In order to investigate the origin of quasars, we estimate the bias factor for low-luminosity quasars at high redshift for the first time. In this study, we use the two-point angular cross-correlation function (CCF) for both low-luminosity quasars at
$-24<M_{rm 1450}<-22$ and Lyman-break galaxies (LBGs). Our sample consists of both 25 low-luminosity quasars (16 objects are spectroscopically confirmed low-luminosity quasars) in the redshift range $3.1<z<4.5$ and 835 color-selected LBGs with $z^{prime}_{rm LBG}<25.0$ at $zsim4$ in the COSMOS field. We have made our analysis for the following two quasar samples; (1) the spectroscopic sample (the 16 quasars confirmed by spectroscopy), and (2) the total sample (the 25 quasars including 9 quasars with photometric redshifts). The bias factor for low-luminosity quasars at $zsim4$ is derived by utilizing the quasar-LBG CCF and the LBG auto-correlation function. We then obtain the $86%$ upper limits of the bias factors for low-luminosity quasars, that are 5.63 and 10.50 for the total and the spectroscopic samples, respectively. These bias factors correspond to the typical dark matter halo masses, log $(M_{rm DM}/(h^{-1}M_{odot}))=$$12.7$ and $13.5$, respectively. This result is not inconsistent with the predicted bias for quasars which is estimated by the major merger models.
We present the result of our low-luminosity quasar survey in the redshift range of 4.5 < z < 5.5 in the COSMOS field. Using the COSMOS photometric catalog, we selected 15 quasar candidates with 22 < i < 24 at z~5, that are ~ 3 mag fainter than the SD
SS quasars in the same redshift range. We obtained optical spectra for 14 of the 15 candidates using FOCAS on the Subaru Telescope and did not identify any low-luminosity type-1 quasars at z~5 while a low-luminosity type-2 quasar at z~5.07 was discovered. In order to constrain the faint end of the quasar luminosity function at z~5, we calculated the 1sigma confidence upper limits of the space density of type-1 quasars. As a result, the 1sigma confidence upper limits on the quasar space density are Phi< 1.33*10^{-7} Mpc^{-3} mag^{-1} for -24.52 < M_{1450} < -23.52 and Phi< 2.88*10^{-7} Mpc^{-3} mag^{-1} for -23.52 < M_{1450} < -22.52. The inferred 1sigma confidence upper limits of the space density are then used to provide constrains on the faint-end slope and the break absolute magnitude of the quasar luminosity function at z~5. We find that the quasar space density decreases gradually as a function of redshift at low luminosity (M_{1450} ~ -23), being similar to the trend found for quasars with high luminosity (M_{1450}<-26). This result is consistent with the so-called downsizing evolution of quasars seen at lower redshifts.
We searched for quasars that are ~ 3 mag fainter than the SDSS quasars in the redshift range 3.7 < z < 4.7 in the COSMOS field to constrain the faint end of the quasar luminosity function. Using optical photometric data, we selected 31 quasar candida
tes with 22 < i < 24 at z ~ 4. We obtained optical spectra for most of these candidates using FOCAS on the Subaru telescope, and identified 8 low-luminosity quasars at z ~ 4. In order to derive the quasar luminosity function (QLF) based on our spectroscopic follow-up campaign, we estimated the photometric completeness of our quasar survey through detailed Monte Carlo simulations. Our QLF at z ~ 4 has a much shallower faint-end slope beta = -1.67^{+0.11}_{-0.17} than that obtained by other recent surveys in the same redshift. Our result is consistent with the scenario of downsizing evolution of active galactic nuclei inferred by recent optical and X-ray quasar surveys at lower redshifts.
