Radio continuum size evolution of star-forming galaxies over 0.35 < z <
  2.25
  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 $\rm rms=2.3\,\mu Jy \,beam^{-1}$, $\theta_{\rm
beam}=0.75\,\rm arcsec$) to construct a mass-complete sample of 3184
radio-selected SFGs that reside on and above the main-sequence (MS) of SFGs. We
find no clear dependence between the radio size and stellar mass, $M_{\star}$,
of SFGs with $10.5\lesssim\log(M_\star/\rm M_\odot)\lesssim11.5$. 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
$R_{\rm eff}=1.5\pm0.2$ ($1.0\pm0.2$) kpc remains nearly constant with cosmic
time; a parametrization of the form $R_{\rm eff}\propto(1+z)^\alpha$ yields a
shallow slope of only $\alpha=-0.26\pm0.08\,(0.12\pm0.14)$ 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 $\sim$2 (1.3) at $z=0.5\,(2)$,
indicating star formation is enhanced at small radii. The galactic-averaged
star formation rate surface density $(\Sigma_{\rm SFR})$ scales with the
distance to the MS, except for a fraction of MS galaxies ($\lesssim10\%$) that
harbor starburst-like $\Sigma_{\rm SFR}$. 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 $\Sigma_{\rm SFR}$ and distance to
the MS to better identify the galaxy population undergoing a starbursting
phase.
