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Light or heavy supermassive black hole seeds: the role of internal rotation in the fate of supermassive stars

Supermassive black holes are a key ingredient of galaxy evolution. However, their origin is still highly debated. In one of the leading formation scenarios, a black hole of $\sim100$ M$_{\odot}$ results from the collapse of the inner core of a supermassive star ($\gtrsim 10^{4-5}$ M$_{\odot}$), created by the rapid accumulation ($\gtrsim 0.1 $ M$_{\odot}$ yr$^{-1}$) of pristine gas at the centre of newly formed galaxies at $z\sim 15$. The subsequent evolution is still speculative: the remaining gas in the supermassive star can either directly plunge into the nascent black hole, or part of it can form a central accretion disc, whose luminosity sustains a surrounding, massive, and nearly hydrostatic envelope (a system called a "quasi-star"). To address this point, we consider the effect of rotation on a quasi-star, as angular momentum is inevitably transported towards the galactic nucleus by the accumulating gas. Using a model for the internal redistribution of angular momentum that qualitative matches results from simulations of rotating convective stellar envelopes, we show that quasi-stars with an envelope mass greater than a few $10^{5}$ M$_{\odot} \times (\rm black~hole~mass/100 M_{\odot})^{0.82}$ have highly sub-keplerian gas motion in their core, preventing gas circularisation outside the black hole's horizon. Less massive quasi-stars could form but last for only $\lesssim 10^4$ years before the accretion luminosity unbinds the envelope, suppressing the black hole growth. We speculate that this might eventually lead to a dual black hole seed population: (i) massive ($>10^{4}$ M$_{\odot}$) seeds formed in the most massive ($> 10^{8}$ M$_{\odot}$) and rare haloes; (ii) lighter ($\sim 10^{2}$ M$_{\odot}$) seeds to be found in less massive and therefore more common haloes.