Geometric approach to the precession of compact binaries

We discuss a geometrical method to define a preferred reference frame for precessing binary systems and the gravitational waves they emit. This minimal-rotation frame is aligned with the angular-momentum axis and fixes the rotation about that axis up to a constant angle, resulting in an essentially invariant frame. Gravitational waveforms decomposed in this frame are similarly invariant under rotations of the inertial frame and exhibit relatively smoothly varying phase. By contrast, earlier prescriptions for radiation-aligned frames induce extraneous features in the gravitational-wave phase which depend on the orientation of the inertial frame, leading to fluctuations in the frequency that may compound to many gravitational-wave cycles. We explore a simplified description of post-Newtonian approximations for precessing systems using the minimal-rotation frame, and describe the construction of analytical/numerical hybrid waveforms for such systems.

Figure 1

Rotation histories for a simple precessing system when $\gamma =0$. The axis denoted by the blue curve (the small, perfectly circular loop) precesses through a single cycle around a fixed axis that is tilted by varying amounts. The other axes of the tracked frame (red and green curves) take paths that depend on the inclination of the axis around which the precession occurs, relative to the inertial frame. The $y$ axis, in particular, is forced to remain on the $x$-$y$ plane of the visualization coordinates, by the choice of $\gamma =0$. If we had defined Euler angles by $z$-$x$-$z$ rotations, as in Ref. 18, then it would have been the $x$ axis that was forced to remain on this plane.

Figure 2

Rotation histories for a simple precessing system when $\gamma$ is set by Eq. (18). These systems are just the same as in Fig. 1, except that the frame is given a final rotation about the $z$ axis to satisfy the minimal-rotation condition. The paths are identical in each case, but tilted with respect to each other. This shows that the minimally rotated frame is essentially invariant under fixed rotations of the inertial frame. Note that the paths in these figures represent many precession cycles, each corresponding to a single cycle of the “zigzag” pattern.

Figure 3

Waveform frequency in various frames aligned with the radiation axis. Here, waveform frequency refers to the time derivative of the phase of the $(\ell ,m)=(2,2)$ mode of the gravitational waveform. The solid red line shows the frequency measured in a frame derived from the inertial frame by a rotation in which the third Euler angle $\gamma$ is set to 0, while the solid blue line shows the same quantity in a frame for which $\gamma$ satisfies the minimal-rotation condition. Clearly, the latter curve is much smoother. We also show as dotted lines the same quantities when the physical system is tilted by 10°. The dotted blue line coincides with the solid blue line, showing the invariance of the waveform in that frame.

Figure 4

Change of phase measured in frames aligned to the radiation axis when the physical system is tilted by 10°. This phase difference is defined by Eq. (21), and plotted for two cases: the first (in red) where ${\varphi }^{2,2}$ refers to phases measured in a frame obtained from the inertial frame with a rotation in which the final Euler angle $\gamma$ is set to 0; the second (in blue) where ${\varphi }^{2,2}$ refers to phases measured in frames satisfying the minimal-rotation condition. The change in phase for the minimal-rotation frames is 0 to within numerical error (roughly a part in ${10}^5$ here). Note that this waveform extends for roughly the length of time such a system would be in the sensitive band of Advanced LIGO if the total mass were about $10M_{\odot }$.