The radio SZ effect as a probe of the cosmological radio background

If there is a substantial cosmological radio background, there should be a radio Sunyaev-Zeldovich (SZ) effect that goes along with it. The radio background Comptonization leads to a slight photon excess at all wavelengths, while Comptonization of the CMB at low frequencies leads to a decrement. For levels of the radio background consistent with observations, these effects cancel each other around $\nu\simeq 735~$MHz, with an excess at lower frequencies and a decrement at higher frequencies. Assuming a purely cosmological origin of the observed ARCADE radio excess, at $\nu \lesssim 20\,{\rm GHz}$ the signal scales as $\Delta T / T_{\rm CMB}\simeq 2\,y\left[ (\nu/735\,{\rm MHz})^{-2.59}-1\right]$ with frequency and the Compton-$y$ parameter of the cluster. For a typical cluster, the total radio SZ signal is at the level of $\Delta T\simeq 1\,{\rm mK}$ around the null, with a steep scaling towards radio frequencies. This is above current raw sensitivity limits for many radio facilities at these wavelengths, providing a unique way to confirm the cosmological origin of the ARCADE excess and probe its properties (e.g., redshift dependence and isotropy). We also give an expression to compute the radio-analogue of the kinematic SZ effect, highlighting that this might provide a new tool to probe large-scale velocity fields and the cosmic evolution of the radio background.