  We construct the average radio spectral energy distribution (SED) of highly
star-forming galaxies (HSFGs) up to z~4. Infrared and radio luminosities are
bound by a tight correlation that is defined by the so-called q parameter. This
infrared-radio correlation provides the basis for the use of radio luminosity
as a star-formation tracer. Recent stacking and survival analysis studies find
q to be decreasing with increasing redshift. It was pointed out that a possible
cause of the redshift trend could be the computation of rest-frame radio
luminosity via a single power-law assumption of the star-forming galaxies'
(SFGs) SED.To test this, we constrained the shape of the radio SED of a sample
of HSFGs. To achieve a broad rest-frame frequency range, we combined previously
published VLA observations of the COSMOS field at 1.4 GHz and 3 GHz with
unpublished GMRT observations at 325 MHz and 610 MHz by employing survival
analysis to account for non-detections in the GMRT maps. We selected a sample
of HSFGs in a broad redshift range (0.3<z<4,SFR>100M0/yr) and constructed the
average radio SED. By fitting a broken power-law, we find that the spectral
index changes from $\alpha_1=0.42\pm0.06$ below a rest-frame frequency of 4.3
GHz to $\alpha_2=0.94\pm0.06$ above 4.3 GHz. Our results are in line with
previous low-redshift studies of HSFGs (SFR>10M0/yr) that show the SED of HSFGs
to differ from the SED found for normal SFGs (SFR<10M0/yr). The difference is
mainly in a steeper spectrum around 10 GHz, which could indicate a smaller
fraction of thermal free-free emission. Finally, we also discuss the impact of
applying this broken power-law SED in place of a simple power-law in
K-corrections of HSFGs and a typical radio SED for normal SFGs drawn from the
literature. We find that the shape of the radio SED is unlikely to be the root
cause of the q-z trend in SFGs.
