Resolving the ISM at the peak of cosmic star formation with ALMA - The distribution of CO and dust continuum in z~2.5 sub-millimetre galaxies

We use ALMA observations of four sub-millimetre galaxies (SMGs) at $z\sim2-3$ to investigate the spatially resolved properties of the inter-stellar medium (ISM) at scales of 1--5 kpc (0.1--0.6$''$). The velocity fields of our sources, traced by the $^{12}$CO($J$=3-2) emission, are consistent with disk rotation to first order, implying average dynamical masses of $\sim$3$\times10^{11}$M$_{\odot}$ within two half-light radii. Through a Bayesian approach we investigate the uncertainties inherent to dynamically constraining total gas masses. We explore the covariance between the stellar mass-to-light ratio and CO-to-H$_{2}$ conversion factor, $\alpha_{\rm CO}$, finding values of $\alpha_{\rm CO}=1.1^{+0.8}_{-0.7}$ for dark matter fractions of 15 \%. We show that the resolved spatial distribution of the gas and dust continuum can be uncorrelated to the stellar emission, challenging energy balance assumptions in global SED fitting. Through a stacking analysis of the resolved radial profiles of the CO(3-2), stellar and dust continuum emission in SMG samples, we find that the cool molecular gas emission in these sources (radii $\sim$5--14 kpc) is clearly more extended than the rest-frame $\sim$250 $\mu$m dust continuum by a factor $>2$. We propose that assuming a constant dust-to-gas ratio, this apparent difference in sizes can be explained by temperature and optical-depth gradients alone. Our results suggest that caution must be exercised when extrapolating morphological properties of dust continuum observations to conclusions about the molecular gas phase of the ISM.