Here we present stringent low-frequency 185MHz limits on coherent radio emission associated with a short gamma-ray burst (SGRB). Our observations of the short GRB 180805A were taken with the upgraded Murchison Widefield Array (MWA) rapid-response system, which triggered within 20s of receiving the transient alert from Swift, corresponding to 83.7s post-burst. The SGRB was observed for 30m, resulting in a 3sigma persistent flux density upper-limit of 40.2mJy/beam. Transient searches were conducted at the Swift position of this GRB on 0.5s, 5s, 30s, and 2m timescales, resulting in 3sigma limits of 570-1830, 270-630, 200-420, and 100-200mJy/beam, respectively. We also performed a dedispersion search for prompt signals at the position of the SGRB with a temporal and spectral resolution of 0.5s and 1.28MHz, resulting in a 6sigma fluence upper-limit range from 570Jyms at DM=3000pc/cm^3 (z~2.5) to 1750Jyms at DM=200pc/cm^3 (z~0.1). We compare the fluence prompt emission limit and the persistent upper-limit to SGRB coherent emission models assuming the merger resulted in a stable magnetar. Our observations were not sensitive enough to detect prompt emission associated with the alignment of magnetic fields of a binary neutron star just prior to the merger, from the interaction between the relativistic jet and the interstellar medium or persistent pulsar-like emission from the spin-down of the magnetar. However, in the case of a more powerful SGRB (a gamma-ray fluence an order of magnitude higher than GRB 180805A and/or a brighter X-ray counterpart), our MWA observations may be sensitive enough to detect coherent radio emission from the jet-ISM interaction and/or the magnetar remnant. Finally, we demonstrate that of all current low-frequency radio telescopes, only the MWA has the sensitivity and response times capable of probing prompt emission models associated with the initial SGRB merger event.
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