Lensing or luck? False alarm probabilities for gravitational lensing of gravitational waves

Strong gravitational lensing of gravitational waves (GWs) has been forecasted to become detectable in the upcoming LIGO/Virgo/KAGRA observing runs. However, definitively distinguishing pairs of lensed sources from random associations is a challenging problem. We investigate the degree to which unlensed events mimic lensed ones because of the overlap of parameters due to a combination of random coincidence and errors in parameter estimation. Lensed events are expected to have consistent masses and sky locations, and constrained relative phases, but may have differing apparent distances due to lensing magnification. We construct a mock catalog of lensed and unlensed events. We find that the probability of a false alarm based on coincidental overlaps of the chirp mass, sky location, and coalescence phase are approximately 9%, 1%, and 10% per pair, respectively. Combining all three parameters, we arrive at an overall false alarm probability per pair of similar to 10-4. We validate our results against the GWTC-2 data, finding that the catalog data is consistent with our simulations. As the number of events, N, in the GW catalogs increases, the number of random pairs of events increases as similar to N2. Meanwhile, the number of lensed events will increase linearly with N, implying that, for sufficiently high N, the false alarms will always dominate over the true lensing events. This issue can be compensated for by placing higher thresholds on the lensing candidates (e.g., selecting a higher signal-to-noise ratio (SNR) threshold, rho thr), which will lead to better parameter estimation and, thus, lower false alarm probabilities per pair-at the cost of dramatically decreasing the size of the lensing sample (proportional to rho-3thr). We show that with our simple overlap criteria for current detectors at design sensitivity, the false alarms will dominate for realistic lensing rates (less than or similar to 10-3) even when selecting the highest SNR pairs. These results highlight the necessity to design alternative identification criteria beyond simple waveform and sky location overlap for conclusive detection of strong lensing. Future GW detectors such as Cosmic Explorer and Einstein Telescope may provide sufficient improvement in parameter estimation and a commensurate decrease in the incidence of coincidental overlap of parameters, allowing for the conclusive detection of strong lensing of GWs even without additional detection criteria.