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 (σ) 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 ΛCDM 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 years. Since the lifetime becomes comparable to the Salpeter e-folding time at this epoch, the model also implies that the Mbh − σ 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 ρbh ∼ (2.3 −1.5)×10M⊙Mpc, which we have computed by combining the Mbh–σ relation with the measured velocity dispersion function of SDSS galaxies (Sheth et al. 2003). Comparison of the local black-hole mass-function with that inferred from combining the feedback-relation with the halo mass-function suggests that most massive (> 10M⊙) black-holes may have already been in place by z ∼ 6. 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) although the Mbh-σ relation is redshift-independent. Subject headings: black-hole physics quasars: general