Estimating the lensing probability for binary black hole mergers in AGN discs by using mismatch thresholds

Stellar-mass binary black holes (BBH) may form, evolve, and merge within the dense environments of active galactic nuclei (AGN) disks, thereby contributing to the BBH population detected by gravitational wave (GW) observatories. Mergers occurring in AGN disks may be gravitationally lensed by the supermassive black hole at the AGN centre. The probability of such lensing events has been approximately estimated by using the Einstein criterion in previous work. However, a more reasonable approach to calculating the lensing probability should be based on whether the detector can distinguish the lensed GW waveform from the unlensed one. In this work, we calculate the lensing probability of LIGO sources embedded in AGN disk by relating threshold mismatch to the signal-to-noise ratio of the observed events. For the sensitivity of LIGO-Virgo-KAGRA O3 observation runs, our results indicate that the lensing probability is several times higher than previous estimates. If AGNs are indeed the primary formation channel for BBHs, we could quantify the probability of detecting the lensed GW events in such a scenario. The non-detections, on the other hand, will place stricter constraints on the fraction of AGN disk BBHs and even the birthplaces of BBH mergers.