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.