Stability of nonspinning effective-one-body model in approximating two-body dynamics and gravitational-wave emission

The detection of gravitational waves and the extraction of physical information from them requires the prediction of accurate waveforms to be used in template banks. For that purpose, the accuracy of effective-one-body (EOB) waveforms has been improved over the last years by calibrating them to numerical-relativity (NR) waveforms. So far, the calibration has employed a handful of NR waveforms with a total length of $\sim 30$ cycles, the length being limited by the computational cost of NR simulations. Here, we address the outstanding problem of the stability of the EOB calibration with respect to the length of NR waveforms. Performing calibration studies against NR waveforms of nonspinning black-hole binaries with mass ratios 1, 1.5, 5 and 8, and with a total length of $\sim 60$ cycles, we find that EOB waveforms calibrated against either 30 or 60 cycles will be indistinguishable by the advanced detectors Laser Interferometric Gravitational-wave Observatory (LIGO) and Virgo when the signal-to-noise ratio (SNR) is below 110. When extrapolating to a very large number of cycles, using very conservative assumptions, we can conclude that state-of-the-art nonspinning EOB waveforms of any length are sufficiently accurate for parameter estimation with advanced detectors when the SNR is below 20, the mass ratio is below 5 and the total mass is above $20M_{\odot }$. The results are not conclusive for the entire parameter space because of current NR errors.

Figure 1

Contours of the global phase difference $\mathrm{\Delta }{\varphi }_g$ between nonspinning NR waveforms of $N$ GW cycles and EOB waveforms with adjustable parameters $\{A^{(1)},A^{(2)}\}$. The three panels show results for mass ratios $q=1$, 5 and 8 (from left to right). The shaded regions, from inside out, are 0.1, 0.2 and 0.5 radian contours for comparisons with $N_{\max }$ cycles of NR waveforms. The solid, dashed and dotted lines are the same contours for comparisons with 30 cycles of NR waveforms. The connected black dots are the calibrated points $\{{\overline{A}}_N^{(1)}(q),{\overline{A}}_N^{(2)}(q)\}$ for $N$ values changing from 30 to $N_{\max }$. The inset zooms around these points. The NR error box of the calibrated point $\{{\overline{A}}_{N_{\max }}^{(1)}(q),{\overline{A}}_{N_{\max }}^{(2)}(q)\}$ is shown with the dashed ellipse.

Figure 2

$||dh||/||h||$ minimized over time and the phase of coalescence as a function of the hybrid matching frequency ${\omega }_m$ for EOB+NR hybrids where EOB waveforms are generated with the calibrated points $\{{\overline{A}}_{30}^{(1)}(q),{\overline{A}}_{30}^{(2)}(q)\}$ and $\{{\overline{A}}_{N_{\max }}^{(1)}(q),{\overline{A}}_{N_{\max }}^{(2)}(q)\}$. We also show the same quantity for PN+NR hybrids using TaylorT1 and TaylorT4 approximants. The bigger symbol in each data set marks the matching frequency where the hybrid is built using 30 cycles of NR waveforms. The horizontal lines mark the effective signal-to-noise ratios (SNR)s 10, 25 and 100, below which the difference between waveforms cannot be distinguished by advanced LIGO detectors.

Figure 3

$||dh||/||h||$ between EOB waveforms generated with the calibrated points $\{{\overline{A}}_{30}^{(1)}(q),{\overline{A}}_{30}^{(2)}(q)\}$ and $\{{\overline{A}}_N^{(1)}(q),{\overline{A}}_N^{(2)}(q)\}$ as a function of the number of NR cycles $N$. We minimize $||dh||/||h||$ over time and phase of coalescence and use 30 cycles of NR waveforms in the $\mathrm{EOB}+\mathrm{NR}$ hybrids. The horizontal lines mark the effective SNR 100 and 200, below which the difference between waveforms cannot be distinguished by advanced LIGO detectors.