Properties of High Redshift Quasars-I: Evolution of the super-massive   black-hole to halo mass ratio
  In the local universe, the masses of Super-Massive Black-Holes (SMBH) appear to correlate with physical properties of their hosts, including the mass of the dark-matter halos. At higher redshifts, we observe the growth of SMBHs indirectly through identification of high redshift quasars. However information on their hosts is difficult to obtain. In this paper we determine the masses of halos that host high redshift quasars (at z>4) by comparing the rate of growth of quasar density with that predicted by the Press-Schechter mass function. The host mass determined depends on how the ratio between SMBH and host halo mass evolves with redshift. Assuming the ratio between SMBH and halo mass does not evolve with redshift, we find a host halo mass of log(M)=11.7+/-0.3 solar masses. Even if the quasars shine at their Eddington limit, this mass is significantly smaller than that seen in the local universe. Indeed the null-hypothesis, of a constant SMBH to halo mass ratio at all redshifts, can be ruled out at greater than a 5-sigma level. SMBHs must therefore have contributed a larger fraction to the host mass in the past. Including redshift evolution of the SMBH to halo mass ratio, we find larger halo masses of log(M)=12.4+/-0.3 solar masses, and a ratio between SMBH and halo mass that increases with redshift as ~(1+z)^1.5 are required to be consistent with both local and high redshift observations. We investigate restrictions placed on the critical linear overdensity of quasar hosts at their epoch of virialisation and find that it cannot exceed the traditional value of delta_c=1.69 by more than a factor of two. Finally, we find that the high redshift quasars are hosted by fluctuations on scales that have a variance of (delta M/M) = 2-3, corresponding to (3-4.5)-sigma fluctuations in the density field. 
Stuart Wyithe & T. Padmanabhan