Do all QSOs have the same black hole mass?

Quasi-stellar objects (QSOs) from SDSS, 2QZ and 2SLAQ covering an order of magnitude in luminosity at fixed redshift exhibit similar amplitudes of clustering, with the brightest sample showing a clustering length only 11 +/- 9 per cent higher than the faintest sample. In addition, QSO clustering evolution at z > 0.5 is well fitted by a model that assumes a fixed host halo mass. If halo and black hole (BH) masses are related, then this may imply that QSOs occur in a relatively narrow range of halo masses with a correspondingly narrow range of BH mass. Hubble Space Telescope and Gemini high-resolution imaging of QSOs covering a large range in luminosity also show a relatively narrow range in QSO host galaxy luminosity. We argue that the slow evolution of early-type galaxies out to z approximate to 1-2 may also provide further support for a slow evolution of QSO host BH masses. The result would mean that if high-z QSOs radiate at Eddington rates then low-z type 1 Seyfert galaxy must radiate at approximate to 100 times less than Eddington. We discuss the consequences in terms of four empirical models where (i) QSOs radiate at a fixed fraction of L-Edd, (ii) QSO luminosity 'flickers' over time, (iii) QSOs have a single BH mass and (iv) QSOs are long lived and evolve via pure luminosity evolution (PLE). We conclude that the L-Edd model requires M-BH and M-halo to be decoupled to circumvent the clustering results. While the single BH mass and flickering models fit the z > 0.5 clustering results, they appear to be rejected by the M-BH-L relation found from reverberation mapping at z approximate to 0. We find that the inclusion of z < 0.5 QSO clustering data improves the fit of a long-lived QSO model and suggest that the predictions of the PLE model for QSO BH masses agree reasonably with ultraviolet bump and reverberation estimates.