We have conducted the most sensitive low frequency (below 100 MHz) search to date for prompt, low-frequency radio emission associated with short-duration gamma-ray bursts (GRBs), using the Owens Valley Radio Observatory Long Wavelength Array (OVRO-LWA). The OVRO-LWAs nearly full-hemisphere field-of-view ($sim20$,$000$ square degrees) allows us to search for low-frequency (sub-$100$ MHz) counterparts for a large sample of the subset of GRB events for which prompt radio emission has been predicted. Following the detection of short GRB 170112A by Swift, we used all-sky OVRO-LWA images spanning one hour prior to and two hours following the GRB event to search for a transient source coincident with the position of GRB 170112A. We detect no transient source, with our most constraining $1sigma$ flux density limit of $650~text{mJy}$ for frequencies spanning $27~text{MHz}-84~text{MHz}$. We place constraints on a number of models predicting prompt, low-frequency radio emission accompanying short GRBs and their potential binary neutron star merger progenitors, and place an upper limit of $L_text{radio}/L_gamma lesssim 7times10^{-16}$ on the fraction of energy released in the prompt radio emission. These observations serve as a pilot effort for a program targeting a wider sample of both short and long GRBs with the OVRO-LWA, including bursts with confirmed redshift measurements which are critical to placing the most constraining limits on prompt radio emission models, as well as a program for the follow-up of gravitational wave compact binary coalescence events detected by advanced LIGO and Virgo.
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