Properties of high redshift quasars-I. Evolution of the super-massive black-hole to halo mass ratio

In the local Universe, the masses of supermassive black holes (SMBHs) appear to correlate with the physical properties of their hosts, including the mass of the dark matter haloes. At higher redshifts, we observe the growth of SMBHs indirectly through the identification of high-redshift quasars. However, information on their hosts is more difficult to obtain. In this paper, we determine the masses of the haloes that host the 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 the SMBH and the host halo mass evolves with redshift. Under the assumption that the ratio between the SMBH and the halo mass does not evolve with redshift, we find a host halo mass of M= 1011.7±0.3 M∃ . Even if the quasars shine at their Eddington limit, this host mass is significantly smaller than that seen at lower redshifts in the local Universe. Indeed, we find that the null hypothesis, of a constant ratio between the SMBH and the halo mass at all redshifts, can be ruled out at greater than a 5σ level. SMBHs must therefore have contributed a larger fraction to the host mass in the past. This finding is consistent with expectations from models of self-limiting SMBH growth. When we include the redshift evolution of the ratio between the SMBH and the halo mass, we find larger halo masses of M∼ 1012.4±0.3 M∼ , in combination with a ratio between the SMBH and the host halo mass that increases with redshift in proportion to ∼(1 +z)1.5, are required to be consistent with both local and high-redshift observations. We also investigate the restrictions placed on the critical linear overdensity of quasar hosts at their epoch of virialization, and find that it cannot exceed the traditional value of δc= 1.69 by more than a factor of 2. Finally, we find that the high-redshift quasars are hosted by fluctuations on scales that have a variance of δM/M= 2-3, corresponding to (3-4.5)σ fluctuations in the density field