Validating the effective-one-body model of spinning, precessing binary black holes against numerical relativity

In Abbott et al. [Phys. Rev. X 6, 041014 (2016)], the properties of the first gravitational wave detected by LIGO, GW150914, were measured by employing an effective-one-body (EOB) model of precessing binary black holes whose underlying dynamics and waveforms were calibrated to numerical-relativity (NR) simulations. Here, we perform the first extensive comparison of such an EOBNR model to 70 precessing NR waveforms that span mass ratios from 1 to 5, dimensionless spin magnitudes up to 0.5, generic spin orientations, and length of about 20 orbits. We work in the observer’s inertial frame and include all $\ell =2$ modes in the gravitational-wave polarizations. We introduce new prescriptions for the EOB ringdown signal concerning its spectrum and time of onset. For total masses between $10{\mathrm{M}}_{\odot }$ and $200{\mathrm{M}}_{\odot }$, we find that precessing EOBNR waveforms have unfaithfulness within about 3% to NR waveforms when considering the Advanced-LIGO design noise curve. This result is obtained without recalibration of the inspiral-plunge signal of the underlying nonprecessing EOBNR model. The unfaithfulness is computed with maximization over time and phase of arrival, sky location, and polarization of the EOBNR waveform, and it is averaged over sky location and polarization of the NR signal. We also present comparisons between NR and EOBNR waveforms in a frame that tracks the orbital precession.

Figure 1

Observer’s frame, defined by the directions of the initial Newtonian angular momentum ${\mathit{L}}_N(0)$ and separation $\mathit{r}(0)$, and precessing frame, instantaneously aligned with ${\mathit{L}}_N(t)$ and described by the Euler angles $(\alpha ,\beta ,\gamma )$ [see Eq. (a4)].

Figure 2

Impact of the six independent spin degrees of freedom on the waveform: we compare two precessing EOBNR waveforms that differ only by the opening angle between the initial in-plane spins. The initial intrinsic parameters of the two configurations are $q=3$, ${\mathit{\chi }}_1·{\widehat{\mathit{L}}}_N={\mathit{\chi }}_2·{\widehat{\mathit{L}}}_N=0$, ${\chi }_{1\perp }=0.01$, ${\chi }_{2\perp }=0.9$, and different opening angles between the in-plane spins, 104° and 171°, respectively.

Figure 3

Parameters of the SXS NR waveforms that are compared to the precessing EOBNR model. The horizontal axis indicates the SXS ID number of the simulation. In the four panels we show mass ratio $q$, dimensionless spin magnitudes $|{\mathit{\chi }}_{1,2}|$, spin opening angles ${\theta }_{1,2}\equiv \arccos ({\widehat{\mathit{\chi }}}_{1,2}·{\widehat{\mathit{L}}}_N)$, and orbital separation $r$ measured at relaxation time from the NR data. Blue points are for BH 1 (the heavier), while the red ones are for BH 2.

Figure 4

Sky- and polarization-averaged unfaithfulness [Eq. (8)] between NR and EOBNR waveforms of 70 precessing BBHs in Advanced LIGO. Left panels: unfaithfulness as a function of the total mass of the binary for three possible inclinations $\iota =0,\pi /3,\pi /2$. The dashed line corresponds to 3% unfaithfulness. Right panels: histograms of the maximum unfaithfulness over the total mass range $10{\mathrm{M}}_{\odot }\le M\le 200{\mathrm{M}}_{\odot }$.

Figure 5

Comparison of NR and EOBNR+ polarization for a precessing BBH with $q=5$, ${\chi }_1=0.5$, ${\chi }_2=0$, with spin 1 initially in the orbital plane (SXS:BBH:0058). The inclination is $\iota =\pi /3$. The NR (EOBNR) data are shown in blue (dashed red) curves.

Figure 6

Comparison of NR and EOBNR spin evolution for the same run of Fig. 5 (SXS:BBH:0058). We show the time evolution of the Cartesian components of ${\mathit{S}}_1$. The NR (EOBNR) data are shown as solid blue (dashed red) lines. Here the time axis indicates the inertial time coordinate in the vicinity of the BHs.

Figure 7

Comparison of NR and EOBNR maximum-radiation axis ${\widehat{\mathit{e}}}_{(3)}^R$ for a precessing BBH with $q=1.76$, ${\chi }_1=0.41$, ${\chi }_2=0.25$, ${\theta }_1=0.53\pi$, and ${\theta }_2=0.77\pi$ (SXS:BBH:0137). The Cartesian components are given with respect to the inertial frame of an observer, $\{{\widehat{\mathit{e}}}_{(i)}^I\}$. The NR (EOBNR) curves are shown in blue (red). Also shown in green are the components of the EOBNR Newtonian angular momentum ${\widehat{\mathit{L}}}_N={\widehat{\mathit{e}}}_{(3)}^P$. Here the time axis indicates the inertial time coordinate in the vicinity of the BHs.

Figure 8

Comparison of NR and EOBNR maximum-radiation-frame modes $h_{2m}^R$ for the same run of Fig. 7 (SXS:BBH:0137). The NR (EOBNR) curves are shown in solid blue (dashed red).