  The optical and near-IR emission from some classes of supernovae (SNe),
including Type IIn and possibly some super-luminous SNe, is likely powered by a
collision between the SN ejecta and dense circumstellar material (CSM). We
argue that for a range of CSM masses and their radii, a collisionless shock can
form, allowing for efficient cosmic-ray (CR) acceleration. We show that pp
collisions between these newly accelerated CRs and the CSM leads to not only
gamma rays but also secondary electrons and positrons that radiate synchrotron
photons in the high-frequency radio bands. Our estimates imply that various
facilities including the Jansky Very Large Array (VLA) and the Atacama Large
Millimeter/submillimeter Array (ALMA) may observe such SNe at Gpc distances by
followup observations in months-to-years, although the detectability strongly
depends on the CSM density as well as observed frequency. Detecting this signal
would give us a unique probe of CR acceleration at early times, and even
non-detections can put interesting limits on the possibility of CR ion
acceleration. Following our previous work, we also show that GeV gamma rays can
escape from the system without severe attenuation, encouraging point-source and
stacking analyses with Fermi. We provide recipes for diagnosing
interaction-powered SN scenario with multi-messenger (neutrino and gamma-ray)
observations.
