A Physical Model for the Luminosity Function of High‐Redshift Quasars

Abstract

We provide a simple theoretical model for the quasar luminosity function at high redshifts that naturally reproduces the statistical properties of the luminous Sloan Digital Sky Survey (SDSS) quasar sample at redshifts z similar to 4.3 and z greater than or similar to 5.7. Our model is based on the assumptions that quasar emission is triggered by galaxy mergers and that the black hole mass is proportional to a power law in the circular velocity of the host galactic halo, v(c). We assume that quasars shine at their Eddington luminosity over a time proportional to the mass ratio between the small and final galaxies in the merger. This simple model fits the quasar luminosity function at z similar to 2-3, reproduces the normalization and logarithmic slope (beta similar to 2.58) at z similar to 4.3, explains the space density of bright SDSS quasars at z similar to 6.0, reproduces the black hole-halo mass relation for dormant black holes in the local universe, and matches the inferred duty cycle of quasar activity (similar to10(7) yr) in Lyman break galaxies at z similar to 3. An acceptable fit to all of these constraints requires 0.7 less than or similar to sigma(8) less than or similar to 1.0. Based on the derived luminosity function, we predict the resulting gravitational lensing rates for high-redshift quasars. The lens fractions in the SDSS samples are predicted to be similar to2% at z similar to 4.3 and similar to8% at z greater than or similar to 5.7. It is interesting to note that the limiting quasar luminosity in our best-fit relation L proportional to 5 nu(c)(5)/G scales as the binding energy of the host galaxy divided by its dynamical time, implying that feedback is the mechanism that regulates black hole growth in galactic potential wells.