Effective Potentials and Morphological Transitions for Binary Black Hole Spin Precession

We derive an effective potential for binary black hole (BBH) spin precession at second post-Newtonian order. This effective potential allows us to solve the orbit-averaged spin-precession equations analytically for arbitrary mass ratios and spins. These solutions are quasiperiodic functions of time: after a fixed period, the BBH spins return to their initial relative orientations and jointly precess about the total angular momentum by a fixed angle. Using these solutions, we classify BBH spin precession into three distinct morphologies between which BBHs can transition during their inspiral. We also derive a precession-averaged evolution equation for the total angular momentum that can be integrated on the radiation-reaction time and identify a new class of spin-orbit resonances that can tilt the direction of the total angular momentum during the inspiral. Our new results will help efforts to model and interpret gravitational waves from generic BBH mergers and predict the distributions of final spins and gravitational recoils.

Figure 1

Effective potentials ${\xi }_{\pm }(S)$ for the spin precession of BBHs with maximal spins, mass ratio $q=0.8$, $L=0.781M^2$ ($r=10M$), and $J=0.85M^2$. These two functions form a loop enclosing the allowed values of $S$ and $\xi$. Since $\xi$ is conserved during the inspiral, $S$ oscillates between the two roots $S_{\pm }$ of the equation $\xi ={\xi }_{\pm }(S)$ on the precession time. The two roots are degenerate at ${\xi }_{\text{min}}$ and ${\xi }_{\text{max}}$, implying that $S$ is constant: these configurations correspond, respectively, to the $\mathrm{\Delta }\mathrm{\Phi }=0°(\mathrm{\Delta }\mathrm{\Phi }=\pm 180°)$ spin-orbit resonances of Schnittman [30]. The four dotted curves are the contours $\cos {\theta }_i=\pm 1$ given by Eqs. (7a) and (7b); transitions between BBHs for which $\mathrm{\Delta }\mathrm{\Phi }$ circulates and those for which it librates about 0° ($\pm 180°$) occur where these curves are tangent to the potentials ${\xi }_{\pm }(S)$, as indicated by the lower (upper) dashed line $\xi ={\xi }_{c0}$ (${\xi }_{c180}$). The three dot-dashed lines correspond to the three BBH systems shown in Fig. 2 as representative of each morphology.

Figure 2

The three morphologies of BBH spin precession. The angular momenta $\mathbf{J}$, $\mathbf{L}$, and ${\mathbf{S}}_i$ are all in the $xz$ plane at $S=S_{\pm }$. In all three panels, the BBHs have maximal spins, $q=0.8$, $L=0.781M^2$ ($r=10M$), and $J=0.85M^2$ as in Fig. 1. The left, middle, and right panels correspond to $\xi =-0.025$, 0.025, and 0.15, respectively. If the components of ${\mathbf{S}}_i$ perpendicular to $\mathbf{L}$ are aligned with each other at both roots $S_{\pm }$, $\mathrm{\Delta }\mathrm{\Phi }$ librates about 0°. If they are aligned at one root and antialigned at the other, $\mathrm{\Delta }\mathrm{\Phi }$ circulates. If they are antialigned at both roots, $\mathrm{\Delta }\mathrm{\Phi }$ librates about 180°.