We report the first detailed measurement of the shape of the CO luminosity function at high redshift, based on $>$320 hr of the NSFs Karl G. Jansky Very Large Array (VLA) observations over an area of $sim$60 arcmin$^2$ taken as part of the CO Luminosity Density at High Redshift (COLDz) survey. COLDz blindly selects galaxies based on their cold gas content through CO($J$=1$to$0) emission at $z$$sim$2-3 and CO($J$=2$to$1) at $z$$sim$5-7 down to a CO luminosity limit of log($L_{rm CO}$/K km s$^{-1}$ pc$^2$)$simeq$9.5. We find that the characteristic luminosity and bright end of the CO luminosity function are substantially higher than predicted by semi-analytical models, but consistent with empirical estimates based on the infrared luminosity function at $z$$sim$2. We also present the currently most reliable measurement of the cosmic density of cold gas in galaxies at early epochs, i.e., the cold gas history of the universe, as determined over a large cosmic volume of $sim$375,000 Mpc$^3$. Our measurements are in agreement with an increase of the cold gas density from $z$$sim$0 to $z$$sim$2-3, followed by a possible decline towards $z$$sim$5-7. These findings are consistent with recent surveys based on higher-$J$ CO line measurements, upon which COLDz improves in terms of statistical uncertainties by probing $sim$50-100 times larger areas and in the reliability of total gas mass estimates by probing the low-$J$ CO lines accessible to the VLA. Our results thus appear to suggest that the cosmic star-formation rate density follows an increased cold molecular gas content in galaxies towards its peak about 10 billion years ago, and that its decline towards the earliest epochs is likely related to a lower overall amount of cold molecular gas (as traced by CO) bound in galaxies towards the first billion years after the Big Bang.
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