The mass function of black holes at 1<z<4.5: comparison of models with
  observations
  In this paper, we compare the observationally derived black hole mass
function (BHMF) of luminous broad-line quasars (BLQSOs) at 1<z<4.5 drawn from
the Sloan Digital Sky Survey (SDSS) presented in Kelly et al. (2010), with
models of merger driven BH growth in the context of standard hierarchical
structure formation models. In the 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 disks. 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. Our model snapshots are compared to the SDSS derived BHMFs of BLQSOs.
Our key findings are that the duty cycle of SMBHs powering BLQSOs increases
with increasing redshift for all models and models with Pop III remnants as
black hole seeds are unable to fit the observationally derived BHMFs for
BLQSOs, lending strong support for the massive seeding model (abridged).
