  The study of molecular gas is crucial for understanding star formation,
feedback, and the broader ecosystem of a galaxy as a whole. However, we have
limited understanding of its physics and distribution in all but the nearest
galaxies. We present a new technique for studying the composition and
distribution of molecular gas in high-redshift galaxies inaccessible to
existing methods. Our proposed approach is an extension of carbon monoxide
intensity mapping methods, which have garnered significant experimental
interest in recent years. These intensity mapping surveys target the 115 GHz
$^{12}$CO (1-0) line, but also contain emission from the substantially fainter
110 GHz $^{13}$CO (1-0) transition. The method leverages the information
contained in the $^{13}$CO line by cross-correlating pairs of frequency
channels in an intensity mapping survey. Since $^{13}$CO is emitted from the
same medium as the $^{12}$CO, but saturates at a much higher column density,
this cross-correlation provides valuable information about both the gas density
distribution and isotopologue ratio, inaccessible from the $^{12}$CO alone.
Using a simple model of these molecular emission lines, we show that a future
intensity mapping survey can constrain the abundance ratio of these two species
and the fraction of emission from optically thick regions to order $\sim30\%$.
These measurements cannot be made by traditional CO observations, and
consequently the proposed method will provide unique insight into the physics
of star formation, feedback, and galactic ecology at high redshifts.
