The Growth Efficiency of High-Redshift Black Holes

The observational evidence that Super-Massive Black Holes ($M_{\bullet} \sim 10^{9-10} \, \mathrm{M_{\odot}}$) are already in place less than $1 \, \mathrm{Gyr}$ after the Big Bang poses stringent time constraints on the growth efficiency of their seeds. Among proposed possibilities, the formation of massive ($\sim 10^{3-6} \, \mathrm{M_{\odot}}$) seeds and/or the occurrence of super-Eddington ($\dot{M}>\dot{M}_{Edd}$) accretion episodes may contribute to the solution of this problem. In this work we analytically and numerically investigate the accretion flow onto high-redshift ($z \sim 10$) black holes to understand the physical requirements favoring rapid and efficient growth. Our model identifies a "feeding-dominated" accretion regime and a "feedback-limited" one, the latter being characterized by intermittent (duty cycles ${\cal D} \lesssim 0.5$) and inefficient growth, with recurring outflow episodes. We find that low-mass seeds ($\lesssim 10^{3-4} \, \mathrm{M_{\odot}}$) evolve in the feedback-limited regime, while more massive seeds ($\gtrsim 10^{5-6} \, \mathrm{M_{\odot}}$) grow very rapidly as they are found in the feeding-dominated regime. In addition to the standard accretion model with a fixed matter-energy conversion factor ($\epsilon = 0.1$), we have also explored slim disk models, appropriate for super-Eddington accretion, where radiation is trapped in the disk and the radiative efficiency is reduced ($\epsilon \lesssim 0.04$), which may ensure a continuous growth with $\dot{M} \gg \dot{M}_{Edd}$ (up to $\sim 300\dot{M}_{Edd}$ in our simulations). Under these conditions, outflows play a negligible role and a black hole can accrete $80\%-100\%$ of the gas mass of the host halo ($\sim 10^7 \, \mathrm{M_{\odot}}$) in $\sim 10 \, \mathrm{Myr}$, while in feedback-limited systems we predict that black holes can accrete only up to $\sim 15\%$ of the available mass.