Subaru High-$z$ Exploration of Low-Luminosity Quasars (SHELLQs). XVIII. The Dark Matter Halo Mass of Quasars at $z\sim6$

We present, for the first time, dark matter halo (DMH) mass measurement of quasars at $z\sim6$ based on a clustering analysis of 107 quasars. Spectroscopically identified quasars are homogeneously extracted from the HSC-SSP wide layer over $891\,\mathrm{deg^2}$. We evaluate the clustering strength by three different auto-correlation functions: projected correlation function, angular correlation function, and redshift-space correlation function. The DMH mass of quasars at $z\sim6$ is evaluated as $5.0_{-4.0}^{+7.4}\times10^{12}\,h^{-1}M_\odot$ with the bias parameter $b=20.8\pm8.7$ by the projected correlation function. The other two estimators agree with these values, though each uncertainty is large. The DMH mass of quasars is found to be nearly constant $\sim10^{12.5}\,h^{-1}M_\odot$ throughout cosmic time, suggesting that there is a characteristic DMH mass where quasars are always activated. As a result, quasars appear in the most massive halos at $z \sim 6$, but in less extreme halos thereafter. The DMH mass does not appear to exceed the upper limit of $10^{13}\,h^{-1}M_\odot$, which suggests that most quasars reside in DMHs with $M_\mathrm{halo}<10^{13}\,h^{-1}M_\odot$ across most of the cosmic time. Our results supporting a significant increasing bias with redshift are consistent with the bias evolution model with inefficient AGN feedback at $z\sim6$. The duty cycle ($f_\mathrm{duty}$) is estimated as $0.019\pm0.008$ by assuming that DMHs in some mass interval can host a quasar. The average stellar mass is evaluated from stellar-to-halo mass ratio as $M_*=6.5_{-5.2}^{+9.6}\times10^{10}\,h^{-1}M_\odot$, which is found to be consistent with [C II] observational results.