Tracking the precession of compact binaries from their gravitational-wave signal

We present a simple method to track the precession of a black-hole-binary system during the inspiral, using only information from the gravitational-wave (GW) signal. Our method consists of locating the frame from which the magnitudes of the ($\ell =2$, $|m|=2$) modes are maximized, which we denote the “quadrupole-aligned” frame. We demonstrate the efficacy of this method when applied to waveforms from numerical simulations. In the test case of an equal-mass nonspinning binary, our method locates the direction of the orbital angular momentum to within $(\Delta \theta ,\Delta \phi )=(0.05°,0.2°)$. We then apply the method to a $q=M_2/M_1=3$ binary that exhibits significant precession. In general, a spinning binary’s orbital angular momentum $\mathbf{L}$ is not orthogonal to the orbital plane. Evidence that our method locates the direction of $\mathbf{L}$ rather than the normal of the orbital plane is provided by comparison with post-Newtonian results. Also, we observe that it accurately reproduces similar higher-mode amplitudes to a comparable non-precessing binary, and that the frequency of the ($\ell =2$, $|m|=2$) modes is consistent with the “total frequency” of the binary’s motion. The simple form of the quadrupole-aligned waveform may be useful in attempts to analytically model the inspiral-merger-ringdown signal of precessing binaries, and in standardizing the representation of waveforms for studies of accuracy and consistency of source modelling efforts, both numerical and analytical.

Figure 1

Profile of the magnitude of ${\Psi }_{4,22}$ as the system is rotated by the Euler angles $\beta$ and $\gamma$, shown relative to the maximum value. The example is taken from one time step ($t=562M$) of the rotated equal-mass nonspinning case discussed in Sec. 4a. Note that the maximum is clearly defined, which in this case is at $(\beta ,\gamma )=(-10°,-205°)$.

Figure 2

Motion of one black-hole puncture for the reference and rotated cases. The orbital planes are related by a rotation about the $y$-axis of 10°, and about the $z$-axis of 25°.

Figure 3

The left panel shows the amplitude of the ${\tilde{\Psi }}_{4,2m}$ modes for the reference case. The $\ell =2$, $m=1$) mode is zero by symmetry, and we see that the ($\ell =2$, $m=0$) mode is much smaller than the dominant mode, and is essentially noise during most of the inspiral. The right panel shows the corresponding amplitudes for the rotated case. We now see that both subdominant modes have become significant. The amplitude of the ($\ell =2$, $m=0$) mode is oscillatory because it is a purely real function; see text for more details.

Figure 4

Error in the angles for the tilt ($\beta$) and twist ($\gamma$) of the orbital plane, as determined by the maximization procedure.

Figure 5

Motion of the black-hole punctures for the $q=3$ precession simulation. The motion of the small black hole is shown in red, and the large black hole is shown in black. The precession of the orbital plane is clearly visible through the inspiral.

Figure 6

Comparison of the polar angles $\theta$ and $\phi$ for the unit directions of $\mathbf{r}\times \mathbf{v}$ (normal to the orbital plane) and $\mathbf{r}\times \mathbf{p}$ (orbital angular momentum) in a PN calculation. The comparison shows that the direction of $\mathbf{r}\times \mathbf{v}$ exhibits extra oscillations.

Figure 7

Amplitude of raw numerical data for inspiral (left), for the “dominant” mode ${\Psi }_{4,22}^{\prime }$ and the “subdominant” mode ${\Psi }_{4,21}^{\prime }$. The right panel shows the frequency of the ($\ell =2$, $m=2$) mode, which exhibits significant oscillations. (The data are also noisy at early times, but this is typical for such data.)

Figure 8

The Euler angles $(\beta ,\gamma )$ found when the maximization procedure was applied to the $q=3$ precessing-binary waveform. For comparison we show the corresponding angles $(-\theta ,-\phi )$ of the normal to the orbital plane as computed from the NR simulation, and for the angular momentum $\mathbf{L}$ from a PN simulation (as in Fig. 6). We approximately align the PN angles with $\beta$ and $\gamma$ at early times. We clearly see that the orbital-plane angles show additional oscillations that are not present in the (2, 2)-maximization angles.

Figure 9

Amplitude of the ($\ell =2$, $m=2$) mode, before (${\Psi }_{4,22}^{\prime }$) and after (${\Psi }_{4,22}$) the maximization procedure.

Figure 10

Frequency of the ($\ell =2$, $m=2$) mode before (${\Psi }_{4,22}^{\prime }$) and after (${\Psi }_{4,22}$) the maximization procedure. We see that the high-frequency oscillations have been removed. The remaining oscillations are of a lower frequency and much lower amplitude; see text and Fig. 11.

Figure 11

Frequency of the ($\ell =2$, $m=2$) mode after (${\Psi }_{4,22}$) the maximization procedure, compared with the “total frequency” ${\omega }_{\mathrm{tot}}$, which is the orbital frequency with a precession term added according to Eq. (1.1). We also show the frequency that results from rotating the system according to the direction of the Newtonian orbital angular momentum, ${\omega }_N$, i.e., the normal to the orbital plane. The frequencies, in order of increasing magnitude of oscillation, are ${\omega }_{22}$, ${\omega }_{\mathrm{tot}}$ and ${\omega }_N$.

Figure 12

Left: selected modes of the precessing-binary waveform, after being transformed into the nonprecessing frame, i.e., after the system has been rotated by the angles that were found from the (2, 2)-maximization procedure. The right-hand plot shows the same modes for a nonspinning (and therefore nonprecessing) $q=3$ waveform. The agreement is remarkable. Note, in particular, the qualitative agreement of the ($\ell =2$, $m=1$) mode, which is of comparable magnitude to the ($\ell =2$, $m=2$) mode in the raw data (see Fig. 7).