Constraints on black hole duty cycles and the black hole-halo relation from SDSS quasar clustering

We use Shen et al.'s (2009) measurements of luminosity-dependent clustering in the Sloan Digital Sky Survey Data Release 5 quasar catalogue, at redshifts 0.4 ≤ z ≤ 2.5, to constrain the relation between quasar luminosity and host halo mass and to infer the duty cycle f_opt, the fraction of black holes that shine as optically luminous quasars at a given time. We assume a monotonic mean relation between quasar luminosity and host halo mass, with lognormal scatter Σ. For specified f_opt and Σ, matching the observed quasar space density determines the normalization of the luminosity-halo mass relation, from which we predict the clustering bias. The data show no change of bias between the faint and bright halves of the quasar sample but a modest increase in bias for the brightest 10 per cent. At the mean redshift z = 1.45 of the sample, the data can be well described either by models with small intrinsic scatter (Σ = 0.1 dex) and a duty cycle f_opt = 6 × 10^-4 or by models with much larger duty cycles and larger values of the scatter. 'Continuity equation' models of the black hole mass population imply f_opt ≥ 2 × 10^-3 in this range of masses and redshifts, and the combination of this constraint with the clustering measurements implies scatter Σ ≥ 0.4 dex. These findings contrast with those inferred from the much stronger clustering of high-luminosity quasars at z ≈ 4, which require minimal scatter between luminosity and halo mass and duty cycles close to one.