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Relativistic supernovae constitute a sub-class of type Ic supernovae (SNe).Their non-thermal, radio emission differs notably from that of regular type Icsupernovae as they have a fast expansion speed (with velocities $\sim$ 0.6-0.8c) which can not be explained by a "standard", spherical SN explosion butadvocates for a quickly evolving, mildly relativistic ejecta associated withthe SN. In this paper, we compute the synchrotron radiation emitted by thecocoon of a long gamma-ray burst jet (GRB). We show that the energy andvelocity of the expanding cocoon, and the radio non-thermal light curves andspectra are consistent with those observed in relativistic SNe. Thus, the radioemission from this events is not coming from the SN shock front, but from themildly relativistic cocoon produced by the passage of a GRB jet through theprogenitor star. We also show that the cocoon radio emission dominates the GRBemission at early times for GRBs seen off-axis, and the flux can be larger atlate times compared with on-axis GRBs if the cocoon energy is at leastcomparable with respect to the GRB energy.

Radio emission from the cocoon of a GRB jet: implications for relativistic supernovae and off-axis GRB emission