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 , using a set of astrophysically-motivated initial conditions , 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 } \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt \hbox { $ < % $ } } 0.5 ) and inefficient growth , with recurring outflow episodes .
We find that low-mass seeds ( \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt \hbox { $ < $ } } 10 ^ { 3 % -4 } \mathrm { M _ { \odot } } ) evolve in the feedback-limited regime , while more massive seeds ( \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt \hbox { $ > $ } } 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 \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt \hbox { $ < % $ } } 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 .