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

We provide a simple theoretical model for the quasar luminosity function at high redshifts that naturally reproduces the statistical properties of the luminous 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 v of the host galactic halo. 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 (beta -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 estimated duty cycle of quasar activity (~10^7 years) in Lyman-break galaxies at z~3. 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 ~10% at z>5.7. Interestingly, the limiting quasar luminosity in our best-fit relation of L proportional to v^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.