Strong lensing of gravitational waves with modified propagation

We explore the impact of corrections to the propagation on the waveforms of gravitationally lensed gravitational waves under the geometrical optics approximation, focusing on both uniform cosmological modifications and local modifications localized around lensing objects. By adopting a model-independent phenomenological approach, we systematically investigate the effects of these modifications in strong lensing scenarios, where detection of multiple images is expected. Our analysis reveals that cosmological modifications can yield corrections to the time delay that remain to be minor compared with the effects that accumulate over the whole propagation process, which are present also in the unlensed waveform. By contrast, local modifications around lensing objects can alter the image position and also the magnification factor, which is potentially polarization-selective and frequency-dependent. In some case we can have image disappearance as well as signal amplification. Furthermore, we demonstrate that such modifications can cause degradation of waveform match with the templates based on general relativity. This study highlights the importance of considering waveform modifications to search for the signature of modified propagation or the existence of extra polarization modes, and proposes potential observational targets.