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We combine molecular gas masses inferred from CO emission in 500 star forminggalaxies (SFGs) between z=0 and 3, from the IRAM-COLDGASS, PHIBSS1/2 and othersurveys, with gas masses derived from Herschel far-IR dust measurements in 512galaxy stacks over the same stellar mass/redshift range. We constrain thescaling relations of molecular gas depletion time scale (tdepl) and gas tostellar mass ratio (Mmolgas/M*) of SFGs near the star formation main-sequencewith redshift, specific star formation rate (sSFR) and stellar mass (M*). TheCO- and dust-based scaling relations agree remarkably well. This suggests thatthe CO-H2 mass conversion factor varies little within 0.6dex of the mainsequence (sSFR(ms,z,M*)), and less than 0.3dex throughout this redshift range.This study builds on and strengthens the results of earlier work. We find thattdepl scales as (1+z)^-0.3 *(sSFR/sSFR(ms,z,M*))^-0.5, with little dependenceon M*. The resulting steep redshift dependence of Mmolgas/M* ~(1+z)^3 mirrorsthat of the sSFR and probably reflects the gas supply rate. The decreasing gasfractions at high M* are driven by the flattening of the SFR-M* relation.Throughout the redshift range probed a larger sSFR at constant M* is due to acombination of an increasing gas fraction and a decreasing depletion timescale. As a result galaxy integrated samples of the Mmolgas-SFR rate relationexhibit a super-linear slope, which increases with the range of sSFR. Withthese new relations it is now possible to determine Mmolgas with an accuracy of0.1dex in relative terms, and 0.2dex including systematic uncertainties.

Combined CO & Dust Scaling Relations of Depletion Time and Molecular Gas Fractions with Cosmic Time, Specific Star Formation Rate and Stellar Mass