The mass function of black holes 1<z<4.5: comparison of models with observations

In this paper, we compare the observationally derived black hole mass function (BHMF) of luminous (>10451046 erg s-1) broad-line quasars (BLQSOs) at 1 < z < 4.5 drawn from the Sloan Digital Sky Survey (SDSS) presented by Kelly et al., with models of merger-driven black hole (BH) growth in the context of standard hierarchical structure formation models. In these models, we explore two distinct black hole seeding prescriptions at the highest redshifts: light seeds remnants of Population III stars and massive seeds that form from the direct collapse of pre-galactic discs. The subsequent merger triggered mass build-up of the black hole population is tracked over cosmic time under the assumption of a fixed accretion rate as well as rates drawn from the distribution derived by Merloni & Heinz. Four model snapshots at z= 1.25, 2, 3.25 and 4.25 are compared with the SDSS-derived BHMFs of BLQSOs. We find that the light seed models fall short of reproducing the observationally derived mass function of BLQSOs at MBH > 109 M? throughout the redshift range; the massive seed models with a fixed accretion rate of 0.3 Edd, or with accretion rates drawn from the Merloni & Heinz distribution provide the best fit to the current observational data at z > 2, although they overestimate the high-mass end of the mass function at lower redshifts. At low redshifts, a drastic drop in the accretion rate is observed and this is explained as arising due to the diminished gas supply available due to consumption by star formation or changes in the geometry of the inner feeding regions. Therefore, the overestimate at the high-mass end of the black hole mass function for the massive seed models can be easily modified, as the accretion rate is likely significantly lower at these epochs than what we assume. For the Merloni & Heinz model, examining the Eddington ratio distributions fEdd, we find that they are almost uniformly sampled from fEdd= 10-2 to 1 at z? 1, while at high redshift, current observations suggest accretion rates close to Eddington, if not mildly super-Eddington, at least for these extremely luminous quasars. Our key findings are that the duty cycle of super-massive black holes powering BLQSOs increases with increasing redshift for all models and models with Population III remnants as black hole seeds are unable to fit the observationally derived BHMFs for BLQSOs, lending strong support for the massive seeding model.