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Breathless BEARS: [O$_{\rm \,III}$] 88$μ$m Emission of Dusty Star-Forming Galaxies at $z = 3-4$

We present [O$_{\rm \,III}$] 88$μ$m observations towards four ${\it Herschel}$-selected dusty star-forming galaxies (DSFGs; log$_{10}$ $μ$L$_{\rm IR}$/L$_{\odot}$ = 13.5 - 14 at $z = 2.9 - 4$) using the Atacama Compact Array (ACA) in Bands 9 and 10. We detect [O$_{\rm \,III}$] emission in all four targets at >3$σ$, finding line luminosity ratios ($L_{\rm [O_{\rm \,III}]}$ / L$_{\rm IR}$ = 10$^{-4.2}$ to 10$^{-3}$) similar to local spiral galaxies, and an order of magnitude lower when compared with local dwarf galaxies as well as high-redshift Lyman-break galaxies. Using the short-wavelength capabilities of the ACA, these observations bridge the populations of galaxies with [O$_{\rm \,III}$] emission at low redshift from space missions and at high redshift from ground-based studies. The difference in [O$_{\rm \,III}$] emission between these DSFGs and other high-redshift galaxies reflects their more evolved stellar populations (> 10 Myr), larger dust reservoirs (M$_{\rm dust}$ $\sim$ 10$^{9 - 11}$ M$_{\odot}$), metal-rich interstellar medium ($Z \sim 0.5 - 2$ Z$_{\odot}$), and likely weaker ionization radiation fields. Ancillary [C$_{\rm \,II}$] emission on two targets provide $L_{[{\rm O}_{\rm \,III}]} / L_{[{\rm C}_{\rm \,II}]}$ ratios at 0.3 - 0.9, suggesting that ionized gas represents a smaller fraction of the total gas reservoir in DSFGs, consistent with theoretical models of DSFGs as transitional systems between gas-rich, turbulent disks and more evolved, gas-poor galaxies. Expanding samples of DSFGs with [O$_{\rm \,III}$] emission will be key to place this heterogeneous, poorly-understood galactic phase in its astrophysical context.