2021ApJ...914L..34P

The recent report of an association of the gravitational-wave (GW) binary black hole (BBH) merger GW190521 with a flare in the active galactic nuclei (AGNs) J124942.3 + 344929 has generated tremendous excitement. However, GW190521 has one of the largest localization volumes among all of the GW events detected so far. The 90% localization volume likely contains 7400 unobscured AGNs brighter than g <= 20.5 AB mag, and it results in a >=70% probability of chance coincidence for an AGN flare consistent with the GW event. We present a Bayesian formalism to estimate the confidence of an AGN association by analyzing a population of BBH events with dedicated follow-up observations. Depending on the fraction of BBHs arising from AGNs, counterpart searches of O (1)- O (100) GW events are needed to establish a confident association, and more than an order of magnitude more for searches without follow-up (i.e., using only the locations of AGN and GW events). Follow-up campaigns of the top ∼5% (based on volume localization and binary mass) of BBH events with total rest-frame mass >=50 M_☉ are expected to establish a confident association during the next LIGO/Virgo/KAGRA observing run (O4), as long as the true value of the fraction of BBHs giving rise to AGN flares is >0.1. Our formalism allows us to jointly infer cosmological parameters from a sample of BBH events that include chance coincidence flares. Until the confidence of AGN associations is established, the probability of chance coincidence must be taken into account to avoid biasing astrophysical and cosmological constraints.