Here we propose a mechanism for efficiently growing intermediate mass black holes (IMBH) in discs around supermassive black holes. Stellar mass objects can efficiently agglomerate when facilitated by the gas disc. Stars, compact objects and binaries can migrate, accrete and merge within discs around supermassive black holes. While dynamical heating by cusp stars excites the velocity dispersion of nuclear cluster objects (NCOs) in the disc, gas in the disc damps NCO orbits. If gas damping dominates, NCOs remain in the disc with circularized orbits and large collision cross-sections. IMBH seeds can grow extremely rapidly by collisions with disc NCOs at low relative velocities, allowing for super-Eddington growth rates. Once an IMBH seed has cleared out its feeding zone of disc NCOs, growth of IMBH seeds can become dominated by gas accretion from the active galactic nucleus (AGN) disc. However, the IMBH can migrate in the disc and expand its feeding zone, permitting a super-Eddington accretion rate to continue. Growth of IMBH seeds via NCO collisions is enhanced by a pile-up of migrators.
We highlight the remarkable parallel between the growth of IMBH in AGN discs with models of giant planet growth in protoplanetary discs. If an IMBH becomes massive enough it can open a gap in the AGN disc. IMBH migration in AGN discs may stall, allowing them to survive the end of the AGN phase and remain in galactic nuclei. Our proposed mechanisms should be more efficient at growing IMBH in AGN discs than the standard model of IMBH growth in stellar clusters. Dynamical heating of disc NCOs by cusp stars is transferred to the gas in an AGN disc helping to maintain the outer disc against gravitational instability. Model predictions, observational constraints and implications are discussed in a companion paper (Paper II).