Counting on Short Gamma-Ray Bursts: Gravitational-wave Constraints of Jet Geometry

The detections of GW170817 and GRB 170817A revealed that at least some short gamma-ray bursts (sGRB) are associated with the merger of neutron-star compact binaries. The gamma-rays are thought to result from the formation of collimated jets, but the details of this process continue to elude us. One important feature of gamma-ray bursts is the emission profile of the jet as a function of viewing angle. We present two related methods to measure the effective angular width, θ_B, of sGRB jets using gravitational-wave (GW) and gamma-ray data, assuming all sGRBs have the same angular dependence for their luminosities. The first is a counting experiment that requires minimal knowledge about each event, beyond whether or not they were detected in gamma-rays. The second method uses GW and electromagnetic data to estimate parameters of the source. We additionally outline a model-independent method to infer the full jet structure of sGRBs using a nonparametric approach. Applying our methods to GW170817 and GW190425, we find weak constraints on the sGRB luminosity profile. We project that with 5 and 100 binary neutron star detections, the counting method would constrain the relative uncertainty in θ_B to within 51% and 12%, respectively. Incorporating GW parameter estimation provides only marginal improvements. We conclude that the majority of the information about jet structure comes from the relative sensitivities of GW and gamma-ray detectors as encoded in simple counting experiments.