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Binary black hole (BBH) mergers are the primary sources of gravitational wave(GW) events detected by LIGO/Virgo. Binary black holes embedded in theaccretion discs of active galactic nuclei (AGN) are possible candidates forsuch GW events. We develop an idealised analytic model for the orbitalevolution of BBHs in AGN accretion discs by combining the evolution equationsof disc-binary interaction and GW inspiral. We investigate the coupled`disc+GW'-driven evolution of BBHs transitioning from the disc-driven regime atlarge orbital separations into the GW-driven regime at small separations. Inthis evolution channel, BBH mergers are accelerated by a combination of orbitaldecay and orbital eccentricity growth in the disc-dominated regime. We providea quantification of the resulting merger timescale $\tau_\text{merger}$, andanalyse its dependence on both the accretion disc and binary orbitalparameters. By computing the evolution of the orbital eccentricity as afunction of the GW frequency, we predict that most binaries in AGN discs shouldhave significant residual eccentricities ($e \sim 0.01-0.1$), potentiallydetectable by LISA. We further discuss the potentials and caveats of thisparticular BBH-in-AGN channel in the framework of binary evolutionary paths.

Evolution of binary black holes in AGN accretion discs: Disc-binary interaction and gravitational wave emission