Radio spectra and sizes of ALMA-identified submillimetre galaxies: evidence of age-related spectral curvature and cosmic ray diffusion?

We analyse the multi-frequency radio spectral properties of $41$ 6GHz-detected ALMA-identified, submillimetre galaxies (SMGs), observed at 610MHz, 1.4GHz, 6GHz with GMRT and the VLA. Combining high-resolution ($\sim0.5''$) 6GHz radio and ALMA $870\,\mu$m imaging (tracing rest-frame $\sim20$GHz, and $\sim250\,\mu$m dust continuum), we study the far-infrared/radio correlation via the logarithmic flux ratio $q_{\rm IR}$, measuring $\langle q_{\rm IR}\rangle=2.19\pm 0.07$ for our sample. We show that the high-frequency radio sizes of SMGs are $\sim1.8\pm 0.4\times$ ($\sim2$--$3$kpc) larger than those of the cool dust emission, and find evidence for a subset of our sources being extended on $\sim 10$kpc scales at 1.4GHz. By combining radio flux densities measured at three frequencies, we can move beyond simple linear fits to the radio spectra of high-redshift star-forming galaxies, and search for spectral curvature, which has been observed in local starburst galaxies. At least a quarter (10/41) of our sample show evidence of a spectral break, with a median $\langle\alpha^{1.4\,{\rm GHz}}_{610\,{\rm GHz}}\rangle=-0.61\pm 0.05$, but $\langle\alpha^{6\,{\rm GHz}}_{1.4\,{\rm GHz}}\rangle=-0.91\pm 0.05$ -- a high-frequency flux deficit relative to simple extrapolations from the low-frequency data. We explore this result within this subset of sources in the context of age-related synchrotron losses, showing that a combination of weak magnetic fields ($B\sim35\,\mu$G) and young ages ($t_{\rm SB}\sim40$--$80\,$Myr) for the central starburst can reproduce the observed spectral break. Assuming these represent evolved (but ongoing) starbursts and we are observing these systems roughly half-way through their current episode of star formation, this implies starburst durations of $\lesssim100$Myr, in reasonable agreement with estimates derived via gas depletion timescales.