  Motivated by the increasing number of detections of merging black holes by
LIGO-VIRGO-KAGRA, black hole (BH) binary mergers in the discs of active
galactic nuclei (AGN) is investigated as a possible merger channel. In this
pathway, BH encounters in the gas disc form mutually bound black hole binary
systems through interaction with the gas in the disc and subsequently inspiral
through gravitational torques induced by the local gas. To determine the
feasibility of this merger pathway, we present the first 3D global hydrodynamic
simulations of the formation and evolution of a stellar-mass BH binaries AGN
discs with three different AGN disc masses and five different initial radial
separations. These 15 simulations show binary capture of prograde and
retrograde binaries can be successful in a range of disc densities including
cases well below that of a standard radiatively efficient alpha disc,
identifying that the majority of these captured binaries are then subsequently
hardened by the surrounding gas. The eccentricity evolution depends strongly on
the orbital rotation where prograde binaries are governed by gravitational
torques form their circumbinary mini-disc, with eccentricities being damped,
while for retrograde binaries the eccentricities are excited to > 0.9 by
accretion torques. In two cases, retrograde binaries ultimately undergo a close
periapsis passage which results in a merger via gravitational waves after only
a few thousand binary orbits. Thus, the merger timescale can be far shorter
than the AGN disc lifetime. These simulations support an efficient AGN disc
merger pathway for BHs.
