A$^{3}$COSMOS: A census on the molecular gas mass and extent of main-sequence galaxies across cosmic time

To constrain for the first time the mean mass and extent of the molecular gas of a mass-complete sample of $>10^{10}$M$_{\odot}$ main-sequence (MS) galaxies at $0.4<z<3.6$. We apply an innovative $uv$-based stacking analysis to a large set of archival Atacama Large Millimeter/submillimeter Array (ALMA) observations. This stacking analysis provides measurements of the mean mass and extent of the molecular gas of galaxy populations. The molecular gas mass of MS galaxies evolves with redshift and stellar mass. At all stellar masses, the molecular gas fraction decreases by a factor of 24 from $z\sim3.2$ to $z\sim0$. At a given redshift, the molecular gas fraction of MS galaxies decreases with stellar mass, at roughly the same rate as their specific star formation rate decreases. The molecular gas depletion time of MS galaxies remains roughly constant at $z>0.5$ with a value of 300--500 Myr, but increases by a factor of 3 from $z\sim0.5$ to $z\sim0$. This evolution of the molecular gas depletion time of MS galaxies can be predicted from the evolution of their molecular gas surface density and a seemingly universal MS-only $\Sigma_{M_{\rm mol}}-\Sigma_{\rm SFR}$ relation with an inferred slope of 1.13, i.e., the so-called KS relation. The far-infrared size of MS galaxies shows no significant evolution with redshift or stellar mass, with a mean circularized half-light radius of 2.2 kpc. Finally, our mean molecular gas masses are lower than previous estimates, likely caused by the fact that literature studies were biased towards individually-detected MS galaxies with massive gas reservoirs. To first order, the molecular gas content of MS galaxies regulates their star formation across cosmic time, while variation of their star formation efficiency plays a secondary role. Despite a large evolution of their gas content and SFRs, MS galaxies evolved along a seemingly universal MS-only KS relation.