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.