We investigate the constraints on models of supermassive black hole ( SMBH ) and quasar formation obtainable from two recent observational developments : the discovery of luminous quasars at z \sim 6 , and estimates of the local mass density of SMBHs .
If \sim 90 per cent of this mass was accreted at redshifts z \la 3 , as suggested by the observed quasar luminosity functions , these joint constraints pose a challenge for models , which must account for the observed luminous quasar population at z \sim 6 within a very limited ‘ mass budget ’ .
We investigate a class of models based within the hierarchical structure formation scenario , in which major mergers lead to black hole formation and fuelling , and the resulting quasars shine at their Eddington-limited rate until their fuel is exhausted .
We show that the simplest such model , in which a constant fraction of the gas within the halo is accreted in each major merger , can not satisfy both constraints simultaneously .
When this model is normalized to reproduce the number density of luminous quasars at z \sim 6 , the mass budget is grossly exceeded due to an overabundance of lower mass SMBHs .
We explore a range of modifications to the simple model designed to overcome this problem .
We show that both constraints can be satisfied if the gas accretion fraction scales as a function of the halo virial velocity .
Similar scalings have been proposed in order to reproduce the local M _ { \bullet } - \sigma relation .
Successful models can also be constructed by restricting the formation of seed black holes to redshifts above z _ { crit } \sim 11.5 or to haloes above a velocity threshold v _ { crit } \sim 55 { km } { s } ^ { -1 } , or assuming that only a fraction of major mergers result in formation of a seed SMBH .
We also briefly discuss the issue of trying to assume a ‘ universal M _ { \bullet } - \sigma relation ’ within the framework of simple Press–Schechter models , and further show that a fixed universal relation between SMBH mass and host halo mass is unlikely .