Virial black-hole mass estimates are presented for 12698 quasars in the redshift interval 0.1 \leq z \leq 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 \simeq 10 ^ { 7 } \hbox { $ \thinspace M _ { \odot } $ } and an upper limit of \simeq 3 \times 10 ^ { 9 } \hbox { $ \thinspace M _ { \odot } $ } , entirely consistent with the largest black-hole masses found to date in the local Universe .
The estimated Eddington ratios of the broad-line quasars ( FWHM \geq 2000 km s ^ { -1 } ) 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 broad-line 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-type galaxies 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 } \hbox { $ \thinspace M _ { \odot } $ } ) .
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 } \hbox { $ \thinspace M _ { \odot } $ } are in place by \simeq 2 .