Inspiral-merger-ringdown waveforms of spinning, precessing black-hole binaries in the effective-one-body formalism

We describe a general procedure to generate spinning, precessing waveforms that include inspiral, merger, and ringdown stages in the effective-one-body (EOB) approach. The procedure uses a precessing frame in which precession-induced amplitude and phase modulations are minimized, and an inertial frame, aligned with the spin of the final black hole, in which we carry out the matching of the inspiral-plunge to merger-ringdown waveforms. As a first application, we build spinning, precessing EOB waveforms for the gravitational modes $\ell =2$ such that in the nonprecessing limit those waveforms agree with the EOB waveforms recently calibrated to numerical-relativity waveforms. Without recalibrating the EOB model, we then compare EOB and post-Newtonian precessing waveforms to two numerical-relativity waveforms produced by the Caltech-Cornell-CITA collaboration. The numerical waveforms are strongly precessing and have 35 and 65 gravitational-wave cycles. We find a remarkable agreement between EOB and numerical-relativity precessing waveforms and spins’ evolutions. The phase difference is $\sim 0.2\mathrm{rad}$ rad at merger, while the mismatches, computed using the advanced-LIGO noise spectral density, are below 2% when maximizing only on the time and phase at coalescence and on the polarization angle.

Figure 1

We show the radiation frame $\{{\mathit{e}}_1^R,{\mathit{e}}_2^R,{\mathit{e}}_3^R\}$, the inertial source frame $\{{\mathit{e}}_1^S,{\mathit{e}}_2^S,{\mathit{e}}_3^S\}$, and the precessing source frame $\{{\mathit{e}}_1^{L_N},{\mathit{e}}_2^{L_N},{\mathit{e}}_3^{L_N}\}$ employed to describe a precessing BH binary and its GW radiation.

Figure 2

We show inspiraling, precessing PN waveforms decomposed in the inertial source frame aligned with the initial total angular momentum ${\mathit{J}}_0$ and in the precessing source frame aligned with the Newtonian orbital angular momentum ${\mathit{L}}_N(t)$. For comparison, we show also the quasinonprecessing PN waveforms defined in Sec. 2b. They lie on top of the ${\mathit{L}}_N(t)$-frame waveforms. The three panels use the same scale on the $y$ axis so that the amplitudes of the modes $h_{22}$, $h_{21}$, and $h_{20}$ can be easily compared.

Figure 3

We show the projections of $\hat{\mathit{J}}\left(t\right)$, $\hat{\mathit{L}}\left(t\right)$, ${\hat{\mathit{L}}}_N\left(t\right)$, ${\hat{\mathit{S}}}_1\left(t\right)$, and ${\hat{\mathit{S}}}_2\left(t\right)$ on the $x$-$y$ plane of the inertial frame for which the $z$ axis is aligned with $\mathit{J}(t_{\mathrm{\Omega }\text{peak}}^{\mathrm{EOB}})$. In the top and bottom panels, we show trajectories of these unit vectors for cases 1 and 2 of Table 1, respectively. The initial point of each trajectory is marked by its name. The trajectory of $\hat{\mathit{J}}\left(t\right)$ ends at the origin, by definition. The trajectory of ${\hat{\mathit{L}}}_N\left(t\right)$ follows that of $\hat{\mathit{L}}\left(t\right)$ with oscillations due to nutation.

Figure 4

We show the ${\hat{\mathit{L}}}_N\left(t\right)$-frame and the $\hat{\mathit{L}}\left(t\right)$-frame precessing waveforms, as well as their relative amplitude and phase differences. The top and bottom panels are waveforms for cases 1 and 2 of Table 1. The left and right panels show the inspiral and the plunge-merger-ringdown stages of the waveforms, respectively.

Figure 5

For case 2 of Table 1, we show the ${\hat{\mathit{L}}}_N\left(t\right)$-frame and $\hat{\mathit{L}}\left(t\right)$-frame waveforms in the top panel and their phase evolutions in the bottom panel, over a short time period from $t=21000M$ to $21500M$. The vertical lines mark the time when the dominant quadrature (the imaginary part for this specific instance) of any waveform becomes zero. It coincides with the time when the corresponding phase evolution in the bottom panel experiences a rapid growth. The absolute phase values are not relevant.

Figure 6

We show evolutions of the dimensionless spin vectors ${\mathit{\chi }}_1={\mathit{S}}_1(t)/m_1^2$ and ${\mathit{\chi }}_2={\mathit{S}}_2(t)/m_2^2$ of the NR simulation and the EOB and ST4 models. Specifically, we show the projections of ${\mathit{\chi }}_1$ and ${\mathit{\chi }}_2$ on the basis vectors of the inertial source frame $\{{\mathit{e}}_1^S,{\mathit{e}}_2^S,{\mathit{e}}_3^S\}$ that is aligned with the initial orbital orientation $[{\widehat{\mathit{L}}}_N{]}_0$ (see . 1). The top two panels show ${\mathit{\chi }}_1$ and ${\mathit{\chi }}_2$ for case 3 of Table 1. The bottom panel shows ${\mathit{\chi }}_1$ (${\mathit{\chi }}_2=0$) for case 4 of Table 1. The EOB and ST4 data start at the after-junk-radiation time in the NR simulations, which are $t=230M$ and $t=160M$ for cases 3 and 4, respectively.

Figure 7

We show for case 3 of Table 1 the $h_{22}$ mode decomposed in the inertial source frame $\{{\mathit{e}}_1^S,{\mathit{e}}_2^S,{\mathit{e}}_3^S\}$ that is aligned with the initial orbital orientation $[{\widehat{\mathit{L}}}_N{]}_0$ (see . 1). For clarity, we show the NR and EOB $h_{22}$ in the top panel and the NR and ST4 $h_{22}$ in the bottom panel. The EOB and ST4 data start at the after-junk-radiation time of $t=230M$.

Figure 8

We show for case 4 of Table 1 the $h_{22}$ mode decomposed in the inertial source frame $\{{\mathit{e}}_1^S,{\mathit{e}}_2^S,{\mathit{e}}_3^S\}$ that is aligned with the initial orbital orientation $[{\widehat{\mathit{L}}}_N{]}_0$ (see . 1). For clarity, we show the NR and EOB $h_{22}$ in the top panel and the NR and ST4 $h_{22}$ in the bottom panel. The EOB and ST4 data start at the after-junk-radiation time of $t=160M$.

Figure 9

We show for cases 3 and 4 of Table 1 the GW polarization $h_{+}$, containing contributions from $\ell =2$ modes, that propagates along a direction $\widehat{\mathit{N}}$ specified by spherical coordinates $\theta =\pi /3$ and $\varphi =\pi /2$ associated with the inertial source frame $\{{\mathit{e}}_1^S,{\mathit{e}}_2^S,{\mathit{e}}_3^S\}$. The EOB waveforms start at the after-junk-radiation times of $t=230M$ and $t=160M$, respectively.

Figure 10

Unfaithfulness of the EOB waveforms when compared to NR waveforms as a function of the total binary mass. Shown are cases 3 and 4 of Table 1. The error bars are estimates of numerical errors. The direction of GW propagation $\widehat{\mathit{N}}$ is specified by the spherical coordinates $\theta =\pi /3$ and $\varphi =\pi /2$.