Identifying Strong Gravitational-wave Lensing during the Second Observing Run of Advanced LIGO and Advanced Virgo

We perform Bayesian model selection with parameter estimation to identify potentially lensed gravitational-wave images from the second observing run (O2) of Advanced LIGO and Advanced Virgo. Specifically, we compute the Bayesian evidence for a pair of events being lensed or not lensed (unlensed) using nested sampling. We consider in the model selection the discrete coalescence phase shifts that can be induced if the gravitational-wave signal intersects with the lens caustics. We find that the pair of events, GW170104 and GW170814 with a π/2 coalescence phase shift, has a significant Bayes factor ($B_U^L\sim 1.98\times {10}^4$) favoring the lensing hypothesis. However, after taking into account the long time delay of approximately 7 months between events, the timing Bayes factor is significantly small (B _t ∼ 8.7 × 10⁻²). The prior probability for detecting strongly lensed pairs at O2 sensitivity is exceedingly small for both galaxy and galaxy cluster lensing. Combining the lensing and timing Bayes factors with the prior odds on lensing gives an odds ratio of $O_U^L\sim 20$. With the value of the odds ratio after including model dependence of the timing and prior odds factors, we do not have strong evidence to demonstrate that the aforementioned pair is strongly lensed.