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We use the results from the ALMA large program ASPECS, the spectroscopicsurvey in the Hubble Ultra Deep Field (HUDF), to constrain CO luminosityfunctions of galaxies and the resulting redshift evolution of $\rho$(H$_2$).The broad frequency range covered enables us to identify CO emission lines ofdifferent rotational transitions in the HUDF at $z>1$. We find strong evidencethat the CO luminosity function evolves with redshift, with the knee of the COluminosity function decreasing in luminosity by an order of magnitude from$\sim$2 to the local universe. Based on Schechter fits, we estimate that ourobservations recover the majority (up to $\sim$90%, depending on theassumptions on the faint end) of the total cosmic CO luminosity at $z$=1.0-3.1.After correcting for CO excitation, and adopting a Galactic CO-to-H$_2$conversion factor, we constrain the evolution of the cosmic molecular gasdensity $\rho$(H$_2$): this cosmic gas density peaks at $z\sim1.5$ and drops byfactor of $6.5_{-1.4}^{+1.8}$ to the value measured locally. The observedevolution in $\rho$(H$_2$) therefore closely matches the evolution of thecosmic star formation rate density $\rho_{\rm SFR}$. We verify the robustnessof our result with respect to assumptions on source inclusion and/or COexcitation. As the cosmic star formation history can be expressed as theproduct of the star formation efficiency and the cosmic density of moleculargas, the similar evolution of $\rho$(H$_2$) and $\rho_{\rm SFR}$ leaves onlylittle room for a significant evolution of the average star formationefficiency in galaxies since $z\sim 3$ (85% of cosmic history).

The ALMA Spectroscopic Survey in the HUDF: CO luminosity functions and the molecular gas content of galaxies through cosmic history