The Molecular Mass function of the Local Universe.

We construct the molecular mass function using the bivariate Kband-Mass Function of the Herschel Reference Survey, a volume-limited sample already widely studied at the entire electromagnetic spectrum. The molecular mass function is derived from the K-band and the gas mass cumulative distribution using a copula method described in detail in our previous papers. The H2 mass is relatively strong correlated with the K-band luminosity because of the tight relation between the stellar mass and the molecular gas mass within the sample with a scatter likely due to those galaxies which have lost their molecular content because of environmental effects or because of a larger gas consumption due to past star formation processes. The derived H2 Mass Function samples the molecular mass range from 4 10^6 to 10^10 solar masses, and when compared with theoretical models, it agrees well with the theoretical predictions at the lower end of the mass values, while at masses larger than 10^10 solar masses the HRS sample may miss galaxies with large content of molecular hydrogen and the outcomes are not conclusive. The value of the local density of the molecular gas mass inferred from our analysis is ~1.5x10^7 Mo Mpc^-3, and it is compared with the results at larger redshifts, confirming the lack of strong evolution of the molecular mass density between z=0 and z=4. This is the first Molecular Mass Function derived on a complete sample in the local Universe, which can be used as a reliable calibration at redshift $z$=0 for models aiming at predicting the evolution of the molecular mass density