Super-critical accretion of medium-weight seed black holes in gaseous proto-galactic nuclei

Accretion at sustained or episodic super-Eddington (SE) rates has been proposed as a pathway to grow efficiently light seeds produced by Pop-III stars. We investigate if SE accretion can be sustained onto a black hole (BH) with $M_{\odot} \sim 10^3$~M$_{\odot}$ in the centre of a gas-rich proto-galaxy at $z=15$. We perform high-resolution smoothed-particle hydrodynamical simulations, including two different sub-grid models for SE accretion, one based on the slim disc paradigm, and one inspired by recent radiation-magnetohydrodynamical simulations by Jiang and collaborators. Radiative feedback has the form of a thermal dump to surrounding gas particles, with the radiative efficiency being set according to the different SE accretion models. We find that, in all simulations, star formation, BH feedback, and interactions between clumps and the BH rapidly quench accretion after $\sim$1~Myr, irrespective of the sub-grid model used for accretion. Quenching is stronger in the model based on the simulations of Jiang and collaborators relative to the slim disc model because of its higher radiative efficiency. The SE growth phase is always very brief, lasting a few 0.1~Myr. In the most optimistic case, the BH reaches a mass of $\sim$10$^4$~M$_{\odot}$. We extrapolate the final BH masses from $z=15$ to $z\sim6$, assuming subsequent galaxy mergers will replenish the gas reservoir and trigger new cycles of SE accretion. We find that at most BH seeds would grow to $\sim$10$^6$~M$_{\odot}$, comparable to the mass of massive BHs in spiral galaxies such as the Milky Way, but falling short of the mass of the high-redshift quasars.