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 the discrete coalescence phase shifts that can be induced if the gravitational-wave signal interacts with the lens caustics in the model selection. We find that the pair of events, GW170104 and GW170814 with a $\pi/2$ coalescence phase shift, has a significant Bayes factor ($B^L_{U}$ $\sim 1.98 \times 10^4$) 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 ($B_t \sim 8.7\times10^{-2}$). The prior probability for detecting strongly lensed pairs at O2 sensitivity are 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 $O^L_{U} \sim 20$. However, the model dependence of the timing and prior odds factors does not provide strong evidence to demonstrate that the pair is strongly lensed.