The First Robust Constraints on the Relationship Between Dust-to-Gas Ratio and Metallicity in Luminous Star-forming Galaxies at High Redshift.

We present rest-optical spectroscopic properties of a sample of four galaxies in the Atacama Large Millimeter/submillimeter Array Hubble Ultra Deep Field (ALMA HUDF). These galaxies span the redshift range $1.41 \leq z \leq 2.54$ and the stellar mass range $10.36\leq\log(M_*/{\rm M}_{\odot})\leq10.91$. They have existing far-infrared and radio measurements of dust-continuum and molecular gas emission from which bolometric star-formation rates (SFRs), dust masses, and molecular gas masses have been estimated. We use new $H$- and $K$-band near-infrared spectra from the Keck/MOSFIRE spectrograph to estimate SFRs from dust-corrected H$\alpha$ emission (SFR(H$\alpha$)) and gas-phase oxygen abundances from the ratio [NII]$\lambda 6584$/H$\alpha$. We find that the dust-corrected SFR(H$\alpha$) is systematically lower than the bolometric SFR by a factor of several, and measure gas-phase oxygen abundances in a narrow range, $12+\log(\mbox{O/H})=8.59-8.69$ ($0.8-1.0\: (\mbox{O/H})_{\odot}$). Relative to a large $z\sim 2$ comparison sample from the MOSDEF survey, the ALMA HUDF galaxies scatter roughly symmetrically around the best-fit linear mass-metallicity relation, providing tentative evidence for a flattening in the SFR dependence of metallicity at high stellar mass. Combining oxygen abundances with estimates of dust and molecular gas masses, we show that there is no significant evolution in the normalization of the dust-to-gas ratio DGR vs. metallicity relation from $z\sim0$ to $z\sim2$. This result is consistent with some semi-analytic models and cosmological simulations describing the evolution of dust in galaxies. Tracing the actual form of the DGR vs. metallicity relation at high redshift now requires combined measurements of dust, gas, and metallicity over a significantly wider range in metallicity.