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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.