Accurate Phenomenological Waveform Models for BH Coalescence in the Frequency Domain

Transcription

1 Accurate Phenomenological Waveform Models for BH Coalescence in the Frequency Domain Goal: synthesize inspiral-merger-ringdown models of the complete WF of Compact Binary Coalescence from pn, NR, BH perturbation theory, self-force, S. Husa, Universitat de les Illes Balears 28th Texas Symposium, 12/2015

2 Accurate Phenomenological Waveform Models for BH Coalescence in the Frequency Domain Goal: synthesize inspiral-merger-ringdown models of the complete WF of Compact Binary Coalescence from pn, NR, BH perturbation theory, self-force, S. Husa, Universitat de les Illes Balears 28th Texas Symposium, 12/2015

3 Accurate Phenomenological Waveform Models for BH Coalescence in the Frequency Domain Goal: synthesize inspiral-merger-ringdown models of the complete WF of Compact Binary Coalescence from pn, NR, BH perturbation theory, self-force, Frequency-domain gravitational waves from non-precessing black-hole binaries - I. New numerical waveforms and anatomy of the signal II. A phenomenological model for the advanced detector era arxiv: , SH, S Khan, M Hannam, M Pürrer, F Ohme, X Jiménez Forteza, A Bohé arxiv: , S Khan, SH, M Hannam, F Ohme, M Pürrer, X Jiménez Forteza, A Bohé New work with X Jiménez Forteza & D Keitel S. Husa, Universitat de les Illes Balears 28th Texas Symposium, 12/2015

4 Motivation Optimal analysis of data from GW detectors relies on matched filtering with accurate template waveforms. hh 1,h 2 i = max SNR: 0,t 0 4< = h Z f2 f 1 h1 (f) h 2(f) S n (f) df M =1 hh 1,h 2 i/( h 1 h 2 ) 2005 breakthrough in NR: Pretorius, NASA Goddard/Brownsville short time scale to explore consequences for GW data analysis Applications of waveforms: Injections Searches + Bayesian parameter estimation

5 Motivation Optimal analysis of data from GW detectors relies on matched filtering with accurate template waveforms breakthrough in NR: Pretorius, NASA Goddard/Brownsville short time scale to explore consequences for GW data analysis Applications of waveforms: Injections Searches + Bayesian parameter estimation

6 Phenomenological modelling of IMR waveforms Key design ideas [alternative choices: Effective One Body ] phenomenological : minimal assumptions - look at waveforms and describe what we see. [EOB-model] Frequency domain: matched filter calculations in Freq. domain [time domain] Explicit expression in terms of elementary functions -> fast, simple [ODEs + optional ROM acceleration] Minimal ingredients: PN approximate to describe low frequencies: uncalibrated EOB Set of NR WFs: SXS + BAM Prediction for BH remnant: New fits for final mass & spin -> QNM freq.

7 Phenomenological modelling of IMR waveforms Key design ideas [alternative choices: Effective One Body ] phenomenological : minimal assumptions - look at waveforms and describe what we see. [EOB-model] Frequency domain: matched filter calculations in Freq. domain [time domain] Explicit expression in terms of elementary functions -> fast, simple [ODEs + optional ROM acceleration] Talk about l= m =2 Minimal ingredients: mode only! PN approximate to describe low frequencies: uncalibrated EOB Set of NR WFs: SXS + BAM Prediction for BH remnant: New fits for final mass & spin -> QNM freq.

8 New NR Waveforms: m1/m2 = 4, 8,18 BAM code: moving puncture finite difference mesh refinement 2 resolutions for q=18, spin 0.4

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11 NR Waveforms: SXS catalogue + new BAM WFs BAM code: moving puncture finite difference mesh refinement, BSSN formulation of Einstein Equations χ PhenC (2010) SEOBNRv2 (2014) PhenD (2015) - η = m 1 m 2 (m 1 + m 2 ) 2 Split WFs into calibration & verification data sets: calibration: 10 BAM, 9 SXS verification: 23 SXS, 6 BAM

12 Choice of inspiral approximate: uncalibrated SEOB Compare PN approximants in hybridization procedure -> decide for uncalibrated SEOBNRv2. S n(f) and 2 s(f) f Early aligo ASD Zero Det. High Power ASD 242M, ρ =26 61M, ρ = M, ρ =8.6 q18, S= Frequency f (Hz)

13 Hybrid waveforms: corner cases / / / - - / - _ + _ / / / /

14 Final state Kerr BH perturbation theory -> complex frequencies of spheroidal harmonic QNMs, functions of final mass & final spin. polynomial fit Amplitudes and relative phases of different harmonics computed in NR. SXS, RIT + BAM q 18 data => Effective spin fits for final spin & radiated energy (final mass) Hierarchical fitting approach by subspaces: no spin / equal mass / full rational function fit

15 Final state Kerr BH perturbation theory -> complex frequencies of spheroidal harmonic QNMs, functions of final mass & final spin. polynomial fit Amplitudes and relative phases of different harmonics computed in NR. SXS, RIT + BAM q 18 data => Effective spin fits for final spin & radiated energy (final mass) Hierarchical fitting approach by subspaces: no spin / equal mass / full Need more high spin data points rational function fit

16 Update on final state results: unequal spins Final spin - extension to unequal spins: Guess ansatz for f(η) from inspecting data: a f = a Eq + f( )( f 1 2 ) At fixed η, difference with equal spin fit well approximated by plane -> determine coefficients, plot in 1D. f( ) =a 0 p (1 4 ) q Compare with fit to full data set. af(χ 1 -χ 2 ) D fits direct 3D fit RMS error: > similar for radiated energy η

17 Update on final state results: precession Based on PhenomP approximation: emission in comoving frame = no precession preserve total spin projections unto & L => radiated energy should depend only weakly on precession. Final spin: a fin = s S 2? M 2 fin 2 + a 2 fin PhenD PhenD λ=mfin^2 PhenD λ=1 AEI 0.10 choose fudge parameter = M 2 fin afin - afinnr as in 2007 AEI fit [Rezzolla+,PRD78,2008] case index

18 Splitting into amplitude/phase & frequency regions Divide and conquer: Split waveform into amplitude and phase, model simple non-oscillatory functions. Simplicity of modelling increases with the number of frequency-regions. Simplest: tens of points, cubic spline. Our choice - 3 regions: inspiral (use PN intuition) merger-ringdown (use QNM intuition) intermediate

19 Amplitude inspiral model Mf apple : h insp =PN+ f 7/3 + f 8/3 + f 3 For each WF fit for,, ()*+,-./ +0*1" 23!"!%!!"!$#!"!$!!"!!#!"!!! -!"!!# -!"!$!!"!!#!"!$!!"!$#!"!%! & ' PN terms have alternate signs, converge slowly -> represent curve by 3 equispaced data points Parameterize (, eff ) P. Ajith, Phys. Rev. D 84, (2011) parameter space and interpolate with polynomial. = m1m2 (m1+m2) 2, eff = m m 2 2 m 1 + m ( )

20 Amplitude inspiral model Mf apple : h insp =PN+ f 7/3 + f 8/3 + f 3 For each WF fit for,, ()*+,-./ +0*1" 23!"!%!!"!$#!"!$!!"!!#!"!!! -!"!!# -!"!$!!"!!#!"!$!!"!$#!"!%! & ' PN terms have alternate signs, converge slowly -> represent curve by 3 equispaced data points Parameterize (, eff ) P. Ajith, Phys. Rev. D 84, (2011) parameter space and interpolate with polynomial. = m1m2 (m1+m2) 2, eff = m m 2 2 m 1 + m ( )

21 Complete amplitude model Deal with smooth functions -> high frequency falloff faster than polynomial. Previous Phenom ringdown based on Lorentzian, now multiply with exponential. h RD = ae (f f ring ) (f f ring ) < f < (local maximum of ringdown): rational function connected C 1 to inspiral and ringdown with 1(2) further parameters, or polynomial. / SXS:q SXS:q BAM:q BAM:q18-0.8_0 BAM:q18+0.4_0

22 Complete amplitude model Deal with smooth functions -> high frequency falloff faster than polynomial. Previous Phenom ringdown based on Lorentzian, now multiply with exponential. h RD = ae (f f ring ) (f f ring ) < f < (local maximum of ringdown): rational function connected C 1 to inspiral and ringdown with 1(2) further parameters, or polynomial.

23 Complete amplitude model Deal with smooth functions -> high frequency falloff faster than polynomial. Previous Phenom ringdown based on Lorentzian, now multiply with exponential. h RD = ae (f f ring ) (f f ring ) < f < (local maximum of ringdown): rational function connected C 1 to inspiral and ringdown with 1(2) further parameters, or polynomial.

24 F Modelling the Fourier domain phase Bad news: Freedom in initial phase & time shift: M f Look at first derivative: 2 nd derivative often too noisy. Can you spot the ringdown frequency? MRD-Ansatz: F d (f) df f (f)! (f) t ringdown frequency

25 F Modelling the Fourier domain phase Bad news: Freedom in initial phase & time shift: M f Look at first derivative: 2 nd derivative often too noisy. Can you spot the ringdown frequency? MRD-Ansatz: F d (f) df f (f)! (f) t ringdown frequency

26 F Modelling the Fourier domain phase Bad news: Freedom in initial phase & time shift: M f Look at first derivative: 2 nd derivative often too noisy. Can you spot the ringdown frequency? MRD-Ansatz: F d (f) df f ϕ/ - (f)! (f) t ringdown frequency

27 Phase model Inspiral: as for amplitude, PN + 3 higher order terms + 0 Ringdown: 0 MR = f f 1/4 + 4 f damp f 2 damp +(f 5f RD ) 2 Intermediate: 0 Int = f f 4 Phase & residuals example: intermediate freq. 16

28 Phase model Inspiral: as for amplitude, PN + 3 higher order terms + 0 Ringdown: 0 MR = f f 1/4 + 4 f damp f 2 damp +(f 5f RD ) 2 Intermediate: 0 Int = f f 4 Phase & residuals example: intermediate freq. 16

29 Phase coefficients as functions of, ˆ

30 PhenomD mismatches against all 48 hybrids early aligo noise curve, low freq. 30 Hz

31 Mismatches between models: SEOBNRv2 vs PenD PhenD/SEOBNRv2_ROM PhenD+EOB insp. PhenD+TaylorF2

32 Matches (Faithfulness) vs. hybrids & between models

33 Time domain waveforms

34 Summary PhenomD: very accurate WFs in time & frequency domain. Open source C implementation (LAL); Mathematica on request. Builds upon EOB inspiral description & detailed study of WF anatomy. Phenom* & SEOBNR agree extremely well in their calibration regions. Need more NR simulations for large spins orbital ang. momentum. PhenomD is modular, e.g. inspiral and MRD can be tuned from different waveform sets, variations of Phen* models easy to generate. Application to precession -> Mark s talk Similar modifications may be possible for modgr, eccentricity