  Radio emission at cm wavelengths from highly star-forming galaxies, such as
SMGs, is dominated by synchrotron radiation arising from supernova activity.
Using deep, high-resolution ($1\sigma=2.3$ $\mu$Jy beam$^{-1}$; $0.75^{"}$) cm
radio-continuum observations taken by the VLA-COSMOS 3 GHz Large Project, we
studied the radio-emitting sizes of a flux-limited sample of SMGs in the COSMOS
field. Of the 39 SMGs studied here, 3 GHz emission was detected towards 18 of
them ($\sim46\pm11\%$) with S/N ratios in the range of ${\rm S/N=4.2-37.4}$.
Using 2D elliptical Gaussian fits, we derived a median deconvolved major axis
FWHM size of $0.54^{"}\pm 0.11^{"}$ for our 18 SMGs detected at 3 GHz. For the
15 SMGs with known redshift we derived a median linear major axis FWHM of
$4.2\pm0.9$ kpc. No clear correlation was found between the radio-emitting size
and the 3 GHz or submm flux density, or the redshift of the SMG. However, there
is a hint of larger radio sizes at $z\sim2.5-5$ compared to lower redshifts.
The sizes we derived are consistent with previous SMG sizes measured at 1.4 GHz
and in mid-$J$ CO emission, but significantly larger than those seen in the
(sub)mm continuum emission. One possible scenario is that SMGs have i) an
extended gas component with a low dust temperature, and which can be traced by
low- to mid-$J$ CO line emission and radio continuum emission, and ii) a
warmer, compact starburst region giving rise to the high-$J$ line emission of
CO, which could dominate the dust continuum size measurements. Because of the
rapid cooling of CR electrons in dense starburst galaxies ($\sim10^4-10^5$ yr),
the more extended synchrotron radio-emitting size being a result of CR
diffusion seems unlikely. Instead, if SMGs are driven by galaxy mergers the
radio synchrotron emission might arise from more extended magnetised ISM around
the starburst region.
