Self-Regulated Growth of Supermassive Black Holes in Galaxies as the Origin of the Optical and X-ray Luminosity Functions of Quasars

We postulate that supermassive black-holes grow in the centers of galaxies until they unbind the galactic gas that feeds them. We show that the corresponding self-regulation condition yields a correlation between black-hole mass (Mbh) and galaxy velocity dispersion (sigma) as inferred in the local universe, and recovers the observed optical and X-ray luminosity functions of quasars at redshifts up to z~6 based on the hierarchical evolution of galaxy halos in a Lambda-CDM cosmology. With only one free parameter and a simple algorithm, our model yields the observed evolution in the number density of optically bright or X-ray faint quasars between 2<z<6 across 3 orders of magnitude in bolometric luminosity and 3 orders of magnitude in comoving density per logarithm of luminosity. The self-regulation condition identifies the dynamical time of galactic disks during the epoch of peak quasar activity (z~2.5) as the origin of the inferred characteristic quasar lifetime of ~10 million years. Since the lifetime becomes comparable to the Salpeter e-folding time at this epoch, the model also implies that the Mbh-sigma relation is a product of feedback regulated accretion during the peak of quasar activity. The mass-density in black-holes accreted by that time is consistent with the local black-hole mass density of ~(0.8-6.3) times 10^5 solar masses per cubic Mpc, which we have computed by combining the Mbh-sigma relation with the measured velocity dispersion function of SDSS galaxies (Sheth et al.~2003). Applying a similar self-regulation principle to supernova-driven winds from starbursts, we find that the ratio between the black hole mass and the stellar mass of galactic spheroids increases with redshift as (1+z)^1.5 although the Mbh-sigma relation is redshift-independent.