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 ( \sigma ) as inferred in the local universe , and recovers the observed optical and X-ray luminosity functions of quasars at redshifts up to z \sim 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 \lesssim z \lesssim 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 \sim 2.5 ) as the origin of the inferred characteristic quasar lifetime of \sim 10 ^ { 7 } years .
Since the lifetime becomes comparable to the Salpeter e -folding time at this epoch , the model also implies that the M _ { bh } - \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 \rho _ { bh } \sim ( 2.3 ^ { +4.0 } _ { -1.5 } ) \times 10 ^ { 5 } M _ { \odot } Mpc ^ { -3 } , which we have computed by combining the M _ { bh } – \sigma 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 ( > 10 ^ { 9 } M _ { \odot } ) black-holes may have already been in place by z \sim 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 ) ^ { 3 / 2 } although the M _ { bh } - \sigma relation is redshift-independent .