ork Le Tiec ea NCSA-WashU 2009-11 kinematics and one orbit 2007 SXS conformal thin self-force studies dynamics of GR improved Bishop, ... sandwich ID 2003-08 PN-NR head-on collision 2013 1989-95 Cauchy comparison Cook, Pfeiffer ea GaTech; SXS characteristic Bona-Masso improved ID extraction Precessing 2007-11 modified ADM, 1999-2005 York, parameter 2000 Ashtekar (hyperbolicity) 2010 RIT; Jena; AEI;… Cornell, Caltech, LSU studies isolated horizons Courtesy Carlos Lousto, BBH superkicks Bernuzzi ea hyperbolic formulations updated by HP C4z 9 H. Pfeiffer
2005: First working BBH inspirals Campanelli+06 Baker+06 Pretorius 05 Important early result: Simplicity of merger Continuous transition Baker+07 inspiral → ringdown 10 H. Pfeiffer
Two approaches towards BBH simulations “BSSN & Moving punctures” “generalized harmonic & spectral” LazEv, Maya, BAM, Goddard SpEC (SXS collaboration) Puncture initial-data Quasi-equilibrium excision data (but see Zlochower+ 17) χ ≲ 0.9 χ ≲ 0.999 BSSN or CC4z Generalized-Harmonic Evolution System Moving puncture BH excision mergers “easy” mergers di ffi cult Sommerfeld outer BC Constraint preserving, minimally reflective outer BC 4th to 8th order finite-di ff erence Spectral methods BHs advect through static grids Moving grid long, phase-accurate inspirals GW extrapolation GW extrapolation & COM correction (Healy,Lousto ’20 for LazEv COM correction) Cauchy-characteristic extraction accurate m=0 modes, GW memory 11 H. Pfeiffer
Spectral Einstein Code (SpEC) Simulating eXtreme Spacetimes collaboration http://www.black-holes.org/SpEC.html 12 H. Pfeiffer
Spectral methods Spectral • Expand in basis-functions, solve for coefficients N � u ( x, t ) = u ( t ) k Φ k ( x ) ˜ k =1 • Compute derivatives exactly N X u 0 ( x, t ) = u ( t ) k Φ 0 ˜ k ( x ) Finite differences k =1 • Compute nonlinearities in physical space • For smooth problems, exponential convergence 13 H. Pfeiffer
Domain-decomposition • Many sub-domains, each with own basis-functions - Spheres - Blocks - Cylinders • Advantages: - Excision of BH singularities - Adaptive Resolution - Parallelization http://www.black-holes.org/SpEC.html 14 H. Pfeiffer
Domain-decomposition • Many sub-domains, each with own basis-functions - Spheres - Blocks - Cylinders • Advantages: - Excision of BH singularities - Adaptive Resolution - Parallelization http://www.black-holes.org/SpEC.html 14 H. Pfeiffer
Domain-decomposition • Many sub-domains, each with own basis-functions - Spheres - Blocks - Cylinders • Advantages: - Excision of BH singularities - Adaptive Resolution - Parallelization http://www.black-holes.org/SpEC.html 14 H. Pfeiffer
Domain-decomposition • Many sub-domains, each with own basis-functions - Spheres - Blocks - Cylinders • Advantages: - Excision of BH singularities - Adaptive Resolution - Parallelization http://www.black-holes.org/SpEC.html 14 H. Pfeiffer
Einstein constraints: Formalism g = ψ 4 ˜ g R + (tr K ) 2 � K 2 = 0 ˜ r 2 ψ = . . . Lichnerowicz 44 r · ( K � g tr K ) = 0 ˜ σ ˜ r · ( 1 L V ) = . . . ˜ coupled nonlinear elliptic PDEs in 3D K = 1 3 tr K g + A conformal scaling A ˜ A A = ψ − 10 ˜ A conformal TT TT decomp. decomp. conformal scaling A TT + 1 A = A TT + 1 A = ˜ ˜ σ (˜ L V ) σ ( L V ) A TT = ψ − 10 ˜ ˜ A TT σ = ψ 6 ˜ σ Hamiltonian picture ≡ Lagrangian picture York(+) 72;74;99, HP ,York 03 15 H. Pfeiffer
Applied to binary black holes ˜ r 2 ψ = . . . • Asymptotics/boundary conditions ˜ N ˜ r · ( 1 Brandt, Brügmann 97; Cook,HP 04 L β )= . . . ˜ • Elliptic solver r 2 ˜ ˜ N = . . . HP+ 02, Ansorg 04 • Spins > 0.9 • Control eccentricity Lovelace..HP+ 08 1 E rot / E rot, max 0.8 0.6 w/ conformal flatness 0.4 0.9995 0.2 0 0 0.25 0.5 0.75 1 HP+ 05; Buonanno..HP+ 08 2 S/M Chatziiouannou, HP+ (in prep) 16 H. Pfeiffer
Einstein Evolution Equations 17 H. Pfeiffer
BH Excision • Excise inside BH horizons • Domain-decomposition follows BHs continuously , conforms to shape of AH al., 2006 t t Horizon Horizon Horizon Horizon Outside Outside Outside Outside Horizon Horizon Horizon Horizon x x x x Scheel, HP+ 08, Szilagyi+ 08, Hemberger+ 13 18 H. Pfeiffer
Outer boundary • In SpEC: - Constraint preserving - Minimally reflective Lindblom, Rinne+ 06 • Causally connected for Δ GW-phase (radians) long simulations Buchman, HP , Scheel, Szilagyi, 2012 19 H. Pfeiffer
Accuracy of SpEC GW precision data Scheel,HP+ 09 20 H. Pfeiffer
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