In-situ extreme mass ratio inspirals via sub-parsec formation and migration of stars in thin, gravitationally unstable AGN discs

We investigate the properties of stars born via gravitational instability in accretion discs around supermassive black holes (SMBHs) in active galactic nuclei (AGN), and how this varies with the SMBH mass, accretion rate, or viscosity. We show with geometrically thin, steady-state disc solutions that fragmentation results in different populations of stars when one considers the initial conditions (e.g. density and temperature of the gravitationally unstable regions). We find that opacity gaps in discs around $10^6 \rm M_{\odot}$ SMBHs can trigger fragmentation at radii $\lesssim 10^{-2}$ pc, although the conditions lead to the formation of initially low stellar masses around $0.1-0.5 \rm M_{\odot}$. Discs around more massive SMBHs ($M_{\rm BH} =10^{7-8} \rm M_{\odot}$) form moderately massive or supermassive stars (the majority at $10^{0-2} \rm M_{\odot}$). Using linear migration estimates, we discuss three outcomes: stars migrate till they are tidally destroyed, accreted as extreme mass ratio inspirals (EMRIs), or leftover after disc dispersal. For a single AGN activity cycle, we find a lower-limit for the EMRI rate $R_{\rm emri}\sim 0-10^{-4} \rm yr^{-1}$ per AGN assuming a SF efficiency $\epsilon=1-30\%$. In cases where EMRIs occur, this implies a volumetric rate up to $0.5-10 \rm yr^{-1} Gpc^{-3}$ in the local Universe. The rates are particularly sensitive to model parameters for $M_{\rm BH}=10^6 \rm M_{\odot}$, for which EMRIs only occur if stars can accrete to $10$s of solar masses. Our results provide further evidence that gas-embedded EMRIs can contribute a substantial fraction of events detectable by milliHz gravitational wave detectors such as LISA. Our disc solutions suggest the presence of migration traps, as has been found for more massive SMBH discs. Finally, the surviving population of stars after the disc lifetime leaves implications for stellar discs in galactic nuclei.