Constraining the Redshift z=6 Quasar Luminosity Function Using   Gravitational Lensing
  Recent discoveries by the Sloan Digital Sky Survey (SDSS) of four bright redshift z=6 quasars could constrain the mechanism by which the supermassive black holes powering these sources are assembled. Here we compute the probability that the fluxes of the quasars are strongly amplified by gravitational lensing, and therefore the likelihood that the black hole masses are overestimated when they are inferred assuming Eddington luminosities. The poorly-constrained shape of the intrinsic quasar luminosity function (LF) at redshift z=6 results in a large range of possible lensing probabilities. If the LF is either steep, or extends to faint magnitudes, the probability for amplification by a factor of ten ore more (and with only one image detectable by SDSS) can reach essentially 100 percent. We show that future observations, in particular, either of the current four quasars at the high angular resolution provided by the Hubble Space Telescope, or of an increased sample of about twenty redshift six quasars at the current angular resolution, should either discover several gravitational lenses, or else provide interesting new constraints on the shape of the z=6 quasar LF. 
Julia M. Comerford, Zoltan Haiman (Princeton University), and Joop   Schaye (IAS)