Numerical relativity simulations for GW Astrophysics Harald - PowerPoint PPT Presentation

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