Spin-Orbit Misalignments in Tertiary-Induced Black-Hole Binary Mergers: Theoretical Analysis

TL;DR

This paper develops an analytical framework to understand the spin dynamics and misalignments of black-hole binaries undergoing Lidov-Kozai oscillations induced by tertiary companions, explaining the observed '90-degree attractor' in spin orientations.

Contribution

The paper introduces a novel analytical formalism for predicting the spin-orbit misalignment in tertiary-induced black-hole mergers, including conditions for adiabatic invariance and breakdown.

Findings

The final spin-orbit misalignment often relates to the initial spin orientation via an adiabatic invariant.

The '90-degree attractor' explains small effective spins in certain merger scenarios.

Constraints on adiabaticity breakdown due to resonant interactions are derived.

Abstract

Black-hole (BH) binary mergers driven by gravitational perturbations of tertiary companions constitute an important class of dynamical formation channels for compact binaries detected by LIGO/VIRGO. Recent works have examined numerically the combined orbital and spin dynamics of BH binaries that undergo large Lidov-Kozai (LK) eccentricity oscillations induced by a highly inclined companion and merge via gravitational wave radiation. However, the extreme eccentricity variations make such systems difficult to characterize analytically. In this paper, we develop an analytical formalism for understanding the spin dynamics of binary BHs undergoing LK-induced mergers. We show that, under certain conditions, the eccentricity oscillations of the binary can be averaged over to determine the long-term behavior of the BH spin in a smooth way. In particular, we demonstrate that the final spin-orbit misalignment angle $\theta_{\rm sl}$ is often related to the binary's primordial spin orientation through an approximate adiabatic invariant. Our theory explains the "$90^\circ$ attractor" (as found in recent numerical studies) for the evolution of $\theta_{\rm sl}$ when the initial BH spin is aligned with the orbital axis and the octupole LK effects are negligible -- such a "$90^\circ$ attractor" would lead to a small binary effective spin parameter $\chi_{\rm eff}\sim 0$ even for large intrinsic BH spins. We calculate the deviation from adiabaticity in closed form as a function of the initial conditions. We also place accurate constraints on when this adiabatic invariant breaks down due to resonant spin-orbit interactions. We consider both stellar-mass and supermassive BH tertiary companions, and provide simple prescriptions for determining analytically the final spin-orbit misalignment angles of the merging BH binaries.