GRB host galaxies with strong H$_2$ absorption: CO-dark molecular gas at
  the peak of cosmic star formation
  We present a pilot search of CO emission in three H$_2$-absorbing,
long-duration gamma-ray burst (GRB) host galaxies at z~2-3. We used the Atacama
Large Millimeter/sub-millimeter Array (ALMA) to target the CO(3-2) emission
line and report non-detections for all three hosts. These are used to place
limits on the host molecular gas masses, assuming a metallicity-dependent
CO-to-H$_2$ conversion factor ($\alpha_{\rm CO}$). We find, $M_{\rm mol} <
3.5\times 10^{10}\,M_{\odot}$ (GRB\,080607), $M_{\rm mol} < 4.7\times
10^{11}\,M_{\odot}$ (GRB\,120815A), and $M_{\rm mol} < 8.9\times
10^{11}\,M_{\odot}$ (GRB\,181020A). The high limits on the molecular gas mass
for the latter two cases are a consequence of their low stellar masses
$M_\star$ ($M_\star \lesssim 10^{8}\,M_{\odot}$) and low gas-phase
metallicities ($Z\sim 0.03\,Z_{\odot}$). The limit on the $M_{\rm mol}/M_\star$
ratio derived for GRB\,080607, however, is consistent with the average
population of star-forming galaxies at similar redshifts and stellar masses. We
discuss the broader implications for a metallicity-dependent CO-to-H$_2$
conversion factor, and demonstrate that the canonical Galactic $\alpha_{\rm
CO}$, will severely underestimate the actual molecular gas mass for all
galaxies at $z>1$ with $M_\star < 10^{10}\,M_\odot$. To better quantify this we
develop a simple approach to estimate the relevant $\alpha_{\rm CO}$ factor
based only on the redshift and stellar mass of individual galaxies. The
elevated conversion factors will make these galaxies appear CO-"dark" and
difficult to detect in emission, as is the case for the majority of GRB hosts.
GRB spectroscopy thus offers a complementary approach to identify
low-metallicity, star-forming galaxies with abundant molecular gas reservoirs
at high redshifts that are otherwise missed by current ALMA surveys.
