We propose a scenario in which massive stars form in a self-gravitating gaseous disc around a supermassive black hole (SMBH). We analyse the dynamics of a disc forming around an SMBH, in which the angular momentum is transported by turbulence induced by the disc's self-gravity. We find that once the surface density of the disc exceeds a critical value, the disc fragments into dense clumps. We argue that the clumps accrete material from the remaining disc and merge into larger clumps; the upper mass of a merged clump is a few tens to a few hundreds of solar mass.
This picture fits well with the observed young stellar discs near the SgrA* black hole in the Galactic Centre. In particular, we show how the masses and spatial distribution of the young stars, and the total mass in the Galactic Centre discs can be explained. However, explaining the origin of the several young stars closest to the black hole (the S-stars) is more problematic: their orbits are compact, eccentric, and have random orientation. We propose that the S-stars were born in a previous starburst(s), and then migrated through their parent disc via type-I or runaway migration. Their orbits were then randomized by the Rauch–Tremaine resonant relaxation.
We then explore the consequences of the star formation scenario for AGN discs, which are continuously resupplied with gas. We argue that some compact remnants generated by the starburst will get embedded in the disc. The disc-born stellar mass black holes will interact gravitationally with the massive accretion disc and be dragged towards the central black hole. Merger of a disc-born black hole with the central black hole will produce a burst of gravitational waves. If the central black hole is accreting at a rate comparable to the Eddington limit, the gas drag from the accretion disc will not alter significantly the dynamics of the final year of merger, and the gravitational waves should be observable by Laser Interferometer Space Antenna (LISA). For a reasonable range of parameters such mergers will be detected monthly, and that the gravitational-wave signal from these mergers is distinct from that of other merger scenarios. Also, for some plausible black hole masses and accretion rates, the burst of gravitational waves should be accompanied by a detectable change in the optical luminosity of the central engine.