The growth efficiency of high-redshift black holes

The observational evidence that Super-Massive Black Holes (M_• ∼ 10^9-10 M_⊙) are already in place less than 1 Gyr after the big bang poses stringent time constraints on the growth efficiency of their seeds. Among proposed possibilities, the formation of massive (∼10^3-6 M_⊙) 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, using a set of astrophysically motivated initial conditions, we analytically and numerically investigate the accretion flow on to high-redshift (z ∼ 10) black holes to understand the physical requirements favouring 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 D ≲ 0.5) and inefficient growth, with recurring outflow episodes. We find that low-mass seeds (≲10^3-4 M_⊙) evolve in the feedback-limited regime, while more massive seeds (≳10^5-6 M_⊙) 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 (ɛ = 0.1), we have also explored slim disc models, appropriate for super-Eddington accretion, where radiation is trapped in the disc and the radiative efficiency is reduced (ɛ ≲ 0.04), which may ensure a continuous growth with dot{M} ≫ dot{M}_{Edd} (up to {∼ } 300 dot{M}_{Edd} in our simulations). Under these conditions, outflows play a negligible role and a black hole can accrete 80-100 per cent of the gas mass of the host halo (∼10⁷ M_⊙) in ∼10 Myr, while in feedback-limited systems we predict that black holes can accrete only up to ∼15 per cent of the available mass.