Title: The relationship of SMBHs and host galaxies at $z<4$ in the deep optical variability-selected AGN sample in the COSMOS field

Abstract: We present the study on the relationship between SMBHs and their host galaxies using our variability-selected AGN sample ($i_\mathrm{AB} \leq 25.9,\ z \leq 4.5$) constructed from the HSC-SSP Ultra-deep survey in the COSMOS field. We estimated the BH mass ($M_\mathrm{BH}=10^{5.5-10}\ M_{\odot}$) based on the single-epoch virial method and the total stellar mass ($M_\mathrm{star}=10^{10-12}\ M_{\odot}$) by separating the AGN component with SED fitting. We found that the redshift evolution of the BH-stellar mass ratio ($M_\mathrm{BH}/M_\mathrm{star}$) depends on the $M_\mathrm{BH}$ which is caused by the no significant correlation between $M_\mathrm{BH}$ and $M_\mathrm{star}$. Variable AGNs with massive SMBHs ($M_\mathrm{BH}>10^{9}\ M_{\odot}$) at $1.5<z<3$ show considerably higher BH-stellar mass ratios ($>\sim1\%$) than the BH-bulge ratios ($M_\mathrm{BH}/M_\mathrm{bulge}$) observed in the local universe for the same BH range. This implies that there is a typical growth path of massive SMBHs which is faster than the formation of the bulge component as final products seen in the present day. For the low-mass SMBHs ($M_\mathrm{BH}<10^{8}\ M_{\odot}$) at $0.5<z<3$, on the other hand, variable AGNs show the similar BH-stellar mass ratios with the local objects ($\sim 0.1\%$) but smaller than those observed at $z > 4$. We interpret that host galaxies harboring less massive SMBHs at intermediate redshift have already acquired sufficient stellar mass, although high-z galaxies are still in the early stage of galaxy formation relative to those at the intermediate/local universe.