High-redshift quasars and the supermassive black hole mass budget: constraints on quasar formation models

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 zsimilar to 6, and estimates of the local mass density of SMBHs. If similar to90 per cent of this mass was accreted at redshifts zless than or similar to3, 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 zsimilar to6 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, cannot satisfy both constraints simultaneously. When this model is normalized to reproduce the number density of luminous quasars at zsimilar to 6, the mass budget is grossly exceeded owing 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-circle-sigma relation. Successful models can also be constructed by restricting the formation of seed black holes to redshifts above z(crit)similar to11.5 or to haloes above a velocity threshold v(crit)similar to55 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-circle-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.