Distinguishing between precessing and lensed gravitational waves: Am I seeing double?

Gravitational lensing is a remarkable consequence of general relativity. The first multiply imaged electromagnetic source was discovered half a century ago. Over the past decade, we have begun observing the universe through a new window: gravitational waves (GWs). The current catalog of the observed GW events, consisting mostly of binary black hole mergers, is now up to 218 events. As the LIGO-Virgo network continues to improve in sensitivity and next-generation ground- and space-based detectors come online, the observed GW signals will extend to higher redshifts. This increases the possibility of gravitational lensing of the GW signal. Strong lensing of a GW signal can produce multiple discrete images, each with a time delay and magnification. If these images interfere in the observational band, one might be able to observe modulations in the frequency-domain GW signal parameterized by the time delay and magnification ratios of the images. Astrophysical black holes are generally spinning, and misalignment between the spins and orbital angular momentum may induce precession of the binary orbit. As the orbital angular momentum changes direction due to precession, the GW signal is modulated. The modulations driven by the spin misalignment can be characterized by precession amplitude and precession frequency. To explore the degeneracies and differences between these two modulations, we employ match-filtering and calculate mismatches between the two waveforms, examining the parameter space of precession (precession amplitude and precession frequency) and lensing (time delays and the ratio of magnification between the two images). We identify where in this parameter space lensing and precession are distinguishable or degenerate. We find that for lighter binaries that spend sufficient time in band while inspiraling, the modulation characteristics for the two phenomena are different, lending to their distinguishability. In contrast, for heavier binaries, degeneracies between the two exist for substantial regions in the parameter space as the binary spends less time in the band.