2016ApJ...823..154G

We estimate the long-duration gamma-ray burst (LGRB) progenitor rate using our recent work on the effects of environmental metallically on LGRB formation in concert with supernovae (SNe) statistics via an approach patterned loosely off the Drake equation. Beginning with the cosmic star formation history, we consider the expected number of broad-line Type Ic events (the SNe type associated with LGRBs) that are in low-metallicity host environments adjusted by the contribution of high-metallicity host environments at a much reduced rate. We then compare this estimate to the observed LGRB rate corrected for instrumental selection effects to provide a combined estimate of the efficiency fraction of these progenitors to produce LGRBs and the fraction of which are beamed in our direction. From this we estimate that an aligned LGRB occurs for approximately every 4000±2000 low-metallically broad-lined SNe Ic. Therefore, if one assumes a semi-nominal beaming factor of 100, then only about one such supernova out of 40 produce an LGRB. Finally, we propose an off-axis LGRB search strategy of targeting only broad-line Type Ic events that occur in low-metallicity hosts for radio observation.