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We present an empirical relation between the cold gas surface density($\Sigma_{\rm gas}$) and the optical extinction (${\rm A_V}$) in a sample of103 galaxies from the Extragalactic Database for Galaxy Evolution (EDGE)survey. This survey provides CARMA interferometric CO observations for 126galaxies included in the Calar Alto Legacy Integral Field Area (CALIFA) survey.The matched, spatially resolved nature of these data sets allows us to derivethe $\Sigma_{\rm gas}$-${\rm A_V}$ relation on global, radial, and kpc (spaxel)scales. We determine ${\rm A_V}$ from the Balmer decrement(H$\alpha$/H$\beta$). We find that the best fit for this relation is$\Sigma_{\rm gas} ({\rm M_\odot pc^{-2}})\sim~26~\times~ {\rm A_V}({\rm mag})$,and that it does not depend on the spatial scale used for the fit. However, thescatter in the fits increases as we probe smaller spatial scales, reflectingthe complex relative spatial distributions of stars, gas, and dust. Weinvestigate the $\Sigma_{\rm gas}$/ ${\rm A_V}$ ratio on radial and spaxelscales as a function of ${\rm EW(H\alpha)}$. We find that at larger values of${\rm EW(H\alpha)}$ (i.e., actively star-forming regions) this ratio tend toconverge to the value expected for dust-star mixed geometries ($\sim$ 30$\mathrm{M_{\odot} \,pc^{-2}\,mag^{-1}}$). On radial scales, we do not find asignificant relation between the $\Sigma_{\rm gas}$/${\rm A_V}$ ratio and theionized gas metallicity. We contrast our estimates of $\Sigma_{\rm gas}$ using${\rm A_V}$ with compilations in the literature of the gas fraction on globaland radial scales as well as with well known scaling relations such as theradial star-formation law and the $\Sigma_{\rm gas}$-$\Sigma_*$ relation. Thesetests show that optical extinction is a reliable proxy for estimating$\Sigma_{\rm gas}$ in the absence of direct sub/millimeter observations of thecold gas.

The EDGE-CALIFA Survey: Using Optical Extinction to Probe the Spatially-Resolved Distribution of Gas in Nearby Galaxies