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Search for H$α$ Emitters at $z\sim7.8$: A Constraint on the H$α$-based Star Formation Rate Density

We search for H$α$ emitters at $z\sim7.8$ in four gravitationally lensed fields observed in the Hubble Frontier Fields program. We use the Lyman break method to select galaxies at the target redshift, and make the photometry in {\it Spitzer}/IRAC 5.8 $μ$m band to detect the H$α$ emission from the candidate galaxies. We find no significant detection of counterparts in the IRAC 5.8 $μ$m band, and this gives a constraint on the H$α$ luminosity function (LF) at $z\sim7.8$. We compare the constraint with previous studies on rest-frame UV and FIR observation using the correlation between the H$α$ luminosity and the star formation rate. Additionally, we convert the constraint on the H$α$ LF into an upper limit for the star formation rate density (SFRD) at this epoch assuming the shape of the LF. We examine two types of parameterization of the LF, and obtain an upper limit for the SFRD of $\log_{10}(ρ_{\rm SFR}\ [M_\odot\ \mathrm{yr^{-1}\ Mpc^{-3}}])\lesssim-1.1$ at $z\sim7.8$. With this constraint on the SFRD, we give an independent probe into the total star formation activity including the dust-obscured and unobscured star formation at the Epoch of Reionization.