Constraints on the $z\sim5$ Star-Forming Galaxy Luminosity Function From $\textit{Hubble Space Telescope}$ Imaging of an Unbiased and Complete Sample of Long Gamma-ray Burst Host Galaxies

We present rest-frame UV \textit{Hubble Space Telescope} imaging of the largest and most complete sample of 23 long duration gamma-ray burst (GRB) host galaxies between redshifts 4 and 6. Of these 23, we present new WFC3/F110W imaging for 19 of the hosts, which we combine with archival WFC3/F110W and WFC3/F140W imaging for the remaining four. We use the photometry of the host galaxies from this sample to characterize both the rest-frame UV luminosity function (LF) and the size-luminosity relation of the sample. We find that when assuming the standard Schechter-function parameterization for the UV LF, the GRB host sample is best fit with $\alpha = -1.30^{+0.30}_{-0.25}$ and $M_* = -20.33^{+0.44}_{-0.54}$ mag, which is consistent with results based on $z\sim5$ Lyman-break galaxies. We find that $\sim68\%$ of our size-luminosity measurements fall within or below the same relation for Lyman-break galaxies at $z\sim4$. This study observationally confirms expectations that at $z\sim5$ Lyman-break and GRB host galaxies should trace the same population and demonstrates the utility of GRBs as probes of hidden star-formation in the high-redshift universe. Under the assumption that GRBs unbiasedly trace star formation at this redshift, our non-detection fraction of 7/23 is consistent at the $95\%$-confidence level with $13 - 53\%$ of star formation at redshift $z\sim5$ occurring in galaxies fainter than our detection limit of $M_{1600 A} \sim -18.3$ mag.