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Stars form in cold molecular clouds. However, molecular gas is difficult toobserve because the most abundant molecule (H2) lacks a permanent dipolemoment. Rotational transitions of CO are often used as a tracer of H2, but COis much less abundant and the conversion from CO intensity to H2 mass is oftenhighly uncertain. Here we present a new method for estimating the columndensity of cold molecular gas (Sigma_gas) using optical spectroscopy. Weutilise the spatially resolved H-alpha maps of flux and velocity dispersionfrom the Sydney-AAO Multi-object Integral-field spectrograph (SAMI) GalaxySurvey. We derive maps of Sigma_gas by inverting the multi-freefall starformation relation, which connects the star formation rate surface density(Sigma_SFR) with Sigma_gas and the turbulent Mach number (Mach). Based on themeasured range of Sigma_SFR = 0.005-1.5 M_sol/yr/kpc^2 and Mach = 18-130, wepredict Sigma_gas = 7-200 M_sol/pc^2 in the star-forming regions of our sampleof 260 SAMI galaxies. These values are close to previously measured Sigma_gasobtained directly with unresolved CO observations of similar galaxies at lowredshift. We classify each galaxy in our sample as 'Star-forming' (219) or'Composite/AGN/Shock' (41), and find that in Composite/AGN/Shock galaxies theaverage Sigma_SFR, Mach, and Sigma_gas are enhanced by factors of 2.0, 1.6, and1.3, respectively, compared to Star-forming galaxies. We compare ourpredictions of Sigma_gas with those obtained by inverting the Kennicutt-Schmidtrelation and find that our new method is a factor of two more accurate inpredicting Sigma_gas, with an average deviation of 32% from the actualSigma_gas.

The SAMI Galaxy Survey: a new method to estimate molecular gas surface densities from star formation rates