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

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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 ~ 4.3 and z 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, vc. 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 ~ 2-3, reproduces the normalization and logarithmic slope (β ~ -2.58) at z ~ 4.3, explains the space density of bright SDSS quasars at z ~ 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 (~107 yr) in Lyman break galaxies at z ~ 3. An acceptable fit to all of these constraints requires 0.7 σ8 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 ~2% at z ~ 4.3 and ~8% at z 5.7. It is interesting to note that the limiting quasar luminosity in our best-fit relation L ∝ v/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.
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