ALMA measurements for 93 Herschel -selected galaxies at 1.1 z 1.7 in COSMOS reveal a sizable (> 29%) population with compact star formation (SF) sizes, lying on average > ×3.6 below the optical stellar mass (M )-size relation of disks. This sample widely spans the star-forming Main Sequence (MS), having 10 8 M 10 11.5 M and 20 SF R 680 M 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 λ obs ∼ 1.1−1.3 mm continuum, all tracing the star-forming molecular gas. These compact galaxies have instead normally extended K band sizes, suggesting strong specific SF R gradients. Compact galaxies comprise the 50 ± 18% of MS galaxies at M > 10 11 M . 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 ∼ 50% at the highest M . The presence of large galaxies above the MS demonstrates an overall poor correlation between galaxy SF size and specific SF R.

To better constrain the physical mechanisms driving star formation, we present the first systematic study of the radio continuum size evolution of star-forming galaxies (SFGs) over the redshift range 0.35 < z < 2.25. We use the VLA COSMOS 3GHz map (noise rms = 2.3 µJy beam −1 , θ beam = 0.75 arcsec) to construct a mass-complete sample of 3184 radio-selected SFGs that reside on and above the main-sequence (MS) of SFGs. We constrain the overall extent of star formation activity in galaxies by applying a 2D-Gaussian model to their radio continuum emission. Extensive Monte Carlo simulations are used to validate the robustness of our measurements and characterize the selection function. We find no clear dependence between the radio size and stellar mass, M , of SFGs with 10.5 log(M /M ) 11.5. Our analysis suggests that MS galaxies are preferentially extended, while SFGs above the MS are always compact. The median effective radius of SFGs on (above) the MS of R eff = 1.5 ± 0.2 (1.0 ± 0.2) kpc remains nearly constant with cosmic time; a parametrization of the form R eff ∝ (1 + z) α yields a shallow slope of only α = −0.26 ± 0.08 (0.12 ± 0.14) for SFGs on (above) the MS. The size of the stellar component of galaxies is larger than the extent of the radio continuum emission by a factor ∼2 (1.3) at z = 0.5 (2), indicating star formation is enhanced at small radii. The galactic-averaged star formation rate surface density (Σ SFR ) scales with the distance to the MS, except for a fraction of MS galaxies ( 10%) that harbor starburst-like Σ SFR . These "hidden" starbursts might have experienced a compaction phase due to disk instability and/or merger-driven burst of star formation, which may or may not significantly offset a galaxy from the MS. We thus propose to jointly use Σ SFR and distance to the MS to better identify the galaxy population undergoing a starbursting phase.