Star Formation Rate Function at $z\sim4.5$: An Analysis from rest UV to Optical

We present a star formation rate function (SFRF) at $z\sim4.5$ based on photometric data from rest UV to optical of galaxies in the CANDELS GOODS-South field using spectral energy distribution (SED) fitting. We evaluate the incompleteness of our sample and correct for it to properly confront the SFRF in this study with those estimated based on other probes. The SFRF is obtained down to $\sim10\ M_\odot\ \mathrm{yr}^{-1}$ and it shows a significant excess to that estimated from UV luminosity function and dust correction based on UV spectral slope. As compared with the UV-based SFRF, the number density is larger by $\sim1$ dex at a fixed SFR, or the best-fit Schechter parameter of $\mathrm{SFR}^*$ is larger by $\sim1$ dex. We extensively examine several assumptions on SED fitting to see the robustness of our result, and find that the excess still exist even if the assumptions change such as star formation histories, dust extinction laws, and one- or two-component model. By integrating our SFRF to $0.22\ M_\odot\ \mathrm{yr}^{-1}$, the cosmic star formation rate density at this epoch is calculated to be $4.53^{+0.94}_{-0.87}\times10^{-2}\ M_\odot\ \mathrm{yr}^{-1}\ \mathrm{Mpc}^{-3}$, which is $\sim0.25$ dex larger than the previous measurement based on UV observations. We also find that galaxies with intensive star formation ($>10\ M_\odot\ \mathrm{yr}^{-1}$) occupies most of the cosmic star formation rate density ($\sim80\%$), suggesting that star formation activity at this epoch is dominant by intensively star-forming galaxies.