Phase effects from strong gravitational lensing of gravitational waves

Assessing the probability that two or more gravitational wave (GW) events are lensed images of the same source requires an understanding of the properties of the lensed images. For short enough wavelengths where wave effects can be neglected, lensed images will generically have a fixed relative phase shift that needs to be taken into account in the lensing hypothesis. For nonprecessing, circular binaries dominated by quadrupole radiation, these lensing phase shifts are degenerate with either a shift in the coalescence phase or a detector and inclination-dependent shift in the orientation angle. This degeneracy is broken by the presence of higher harmonic modes with vertical bar m vertical bar not equal 2 in the former and vertical bar m vertical bar not equal 1 in the latter. The presence of precession or eccentricity will also break this degeneracy. This implies that a lensed GW image will not necessarily be consistent with (unlensed) predictions from general relativity (GR). Therefore, unlike the conventional scenario of electromagnetic waves, strong lensing of GWs can lead to images with a modified phase evolution that can be observed. However, we find that for a wide range of parameters, the lensed (phase modified) waveform is similar enough to an unlensed (GR) waveform that GW detection pipelines will still find it. In particular, for present detectors, we find that templates with a shifted detector-dependent orientation angle have a signal-to-noise ratio differences of less than 1% for mass ratios up to 0.1, and less than 5% for precession parameters up to 0.5 and eccentricities up to 0.4 at 20 Hz. In these ranges, the mismatch is lower than 10% with the alternative detector-independent coalescence phase shift. None-theless, for a loud enough source, even with only one image it may be possible to directly identify it as a strongly lensed image from its non-GR, phase-shifted waveform. In more extreme cases, lensing may lead to considerable distortions, and the lensed images may even be undetected with current searches. Nevertheless, an exact template with a phase shift in Fourier space can always be constructed to fit any lensed image. We conclude that an optimal strong lensing search strategy would incorporate phase information in all stages of the identification of strong lensing, with an exact treatment in the final assessment of the probability of multiple lensed events. This work clarifies the role that strong lensing plays in the phase evolution of GWs: how it can lead to apparent deviations from GR, how it can affect the detectability of GW events, and how it can be exploited to help identify cases of strong gravitational lensing of gravitational wave sources.