  Strong gravitational lensing of gravitational waves (GWs) has been forecasted
to become detectable in the upcoming 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. We construct a mock catalog of
lensed and unlensed events. We find that the false alarm probability (FAP)
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, the overall FAP per pair is $\sim10^{-4}$. As the number
of events, $N$, in the GW catalogs increases, the number of random pairs of
events increases as $\sim N^2$. 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), which will lead to
better parameter estimation and, thus, lower FAP per pair -- at the cost of
dramatically decreasing the size of the lensing sample (by $\sim
1/\mbox{SNR}^3$). We show that with our simple overlap criteria for current
detectors at design sensitivity, the false alarms will dominate for realistic
lensing rates ($\lesssim10^{-3}$) even when selecting the highest SNR pairs.
These results highlight the necessity to design alternative identification
criteria beyond simple waveform and sky location overlap. Future GW detectors
Cosmic Explorer and Einstein Telescope may provide sufficient improvement in
parameter estimation, allowing for the conclusive detection of strong lensing
of GWs.
