GOLDRUSH. III. A Systematic Search of Protoclusters at $z\sim4$ Based on the $>100\,\mathrm{deg^2}$ Area

We conduct a systematic search for galaxy protoclusters at $z\sim3.8$ based on the latest internal data release (S16A) of the Hyper SuprimeCam Subaru strategic program (HSC-SSP). In the Wide layer of the HSC-SSP, we investigate the large-scale projected sky distribution of $g$-dropout galaxies over an area of $121\,\mathrm{deg^2}$, and identify 216 large-scale overdense regions ($>4\sigma$ overdensity significance) that are good protocluster candidates. Of these, 37 are located within $8\,\mathrm{arcmin}$ ($3.4\,\mathrm{physicalMpc}$) from other protocluster candidates of higher overdensity, and are expected to merge into a single massive structure by $z=0$. Therefore, we find 179 unique protocluster candidates in our survey. A cosmological simulation that includes projection effects predicts that more than 76\% of these candidates will evolve into galaxy clusters with halo masses of at least $10^{14}\,M_{\odot}$ by $z=0$. The unprecedented size of our protocluster candidate catalog allowed us to perform, for the first time, an angular clustering analysis of the systematic sample of protocluster candidates. We find a correlation length of $35.0\,h^{-1}\,\mathrm{Mpc}$. The relation between correlation length and number density of $z\sim3.8$ protocluster candidates is consistent with the prediction of the $\Lambda$CDM model, and the correlation length is similar to that of rich clusters in the local universe. This result suggests that our protocluster candidates are tracing similar spatial structures as those expected of the progenitors of rich clusters and enhances the confidence that our method to identify protoclusters at high redshifts is robust. In the coming years, our protocluster search will be extended to the entire HSC-SSP Wide sky coverage of $\sim1400\,\mathrm{deg^2}$ to probe cluster formation over a wide redshift range of $z\sim2\mathrm{-}6$.