  Using a new large-scale (~ 0.75 Gpc)^3 hydrodynamic cosmological simulation
we investigate the growth rate of supermassive black holes in the early
universe (z > 4.75). Remarkably, we find a clear peak in the typical Eddington
ratio at black hole masses of 4-8 * 10^7 solar masses (typically found in halos
of ~7 * 10^11 to 10^12 solar masses), independent of redshift and indicative
that most of BH growth occurs in the cold-flow dominated regime. Black hole
growth is by and large regulated by the evolution of gas density. The typical
Eddington ratio at a given mass scales simply as cosmological density (1+z)^3
and the peak is caused by the competition between increased gas density
available in more massive hosts, and a decrease due to strong AGN feedback that
deprives the black hole of sufficient gas to fuel further rapid growth in the
high mass end. In addition to evolution in the mean Eddington ratio, we show
that the distribution of Eddington ratio among both mass-selected and
luminosity-selected samples is approximately log-normal. We combine these
findings into a single log-normal fitting formula for the distribution of
Eddington ratios as a function of (M_BH,z). This formula can be used in
analytic and semi analytic models for evolving black hole populations,
predicting black hole masses of observed quasars, and, in conjunction with the
observed distribution of Eddington ratios, can be used to constrain the black
hole mass function.
