Eddington-limited accretion and the black hole mass function at redshift
  6
  We present discovery observations of a quasar in the Canada-France High-z
Quasar Survey (CFHQS) at redshift z=6.44. We also use near-IR spectroscopy of
nine CFHQS quasars at z~6 to determine black hole masses. These are compared
with similar estimates for more luminous Sloan Digital Sky Survey (SDSS)
quasars to investigate the relationship between black hole mass and quasar
luminosity. We find a strong correlation between MgII FWHM and UV luminosity
and that most quasars at this early epoch are accreting close to the Eddington
limit. Thus these quasars appear to be in an early stage of their life cycle
where they are building up their black hole mass exponentially. Combining these
results with the quasar luminosity function, we derive the black hole mass
function at z=6. Our black hole mass function is ~10^4 times lower than at z=0
and substantially below estimates from previous studies. The main uncertainties
which could increase the black hole mass function are a larger population of
obscured quasars at high-redshift than is observed at low-redshift and/or a low
quasar duty cycle at z=6. In comparison, the global stellar mass function is
only ~10^2 times lower at z=6 than at z=0. The difference between the black
hole and stellar mass function evolution is due to either rapid early star
formation which is not limited by radiation pressure as is the case for black
hole growth or inefficient black hole seeding. Our work predicts that the black
hole mass - stellar mass relation for a volume-limited sample of galaxies
declines rapidly at very high redshift. This is in contrast to the observed
increase at 4<z<6 from the local relation if one just studies the most massive
black holes.
