PHIBSS: Exploring the Dependence of the CO-H$_2$ Conversion Factor on Total Mass Surface Density at ${\it z} < 1.5$

We present an analysis of the relationship between the CO-H$_{2}$ conversion factor ($\alpha_{\rm CO}$) and total mass surface density ($\Sigma_{\rm tot}$) in star-forming galaxies at $z < 1.5$. Our sample, which is drawn from the IRAM Plateau de Bure HIgh-$z$ Blue Sequence Survey (PHIBSS) and the CO Legacy Database for GASS (COLD GASS), includes 'normal,' massive star-forming galaxies that dominate the evolution of the cosmic star formation rate (SFR) at this epoch and probe the $\Sigma_{\rm tot}$ regime where the strongest variation in $\alpha_{\rm CO}$ is observed. We constrain $\alpha_{\rm CO}$ via existing CO observations, measurements of the star formation rate, and an assumed molecular gas depletion time ($t_{\rm dep}$=$M_{\rm gas}$/SFR) --- the latter two of which establish the total molecular gas mass independent of the observed CO luminosity. For a broad range of adopted depletion times, we find that $\alpha_{\rm CO}$ is independent of total mass surface density, with little deviation from the canonical Milky Way value. This runs contrary to a scenario in which $\alpha_{\rm CO}$ decreases as surface density increases within the extended clouds of molecular gas that potentially fuel clumps of star formation in $z\sim1$ galaxies, similar to those observed in local ULIRGs. Instead, our results suggest that molecular gas, both at $z\sim0$ and $z\sim1$, is primarily in the form of self-gravitating molecular clouds. While CO observations suggest a factor of $\sim3$ reduction in the average molecular gas depletion time between $z \sim 0$ and $z\sim1$, we find that, for typical galaxies, the structure of molecular gas and the process of star formation at $z \sim 1$ is otherwise remarkably similar to that observed in local star-forming systems.