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 $SFR$.
