Revisiting the Completeness and the Luminosity Function in High-Redshift
  Low-Luminosity Quasar Surveys
  Recent studies have derived quasar luminosity functions (QLFs) at various
redshifts. However, the faint side of the QLF at high redshifts is still too
uncertain. An accurate estimate of the survey completeness is essential to
derive an accurate QLF for use in studying the luminosity-dependent density
evolution of the quasar population. Here we investigate how the luminosity
dependence of quasar spectra (the Baldwin effect) and the attenuation model for
the inter-galactic medium (IGM) affect the completeness estimates. For this
purpose, we revisit the completeness of quasar surveys specifically at
$z\sim4-5$, using the COSMOS images observed with Subaru/Suprime-Cam. As the
result, we find that the completeness estimates are sensitive to the luminosity
dependence of the quasar spectrum and difference in the IGM attenuation models.
At $z\sim4$, the number density of quasars when we adopt the latest IGM model
and take the luminosity dependence of spectra into account are
$(3.49\pm1.62)\times10^{-7}$ Mpc$^{-3}$ mag$^{-1}$ for $-24.09<M_{1450}<-23.09$
and $(5.24\pm2.13)\times10^{-7}$ Mpc$^{-3}$ mag$^{-1}$ for
$-23.09<M_{1450}<-22.09$, respectively, which are $\sim$$24$$\%$ lower than
that estimated by the conventional method. On the other hand, at $z\sim5$, a
$1\sigma$ confidence upper limit of the number density at
$-24.5<M_{1450}<-22.5$ in our new estimates is $\sim$$43$$\%$ higher than that
estimated previously. The results suggest that the luminosity dependence of the
quasar spectrum and the IGM model are important for deriving accurate number
density of high-$z$ quasars. Even taking these effects into account, the
inferred luminosity-dependent density evolution of quasars is consistent with
the AGN down-sizing evolutionary picture.
