Constraining the beaming of gamma-ray bursts with radio surveys

The degree of beaming in gamma-ray bursts (GRBs) is currently unknown. The uncertainty in the γ-ray-beaming angle, θ_b, leaves the total energy release (∝ θ²_b) and the event rate per galaxy (∝ θ^-2_b) unknown to within orders of magnitude. Since the delayed radio emission of GRB sources originates from a mildly relativistic shock and receives only weak relativistic beaming, the rate of radio-selected transients with no GRB counterparts can be used to set an upper limit on θ^-2_b. We find that a VLA survey with a sensitivity of ∼0.1 mJy at 10 GHz could identify ≳2 × 10⁴(θ_b/10^c)^-2 radio afterglows across the sky if each source is sampled at least twice over a period of 1 month or longer. From the total number of ≳0.1 mJy sources observed at 8.44 GHz and the fraction of fading sources at 1.44 GHz, we get the crude limit θ_b ≳ 6^o.