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 ($\ensuremath{\ell}=2$, $|m|=2$) 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 $(\ensuremath{\Delta}\ensuremath{\theta},\ensuremath{\Delta}\ensuremath{\varphi})=(0.05\ifmmode^\circ\else\textdegree\fi{},0.2\ifmmode^\circ\else\textdegree\fi{})$. 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 ($\ensuremath{\ell}=2$, $|m|=2$) 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.