The Main Sequence at $z \sim 1.3$ contains a sizable fraction of galaxies with compact star formation sizes: a new population of early post-starbursts?

ALMA measurements for 93 $Herschel$-selected galaxies at $1.1 \leqslant z \leqslant 1.7$ in COSMOS reveal a sizable ($>29$\%) population with compact star formation (SF) sizes, lying on average $> \times 3.6$ below the optical stellar mass ($M_{\star}$)-size relation of disks. This sample widely spans the star-forming Main Sequence (MS), having $10^{8} \leqslant M_{\star} \leqslant 10^{11.5} \ M_{\odot}$ and $20 \leqslant SFR \leqslant 680 \ M_{\odot} \rm yr^{-1}$. The 32 size measurements and 61 upper limits are measured on ALMA images that combine observations of CO(5-4), CO(4-3), CO(2-1) and $\lambda_{\rm obs} \sim 1.1-1.3 \ \rm mm$ continuum, all tracing the star-forming molecular gas. These compact galaxies have instead normally extended $K_{band}$ sizes, suggesting strong specific $SFR$ gradients. Compact galaxies comprise the $50\pm18 \%$ of MS galaxies at $M_{\star} > 10^{11} M_{\odot}$. This is not expected in standard bi-modal scenarios where MS galaxies are mostly steadily-growing extended disks. We suggest that compact MS objects are early post-starburst galaxies in which the merger-driven boost of SF has subsided. They retain their compact SF size until either further gas accretion restores pre-merger galaxy-wide SF, or until becoming quenched. The fraction of merger-affected SF inside the MS seems thus larger than anticipated and might reach $\sim 50$\% at the highest $M_{\star}$. The presence of large galaxies above the MS demonstrates an overall poor correlation between galaxy SF size and specific $SFR$.