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Dynamical encounters of stellar-mass black holes (BHs) in a disc of compactobjects around a supermassive BH (SMBH) can accelerate the formation andcoalescence of BH binaries. It has been proposed that binary-single encountersamong BHs in such discs can lead to an excess of highly-eccentric BH mergers.However, previous studies have neglected how the disc velocity dispersion andthe SMBH's tidal field affect the 3-body dynamics. We investigate the outcomesof binary-single encounters considering different values of the disc velocitydispersion, and examine the role of the SMBH's tidal field. We then demonstratehow their inclusion affects the properties of merging BH binaries. We performsimulations of 4-body encounters (i.e. with the SMBH as fourth particle) usingthe highly-accurate, regularized code TSUNAMI, which includes post-Newtoniancorrections up to order 3.5PN. The disc velocity dispersion controls how orbitsin the disc are aligned and circular, and determines the relative velocity ofthe binary-single pair before the encounter. As the velocity dispersiondecreases, the eccentricity of post-encounter binaries transitions from thermalto superthermal, and binaries experience enhanced hardening. The transitionbetween these two regimes happens at disc eccentricities and inclinations oforder e ~ i ~ 10^-4. These distinct regimes correspond to a disc dominated byrandom motions, and one dominated by the Keplerian shear. The inclusion of theSMBH's tidal field and the disc velocity dispersion can significantly affectthe number of GW mergers, and especially the number of highly-eccentricinspirals. These can be up to ~2 times higher at low velocity dispersion, and~12 times lower at high velocity dispersions. The spin-orbit alignment isinfluenced by the tidal field exclusively at high velocity dispersions,effectively inhibiting the formation of anti-aligned binary BHs.

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