# Spin-Orbit Misalignment of Merging Black-Hole Binaries with Tertiary Companions

## TL;DR

This paper investigates how tertiary companions influence the spin-orbit misalignment in merging black-hole binaries, revealing chaotic spin evolution and potential observational signatures in gravitational wave signals.

## Contribution

It provides a detailed analysis of the effects of external companions on black-hole binary evolution, including an analytic expression for final misalignment in non-eccentric cases.

## Key findings

- Close inclined companions induce Lidov-Kozai oscillations, shortening merger times.
- Chaotic spin evolution leads to wide misalignment angles from initially aligned systems.
- Analytic formula for final misalignment in systems without eccentricity excitation.

## Abstract

We study the effect of external companion on the orbital and spin evolution of merging black-hole (BH) binaries. An sufficiently close by and inclined companion can excite Lidov-Kozai (LK) eccentricity oscillations in the binary, thereby shortening its merger time. During such LK-enhanced orbital decay, the spin axis of the BH generally exhibits chaotic evolution, leading to a wide range ($0^\circ$-$180^\circ$) of final spin-orbit misalignment angle from an initially aligned configuration. For systems that do not experience eccentricity excitation, only modest ($\lesssim 20^\circ$) spin-orbit misalignment can be produced, and we derive an analytic expression for the final misalignment using the principle of adiabatic invariance. The spin-orbit misalignment directly impacts the gravitational waveform, and can be used to constrain the formation scenarios of BH binaries and dynamical influences of external companions.