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.
