Three-body encounters in black hole discs around a supermassive black hole: The disc velocity dispersion and the Keplerian tidal field determine the eccentricity and spin-orbit alignment of gravitational wave mergers

Dynamical encounters of stellar-mass black holes (BHs) in a disc of compact objects around a supermassive BH (SMBH) can accelerate the formation and coalescence of BH binaries. It has been proposed that binary-single encounters among BHs in such discs can lead to an excess of highly-eccentric BH mergers. However, previous studies have neglected how the disc velocity dispersion and the SMBH's tidal field affect the 3-body dynamics. We investigate the outcomes of binary-single encounters considering different values of the disc velocity dispersion, and examine the role of the SMBH's tidal field. We then demonstrate how their inclusion affects the properties of merging BH binaries. We perform simulations of 4-body encounters (i.e. with the SMBH as fourth particle) using the highly-accurate, regularized code TSUNAMI, which includes post-Newtonian corrections up to order 3.5PN. The disc velocity dispersion controls how orbits in the disc are aligned and circular, and determines the relative velocity of the binary-single pair before the encounter. As the velocity dispersion decreases, the eccentricity of post-encounter binaries transitions from thermal to superthermal, and binaries experience enhanced hardening. The transition between these two regimes happens at disc eccentricities and inclinations of order e ~ i ~ 10^-4. These distinct regimes correspond to a disc dominated by random motions, and one dominated by the Keplerian shear. The inclusion of the SMBH's tidal field and the disc velocity dispersion can significantly affect the number of GW mergers, and especially the number of highly-eccentric inspirals. These can be up to ~2 times higher at low velocity dispersion, and ~12 times lower at high velocity dispersions. The spin-orbit alignment is influenced by the tidal field exclusively at high velocity dispersions, effectively inhibiting the formation of anti-aligned binary BHs.