A$^{3}$COSMOS: Dissecting the gas content of star-forming galaxies across the main sequence at 1.2 $\leq z$ < 1.6

We aim to understand the physical mechanisms that drive star formation in a sample of mass-complete (>10$^{9.5}M_{\odot}$) star-forming galaxies (SFGs) at 1.2 $\leq z$ < 1.6. We selected SFGs from the COSMOS2020 catalog and applied a $uv$-domain stacking analysis to their archival Atacama Large Millimeter/submillimeter Array (ALMA) data. Our stacking analysis provides precise measurements of the mean molecular gas mass and size of SFGs. We also applied an image-domain stacking analysis on their \textit{HST} $i$-band and UltraVISTA $J$- and $K_{\rm s}$-band images. Correcting these rest-frame optical sizes using the $R_{\rm half-stellar-light}$-to-$R_{\rm half-stellar-mass}$ conversion at rest 5,000 angstrom, we obtain the stellar mass size of MS galaxies. Across the MS (-0.2 < $\Delta$MS < 0.2), the mean molecular gas fraction of SFGs increases by a factor of $\sim$1.4, while their mean molecular gas depletion time decreases by a factor of $\sim$1.8. The scatter of the MS could thus be caused by variations in both the star formation efficiency and molecular gas fraction of SFGs. The majority of the SFGs lying on the MS have $R_{\rm FIR}$ $\approx$ $R_{\rm stellar}$. Their central regions are subject to large dust attenuation. Starbursts (SBs, $\Delta$MS>0.7) have a mean molecular gas fraction $\sim$2.1 times larger and mean molecular gas depletion time $\sim$3.3 times shorter than MS galaxies. Additionally, they have more compact star-forming regions ($\sim$2.5~kpc for MS galaxies vs. $\sim$1.4~kpc for SBs) and systematically disturbed rest-frame optical morphologies, which is consistent with their association with major-mergers. SBs and MS galaxies follow the same relation between their molecular gas mass and star formation rate surface densities with a slope of $\sim1.1-1.2$, that is, the so-called KS relation.