Constraints on Black Hole Growth, Quasar Lifetimes, and Eddington Ratio Distributions from the SDSS Broad Line Quasar Black Hole Mass Function

We present an estimate of the black hole mass function (BHMF) of broad line quasars (BLQSOs) that self-consistently corrects for incompleteness and the statistical uncertainty in the mass estimates, based on a sample of 9886 quasars at 1 < z < 4.5 drawn from the Sloan Digital Sky Survey. We find evidence for `cosmic downsizing' of black holes in BLQSOs, where the peak in their number density shifts to higher redshift with increasing black hole mass. The cosmic mass density for black holes seen as BLQSOs peaks at z ~ 2. We estimate the completeness of the SDSS as a function of black hole mass and Eddington ratio, and find that at z > 1 it is highly incomplete at M_BH < 10^9 M_Sun and L / L_Edd < 0.5. We also estimate a lower limit on the lifetime of a single BLQSO phase and we place constraints on the maximum mass of a black hole in a BLQSO. Our estimated distribution of BLQSO Eddington ratios peaks at L / L_Edd ~ 0.05 and has a dispersion of ~ 0.4 dex, implying that most BLQSOs are not radiating at or near the Eddington limit; however the location of the peak is subject to considerable uncertainty. The steep increase in number density of BLQSOs toward lower Eddington ratios is expected if the BLQSO accretion rate monotonically decays with time. Furthermore, our estimated lifetime and Eddington ratio distributions imply that the majority of the most massive black holes spend a significant amount of time growing in an earlier obscured phase, a conclusion which is independent of the unknown obscured fraction. These results are consistent with models for self-regulated black hole growth, at least for massive systems at z > 1, where the BLQSO phase occurs at the end of a fueling event when black hole feedback unbinds the accreting gas, halting the accretion flow.