Cosmic evolution of molecular gas mass density from an empirical
  relation between $\rm L_{1.4GHz}$ and $\rm L^{\prime}_{CO}$
  Historically, GHz radio emission has been used extensively to characterize
the star-formation activity in galaxies. In this work, we look for empirical
relations amongst the radio luminosity, the infrared luminosity, and the
CO-based molecular gas mass. We assemble a sample of 278 nearby galaxies with
measurements of radio continuum and total infrared emission, and the $^{12}$CO
(J = 1-0) emission line. We find a correlation between the radio continuum and
the CO emission line (with a scatter of 0.36 dex), in a large sample of
different kind of galaxies. Making use of this correlation, we explore the
evolution of the molecular gas mass function and the cosmological molecular gas
mass density in six redshift bins up to $z = 1.5$. These results agree with
previous semi-analytic predictions and direct measurements: the cosmic
molecular gas density increases up to $z=1.5$. In addition, we find a single
plane across five orders of magnitude for the explored luminosities, with a
scatter of 0.27 dex. These correlations are sufficiently robust to be used for
samples where no CO measurements exist.
