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In this paper we use ASPECS, the ALMA Spectroscopic Survey in the {\emHubble} Ultra Deep Field (UDF) in band 3 and band 6, to place blind constraintson the CO luminosity function and the evolution of the cosmic molecular gasdensity as a function of redshift up to $z\sim 4.5$. This study is based ongalaxies that have been solely selected through their CO emission and notthrough any other property. In all of the redshift bins the ASPECS measurementsreach the predicted `knee' of the CO luminosity function (around$5\times10^{9}$ K km/s pc$^2$). We find clear evidence of an evolution in theCO luminosity function with respect to $z\sim 0$, with more CO luminousgalaxies present at $z\sim 2$. The observed galaxies at $z\sim 2$ also appearmore gas-rich than predicted by recent semi-analytical models. The comovingcosmic molecular gas density within galaxies as a function of redshift shows afactor 3-10 drop from $z \sim 2$ to $z \sim 0$ (with significant error bars),and possibly a decline at $z>3$. This trend is similar to the observedevolution of the cosmic star formation rate density. The latter thereforeappears to be at least partly driven by the increased availability of moleculargas reservoirs at the peak of cosmic star formation ($z\sim2$).

ALMA spectroscopic survey in the Hubble Ultra Deep Field: CO luminosity functions and the evolution of the cosmic density of molecular gas