Fast and efficient Bayesian method to search for strongly lensed gravitational waves

A small fraction of the gravitational-wave (GW) signals from binary black holes observable by ground-based detectors will be strongly lensed by intervening objects such as galaxies and clusters. Strong lensing will produce nearly identical copies of the GW signals separated in time. These lensed signals must be identified against a background of unlensed pairs GW events, some of which may appear similar by accident. This is usually done using fast, but approximate methods that, for example, check for the overlap between the posterior distributions of a subset of binary parameters, or using slow, but accurate joint Bayesian parameter estimation. In this work, we present a modified version of the posterior overlap method dubbed "PO2.0" that is mathematically equivalent to joint parameter estimation while still remaining fast. We achieve a significant gain in efficiency by incorporating informative priors about the binary and lensing populations, selection effects, and all the inferred parameters of the binary. For binary black hole signals lensed by galaxies, our improved method can detect 65% lensed events at a pair-wise false alarm probability of $\sim 2\times 10^{-6}$. Consequently, we have a 13% probability of detecting a strongly lensed event above $2.25\sigma$ significance during 18 months of observation by the LIGO-Virgo detectors at their current sensitivity. We also show how we can compute the joint posteriors of the lens and source parameters from a pair of lensed events by reweighting the posteriors of individual events in a computationally inexpensive way.