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 (M_bh) 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 Λ-dominated cold dark matter 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 with 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 ~10⁷ yr. Since the lifetime becomes comparable to the Salpeter e-folding time at this epoch, the model also implies that the M_bh-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 ρ_bh ~ (2.3) × 10⁵ M_☉ Mpc^-3, which we have computed by combining the M_bh-σ relation with the measured velocity dispersion function of Sloan Digital Sky Survey galaxies. 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 (>10⁹ M_☉) 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 local ratio between the black hole mass and the stellar mass of galactic spheroids should be ~0.001, independent of mass and in agreement with observations. This ratio increases with redshift, although the M_bh-σ relation is redshift-independent.