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We report the first detailed measurement of the shape of the CO luminosityfunction at high redshift, based on $>$320 hr of the NSF's Karl G. Jansky VeryLarge Array (VLA) observations over an area of $\sim$60 arcmin$^2$ taken aspart of the CO Luminosity Density at High Redshift (COLDz) survey. COLDz"blindly" selects galaxies based on their cold gas content throughCO($J$=1$\to$0) emission at $z$$\sim$2-3 and CO($J$=2$\to$1) at $z$$\sim$5-7down to a CO luminosity limit of log($L'_{\rm CO}$/K km s$^{-1}$pc$^2$)$\simeq$9.5. We find that the characteristic luminosity and bright endof the CO luminosity function are substantially higher than predicted bysemi-analytical models, but consistent with empirical estimates based on theinfrared luminosity function at $z$$\sim$2. We also present the currently mostreliable measurement of the cosmic density of cold gas in galaxies at earlyepochs, i.e., the cold gas history of the universe, as determined over a largecosmic volume of $\sim$375,000 Mpc$^3$. Our measurements are in agreement withan increase of the cold gas density from $z$$\sim$0 to $z$$\sim$2-3, followedby a possible decline towards $z$$\sim$5-7. These findings are consistent withrecent surveys based on higher-$J$ CO line measurements, upon which COLDzimproves in terms of statistical uncertainties by probing $\sim$50-100 timeslarger areas and in the reliability of total gas mass estimates by probing thelow-$J$ CO lines accessible to the VLA. Our results thus appear to suggest thatthe cosmic star-formation rate density follows an increased cold molecular gascontent in galaxies towards its peak about 10 billion years ago, and that itsdecline towards the earliest epochs is likely related to a lower overall amountof cold molecular gas (as traced by CO) bound in galaxies towards the firstbillion years after the Big Bang.

COLDz: Shape of the CO Luminosity Function at High Redshift and the Cold Gas History of the Universe