The shallow slope of the z ∼ 6 quasar luminosity function: Limits from the lack of multiple-image gravitational lenses

We place a limit on the logarithmic slope of the luminous quasar luminosity function at z ∼ 6 of β ≳ -3.0 (90 per cent) using gravitational lensing constraints to build on the limit of β ≳ -3.3 (90 per cent) derived from an analysis of the luminosity distribution by Fan et al. This tight constraint is obtained by noting that, of the two quasars that are lensed by foreground galaxies, neither is multiply imaged. These observations are surprising if the luminosity function is steep because magnification bias results in an overabundance of multiply imaged relative to singly imaged lensed quasars. Our Bayesian analysis uses the a posteriori information regarding alignments with foreground galaxies of the two lensed quasars, and provides a constraint on β that is nearly independent of the uncertain evolution in the lens population. The results suggest that the bright end of the quasar luminosity function continues to flatten out to z ∼ 6, as is observed between z ∼ 3 and ∼5. Provided that SDSS J1148+5251 at z = 6.37 is magnified by an intervening lens galaxy at z ∼ 5, we also show that the high lens redshift in this system (if it is real) implies a comoving density of massive galaxies that may be close to constant out to high redshift. The combination of constraints on the quasar luminosity function and lens galaxy evolution is used to compute an improved estimate for the z ∼ 6 multiple-image lens fraction of ∼l-3 per cent.