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

Strong gravitational lensing of gravitational wave sources is expected to be detected 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 non-lensed events mimic lensed ones because of the overlap of parameters due to a combination of random coincidence and errors in parameter estimation. We construct a mock catalog of lensed and non-lensed 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 11%, 1%, and 10% per pair, respectively. Combining the three, we arrive at a false alarm probability per pair of 10^-4. As the number of events, N, in the GW catalogs increases, the number of random pairs of events increases as ∼Ν². 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 actual lensing events. This issue can be compensated for by placing higher thresholds on the lensing candidates (e.g., selecting a higher signal-to-noise (SNR) threshold), which will lead to better parameter estimation and thus lower false alarm rates per pair, at the cost of dramatically decreasing the size of the lensing sample (by ∼SNR³).  We show that with our simple overlap criteria for current detectors at design sensitivity, the false alarms will win for realistic lensing rates (≤10^-3) even when selecting the highest SNR pairs. These results highlight the necessity to design alternative identification criteria for conclusive detection of strong lensing.