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range, we combined previously published Very Large Array observations of the COSMOS field at 1.4?GHz and 3?GHz with unpublished Giant Meterwave Radio Telescope (GMRT) observations at 325?MHz and 610?MHz by employing survival analysis to account for non-detections in the GMRT maps.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 ( z ?∈?[0.3,?4],??SFR?≥?100? M _(⊙)?yr~(?1)) and constructed the average radio SED. By fitting a broken power-law, we find that the spectral index changes from α _(1)?=?0.42?±?0.06 below a rest-frame frequency of 4.3?GHz to α _(2)?=?0.94?±?0.06 above 4.3?GHz. Our results are in line with previous low-redshift studies of HSFGs (?SFR?> ?10? M _(⊙)??yr~(?1)) that show the SED of HSFGs to differ from the SED found for normal SFGs (?SFR?< ?10? M _(⊙)?yr~(?1)). 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.
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