Properties of High Redshift Quasars-II: What does the quasar luminosity   function tell us about super-massive black-hole evolution?
  In the local universe, the masses of Super-Massive Black-Holes (SMBH) appear to correlate with the physical properties of their hosts, including the mass of the dark-matter halo. Using these clues as a starting point many studies have produced models that can explain phenomena like the quasar luminosity function. The shortcoming of this approach is that working models are not unique, and as a result it is not always clear what input physics is being constrained. Here we take a different approach. We identify critical parameters that describe the evolution of SMBHs at high redshift, and constrain their parameter space based on observations of high redshift quasars from the Sloan Digital Sky Survey. We find that the luminosity function taken in isolation is somewhat limited in its ability to constrain SMBH evolution due to some strong degeneracies. This explains the presence in the literature of a range of equally successful models based on different physical hypotheses. Including the constraint of the local SMBH to halo mass ratio breaks some of the degeneracies, and our results suggest halo masses at z~4.8 of 10^{12.5+/-0.3}M_solar (with 90% confidence), with a SMBH to halo mass ratio that decreases with time (>99%). We also find a quasar luminosity to halo mass ratio that increases with halo mass (>99%). These features need to be incorporated in all successful models of SMBH evolution. On the other hand current observations do not permit any conclusions regarding the evolution of quasar lifetime, or the SMBH occupation fraction in dark matter halos. 
Stuart Wyithe and T. Padmanabhan