The formation of the most massive quasars observed at high redshifts requires extreme inflows of gas down to the length scales of the central compact object. Here, we estimate the maximum inflow rate allowed by gravity down to the surface of supermassive stars, the possible progenitors of these supermassive black holes. We use the continuity equation and the assumption of free-fall to derive maximum allowed inflow rates for various density profiles. We apply our approach to the mass-radius relation of rapidly accreting supermassive stars to estimate an upper limit to the accretion rates allowed during the formation of these objects. We find that the maximum allowed rate $dot M_{rm max}$ is given uniquely by the compactness of the accretor. For the compactness of rapidly accreting supermassive stars, $dot M_{rm max}$ is related to the stellar mass $M$ by a power-law $dot M_{rm max}propto M^{3/4}$. The rates of atomically cooled halos (0.1 -- 10 M$_odot$ yr$^{-1}$) are allowed as soon as $Mgtrsim1$ M$_odot$. The largest rates expected in galaxy mergers ($10^4-10^5$ M$_odot$ yr$^{-1}$) become accessible once the accretor is supermassive ($Mgtrsim10^4$ M$_odot$). These results suggest that supermassive stars can accrete up to masses $>10^6$ M$_odot$ before they collapse via the general-relativistic instability. At such masses, the collapse is expected to lead to the direct formation of a supermassive black hole even within metal-rich gas, resulting in a black hole seed that is significantly heavier than in conventional direct collapse models for atomic cooling halos.
Download