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. 
Stuart Wyithe and Abraham Loeb