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A Complete Survey of the Transient Radio Sky and Implications for Gamma-Ray Bursts, Supernovae, and other Relativistic Explosions

We had previously reported on a survey for radio transients, used to set an upper limit on the number of orphan gamma-ray burst (GRB) radio afterglows, and thus a lower limit on the typical GRB beaming factor. Here we report radio and optical follow-up observations of these possible transients, achieving the first full characterization of the transient radio sky. We find that only two source are likely to be real radio transients, an optically obscured radio supernova (SN) in the nearby galaxy NGC 4216, and a source not associated with a bright host galaxy, which is too radio luminous to be a GRB afterglow. We speculate that this may be a flare from a peculiar active galactic nucleus, or a burst from an unusual Galactic compact object. We place an upper limit of 65 radio transients above 6 mJy over the entire sky at the 95% confidence level. The implications are as follows. First, we derive a limit on the typical beaming of GRBs; we find f_b^{-1} >~ 60, ~5 times higher than our earlier results. Second, we impose an upper limit on the rate of events that eject >~ 10^{51} erg in unconfined relativistic ejecta, whether or not accompanied by detectable emission in wavebands other than the radio. Our estimated rate, <=1000/y/Gpc, is about two orders of magnitude smaller than the rate of core-collapse SNe (and type Ib/c events in particular), indicating that only a minority of such events eject significant amounts of relativistic material, which are required by fireball models of long-soft GRBs. Finally, we show that future wider and/or deeper radio variability surveys are expected to detect numerous orphan radio GRB afterglows. Our survey also illustrates the great potential of sensitive surveys with new instruments to revolutionize the study of nearby SNe (abridged).