The rest-frame optical sizes of massive galaxies with suppressed star formation at $z\sim 4$

We present the rest-frame optical sizes of massive quiescent galaxies (QGs) at $z\sim 4$ measured at $K^{\mathrm{\prime }}$-band with the Infrared Camera and Spectrograph (IRCS) and AO188 on the Subaru telescope. Based on a deep multi-wavelength catalog in the Subaru XMM-Newton Deep Survey Field (SXDS), covering a wide wavelength range from the $u$-band to the IRAC $8.0\mu m$ over 0.7 deg${}^2$, we evaluate photometric redshift to identify massive ($M_{\star}\sim10^{11}
M_\odot$) galaxies with suppressed star formation. These galaxies show a prominent 4000$\mathring{\mathrm{A}}$ break feature at $z\sim 4$, suggestive of an evolved stellar population. We then conduct follow-up $K^{\mathrm{\prime }}$-band imaging with adaptive optics for the five brightest galaxies ($K_{AB,total}=22.5\sim 23.4$). Compared to lower redshift ones, QGs at $z\sim 4$ have smaller physical sizes of effective radii $r_{eff}=0.2$ to $1.8$ kpc. The mean size measured by stacking the four brightest objects is $r_{eff}=0.7\mathrm{k}\mathrm{p}\mathrm{c}$. This is the first measurement of the rest-frame optical sizes of QGs at $z\sim 4$. We evaluate the robustness of our size measurements using simulations and find that our size estimates are reasonably accurate with an expected systematic bias of $\sim 0.2$ kpc. If we account for the stellar mass evolution, massive QGs at $z\sim 4$ are likely to evolve into the most massive galaxies today. We find their size evolution with cosmic time in a form of $\log(r_e/{\rm kpc})= -0.44+1.77 \log(t/\rm Gyr)$. Their size growth is proportional to the square of stellar mass, indicating the size-stellar mass growth driven by minor dry mergers.