Effective-one-body waveforms calibrated to numerical relativity simulations: coalescence of non-spinning, equal-mass black holes

TL;DR

This work refines the effective-one-body (EOB) framework by calibrating it against a high-precision numerical-relativity (NR) simulation of an equal-mass, non-spinning black-hole binary. By aligning NR and EOB waveforms at low frequency and exploring the degeneracy of the EOB adjustable parameters, the authors identify a minimal parameter set that achieves NR-level phase and amplitude accuracy through inspiral, plunge, and merger, and they validate this accuracy for the dominant and several subdominant modes. The study demonstrates the strong data-analysis viability of the calibrated EOB waveforms, achieving maximal overlaps with LIGO-class noise curves exceeding 0.999 for total masses in the tens to hundreds of solar masses, and it extends the calibration to unequal-mass cases (2:1 and 3:1) via a ν-dependent parameterization. These results offer a robust analytical waveform model capable of interpolating accurate NR simulations, thereby improving GW data-analysis templates for current and next-generation detectors.

Abstract

We calibrate the effective-one-body (EOB) model to an accurate numerical simulation of an equal-mass, non-spinning binary black-hole coalescence produced by the Caltech-Cornell collaboration. Aligning the EOB and numerical waveforms at low frequency over a time interval of ~1000M, and taking into account the uncertainties in the numerical simulation, we investigate the significance and degeneracy of the EOB adjustable parameters during inspiral, plunge and merger, and determine the minimum number of EOB adjustable parameters that achieves phase and amplitude agreements on the order of the numerical error. We find that phase and fractional amplitude differences between the numerical and EOB values of the dominant gravitational wave mode h_{22} can be reduced to 0.02 radians and 2%, respectively, until a time 26 M before merger, and to 0.1 radians and 10%, at a time 16M after merger (during ringdown), respectively. Using LIGO, Enhanced LIGO and Advanced LIGO noise curves, we find that the overlap between the EOB and the numerical h_{22}, maximized only over the initial phase and time of arrival, is larger than 0.999 for equal-mass binary black holes with total mass 30-150 Msun. In addition to the leading gravitational mode (2,2), we compare the dominant subleading modes (4,4) and (3,2) and find phase and amplitude differences on the order of the numerical error. We also determine the mass-ratio dependence of one of the EOB adjustable parameters by fitting to numerical {\it inspiral} waveforms for black-hole binaries with mass ratios 2:1 and 3:1. These results improve and extend recent successful attempts aimed at providing gravitational-wave data analysts the best analytical EOB model capable of interpolating accurate numerical simulations.

Figures (19)

• Figure 1: Numerical error estimates. Phase difference between numerical $\Psi_4^{22}$ waveforms, when aligned using the same procedure as employed for the EOB-NR alignment [see Eq. (\ref{['waveshifts']})]. "N6" and "N5" denote the highest- and next-to-highest numerical resolution, $n$ denotes the order of extrapolation to infinite extraction radius, and "$r=225M$" denotes waves extracted at finite radius $r=225M$. The data are smoothed with a rectangular window of width $10M$; the light grey dots represent the unsmoothed data for the N5-N6 comparison at $r_{\rm ex}=225M$.
• Figure 2: In the parameter space of the EOB-dynamics adjustable parameters $a_5(1/4)$ and $v_{\rm pole}(1/4)$ we show the contours of the time $t_{\rm ref}$ at which the phase difference between the numerical "30c1/N6, n=3" and EOB $\Psi_4^{22}$ becomes larger than $0.02$ radians. Note that the innermost red contours cover two disjoint regions. The inset shows the effect of numerical uncertainty: The filled contours are the $t_{\rm ref}=3850M$ and $3900M$ contours from the main panel. The open contours are identical, except computed using the "30c1/N6, n=2" numerical $\Psi_4^{22}$. The reference model is shown as a black dot.
• Figure 3: For the case $a_5(1/4)=6.344$ and $v_{\rm pole}(1/4)=0.85$ ($A_8=0$, $a_{\rm RR}^{{\cal F}_\Phi}=0$ and $a^{{\cal F}_r}_{\rm RR}=0$), we show the phase difference between the numerical and EOB mode $h_{22}$ versus the numerical GW frequency $M \omega_{22}$ for EOB models in which the GW energy flux is shut down at several EOB orbital frequencies. The vertical line marks the maximum EOB orbital frequency.
• Figure 4: Effect of $a^{{\cal F}_r}_{\rm RR}$ on contours of acceptable EOB parameters. The solid contours are the $t_{\rm ref}=3850M$ and $3900M$ contours from Fig. \ref{['fig:TrefContourPlot4']}. The open contours shifted to the lower-right
• Figure 5: We compare the numerical and EOB $h_{22}$ amplitudes when the EOB model with reference values $a_5(1/4)=6.344$ and $v_{\rm pole}(1/4)=0.85$ are used. We show the EOB amplitudes without the NQC corrections and the EOB amplitude with the NQC terms suggested in Ref. DN2008, where the NQC parameters take the values $a=0.75$ and $\epsilon=0.09$. When the NQC corrections are not included, we show the EOB amplitude of Eq. (\ref{['h22']}) which uses the resummation procedure of Ref. DIN, and also the EOB amplitudes of Eq. (\ref{['h22']}) when the Padé-resummations $P_4^1$ and $P_3^2$ of $\rho_{22}$ suggested in Ref. DIN are applied. Note that in this plot, the EOB amplitudes do not contain the merger-ringdown contribution.