The cosmological evolution of quasar black-hole masses

Virial black-hole mass estimates are presented for more than 12000 quasars in the redshift interval 0.1<z<2.1, based on modelling of spectra from the Sloan Digital Sky Survey (SDSS) first data release. The black-hole masses of the SDSS quasars are found to lie between $\simeq10^{7}\Msun$ and an upper limit of $\simeq 3\times 10^{9}\Msun$, entirely consistent with the largest black-hole masses found to date in the local Universe. Moreover, the quasar Eddington ratios show a clear upper boundary at $L_{bol}/L_{Edd}\simeq 1$, suggesting that the Eddington luminosity is still a relevant physical limit to the accretion rate of luminous quasars at $z\leq 2$. By combining the black-hole mass distribution of the SDSS quasars with the 2dF quasar luminosity function, the number density of active black holes at $z\simeq 2$ is estimated as a function of mass. In addition, we independently estimate the local black-hole mass function for early-types using the $M_{bh}-\sigma$ and $M_{bh}-L_{bulge}$ correlations. Based on the SDSS velocity dispersion function and the 2MASS $K-$band luminosity function, both estimates are found to be consistent at the high-mass end ($M_{bh}\geq 10^{8}\Msun$). By comparing the estimated number density of active black holes at $z\simeq 2$ with the local mass density of dormant black holes, we set lower limits on the quasar lifetimes and find that the majority of black holes with mass $\geq 10^{8.5}\Msun$ are in place by $\simeq 2$.