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 SDSS Data Release 5 Quasar Catalog, 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 log-normal scatter \Sigma. For specified f_opt and \Sigma, 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%. At the mean redshift z=1.45 of the sample, the data can be well described either by models with small intrinsic scatter (\Sigma=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 \Sigma > 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.