Radio synchrotron emission from secondary electrons in interaction-powered supernovae

Several supernovae (SNe) with an unusually dense circumstellar medium (CSM) have been recently observed at radio frequencies. Their radio emission is powered by relativistic electrons that can be either accelerated at the SN shock (primaries) or injected as a by-product (secondaries) of inelastic proton-proton collisions. We investigate the radio signatures from secondary electrons, by detailing a semi-analytical model to calculate the temporal evolution of the distributions of protons, primary and secondary electrons. With our formalism, we track the cooling history of all the particles that have been injected into the emission region up to a given time, and calculate the resulting radio spectra and light curves. For a SN shock propagating through the progenitor wind, we find that secondary electrons control the early radio signatures, but their contribution decays faster than that of primary electrons. This results in a flattening of the light curve at a given radio frequency that depends only upon the radial profiles of the CSM density and of the shock velocity, $v_0$. The relevant transition time at the peak frequency is $\sim 190 \, K_{\rm ep,-3}^{-1} A_{\rm w, 16} \beta_{0, -1.5}^{-2}\,{\rm d}$, where $A_{\rm w}$ is the wind mass-loading parameter, $\beta_0=v_0/c$ and $K_{\rm ep}$ is the electron-to-proton ratio of accelerated particles. We explicitly show that late peak times at 5 GHz (i.e., $t_{\rm pk} \gtrsim 300-1000$ d) suggest a shock wave propagating in a dense wind ($A_{\rm w} \gtrsim 10^{16}-10^{17}$ gr cm$^{-1}$), where secondary electrons are likely to power the observed peak emission.