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Super-Eddington accretion; flow regimes and conditions in high-z galaxies

We review and discuss theoretical studies addressing the possibility of gas accretion onto black holes occurring at rates exceeding the Eddington limit. Our focus is on the applications to the growth of black hole seeds at high redshift. We first present the general notion of Super-Eddington accretion, and then summarize the different models and numerical simulations developed to study such regime. We consider optically thick flows in accretion disks as well as in spherically symmetric envelopes, and devote particular attention to the widely adopted model based on the SLIM disk solution. While attractive for its simplicity, the SLIM disk solution is challenged by the latest generation of three-dimensional radiation (magneto)-hydrodynamical simulations, in which radiative losses can be an order of magnitude higher, and the mechanisms of radiation transport is more complex than straight advection as it takes place in a complex turbulent regime. We then discuss the gas supply rate to the sub-pc scale accretion disk or envelope from larger scales, revisiting gas inflow rates in protogalaxies under various conditions. We conclude that in the dense gaseous nuclei of high-z galaxies the conditions necessary for the onset of Super Eddington accretion regimes, such as a high optical depth and high gas supply rates from large scales, should be naturally met. Feedback from the growing BH seed should not alter significantly such conditions according to the results of radiation magneto-hydrodynamical simulations of super-critical flows in accretion disks. Furthermore, based on the required nuclear gas inflow rates and the tendency of stellar feedback to remove efficiently gas in low mass halos, we argue that super-critical accretion will be more easily achieved in relatively sizable halos, with virial masses $M_{\rm vir} > 10^{10}$ M$_{\odot}$, which become more common at $z < 15$.