An eccentric binary black hole inspiral-mergerringdown gravitational waveform model from post- Newtonian and numerical relativity Ian Hinder Max Planck Institute for Gravitational Physics (Albert Einstein Institute) Potsdam, Germany Simulating extreme Spacetimes TEGrAW, Paris, June 2017
Introduction Eccentric binary systems circularise as E and L are emitted (Peters 1964) Eccentricity of BBH expected to be 0 well before merger Eccentric binaries in LIGO band? Can we measure (bound) eccentricity of GW events such as GW150914? Eccentric waveform model could be compared with GW data to measure/constrain eccentricity) Construct and test such a model using Post-Newtonian approximation and Numerical Relativity Only need late inspiral+merger; e.g. last 5 orbits for GW150914 2
A selection of eccentric Numerical Relativity simulations 19 new accurate NR simulations, ~25 cycles, SpEC code Non-spinning Initial eccentricity e 0.2 q = m1/m2 3 e 0 = 0 e 0 = 0.20 3
Eccentric simulation 4
What does an eccentric BBH merger look like? 0.4 0.2 e 0 = 0 e 0 = 0.2 0.4 0.2 Re[h22] Re[h22] -0.2-0.2-0.4-2000 -1500-1000 -500-0.4-100 -50 0 50 100 t/m t/m Eccentric mergers are circular! 5
What does an eccentric BBH merger look like? Circularisation just before merger for q=1 [Hinder et al. 2008] Now extend to q=3 GW 0.6 0.5 0.4 0.3 Eccentricity lost before merger For all eccentricities, 0.2 Same waveform for t > tpeak - 30 M Merger remnant has same mass and spin Can use circular merger model 0.1-500 -400-300 -200-100 0 (t-t peak )/M GW instantaneous frequency (q=3) independent of e for t > tpeak - 30 M (similar for amplitude) 6
Modelling the inspiral: the building blocks Post-Newtonian model: Conservative motion (without inspiral): constant E and L eccentricity e, semi-major axis a r, φ in E and L (3 PN) Radiation reaction: Adiabatic constants E and L integrated from 2 PN fluxes Waveforms 0 PN (restricted approximation): See [IH et al. 2010] for details Empirically found best agreement with NR for PN expansion variable x (TaylorT4 x when e 0) h+, hx in r, φ 7
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Validation of PN inspiral against Numerical Relativity NR and PN agree well in inspiral for last ~10 orbits 0.4 0.2-0.2-0.4 0 500 1000 1500 2000 Fit best PN (e,x,l,φ) over inspiral PN breaks down near merger 9
Eccentric model construction Use eccentric PN for inspiral (agrees well with NR) Use quasi-circular for merger M GW 0.6 0.5 0.4 0.3 0.2 0.1 e = 0 e = 0.2 Eccentric PN -1000-800 -600-400 -200 0 t/m PN for t < tref? Circular model Use a fitted ansatz for Δt (time to peak) Blend in frequency and amplitude of waveform between eccentric PN and circular 10
Eccentric model construction: Merger Circular Merger Model (CMM): One-parameter (q) family of e=0 waveforms (a) 22 0.6 0.5 0.4 0.3 q = 1.5 q = 2.5 q = 3.0 q = 3.3 Interpolate ω(t) and A(t) for q {1, 2, 4} from SXS public catalogue Test against 4 additional e=0 waveforms Modelling error negligible (e) Re[h22] 0.2 0.1 NR Circular merger model -100-50 0 50 0.4 0.2 (t-t peak )/M Comparison between NR and CMM for 4 quasi-circular waveforms not used to construct the model -0.2-100 -50 0 50 (t-t peak )/M 11
Eccentric model construction: Transition Smoothly blend PN inspiral and circular merger log 10 M -0.2-0.4-0.6-0.8 NR PN Circular merger model IMR model t ref t 2 Δt t circ t peak tpeak -tref 460 440 420 NR Fit -1.0-1.2 2300 2400 2500 2600 2700 2800 Blending parameters from NR simulations Most important: t/m IMR from blend of PN and CMM 39 15 16 21 18 19 20 17 22 45 40 41 43 44 59 61 62 53 52 48 49 50 Δt: where is the peak of the merger waveform? Fit from NR. Fit error ±1 M. 400 380 NR case t(q, e, l) = t 0 + a 1 e + a 2 e 2 + b 1 q + b 2 q 2 + c 1 e cos(l + c 2 ) 12
Results: Waveform comparison Determine PN parameters of NR waveform via 1 orbit fit ~7 cycles before peak Optimise PN ωgw(x0, e0, l0) to get best agreement with NR 13
Results: Waveform comparison Typical case: good agreement over the ~25 cycles of the NR waveform Case 16 (q=1., e 0 =5) 0.4 0.2 NR Model Fit Re[h22] -0.2-0.4-3000-2500-2000-1500-1000-500 0-150-100-50 0 50 (t-t peak )/M 14
Results: Waveform comparison Worst case: dephasing of both Φ(t) (orbital oscillations) and l(t) (eccentric oscillations) Agreement is worse for higher e and higher q 0.4 0.2 Case 52 (q=3., e 0 =9) NR Model Fit Re[h22] -0.2-0.4-3000-2500-2000-1500-1000-500 0-150-100-50 0 50 (t-t peak )/M 15
Results: Faithfulness Target GW150914: O1 Advanced LIGO noise curve with fmin = 30 Hz Short NR waveforms sufficient Label with eref from fit to PN ~7 cycles before the merger Overlap: (h 1 h 2 ) 4 Re Z fmax f min h1 (f) h 2(f) df S n (f) Faithfulness: F = max c,t c (h 1 ( c,t c ) h 2 ) p (h1 h 1 )(h 2 h 2 ) 16
Results: Faithfulness 10-1 10-1 10-2 q=1 10-2 q=3 1-F 10-3 1-F 10-3 e 10-4 e 10-4 10-5 50 100 150 200 250 300 10-5 50 100 150 200 250 300 M/M Eccentric model faithful (97%) with NR for q 3: M/M For e ref < 5, M > 70 M For e ref < 8, M > 93 M Limits on M from: (i) length of NR, (ii) accumulated PN errors (from RR) 17
Conclusions and outlook Eccentric inspiral-merger-ringdown BBH waveform model See [Huerta et al. 2016] for a similar model, not calibrated to NR simulations Non-spinning, q 3, e ref < 0.1 Numerical Relativity for calibration and testing < 3% unfaithfulness to NR for GW150914-like events NR simulations and Mathematica code for model will be public Future: Implications for measurement of e with GW detectors Longer NR waveforms Spin 18