One of the last missing pieces in the puzzle of galaxy formation and evolution through cosmic history is a detailed picture of the role of the cold gas supply in the star-formation process. Cold gas is the fuel for star formation, and thus regulates the buildup of stellar mass, both through the amount of material present through a galaxys gas mass fraction, and through the efficiency at which it is converted to stars. Over the last decade, important progress has been made in understanding the relative importance of these two factors along with the role of feedback, and the first measurements of the volume density of cold gas out to redshift 4, (the cold gas history of the Universe) has been obtained. To match the precision of measurements of the star formation and black-hole accretion histories over the coming decades, a two orders of magnitude improvement in molecular line survey speeds is required compared to what is possible with current facilities. Possible pathways towards such large gains include significant upgrades to current facilities like ALMA by 2030 (and beyond), and eventually the construction of a new generation of radio-to-millimeter wavelength facilities, such as the next generation Very Large Array (ngVLA) concept.
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