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