Probing Dark Matter Substructure with Wave-Optics Distortions of Strongly Lensed LISA Gravitational Waves

Strong lensing changes the phase of a gravitational-wave signal as well as its amplitude and arrival time. We study whether this phase information can distinguish three dark-matter structures in the lens: a Navarro--Frenk--White halo (NFW), a self-interacting halo (SIDM), and a fuzzy-dark-matter field (FDM). We generate waveforms for the detectable lensed massive-black-hole-binary population of a four-year LISA mission and fit every signal with the same smooth singular-isothermal-ellipsoid lens with external shear. In 132 lens systems, 311 images are resolved as separate signals in time. NFW and SIDM produce real waveform changes, but their slowly varying part is largely degenerate with the constant, gradient, and curvature of a smooth lens. The coherent density fluctuations of FDM leave a larger frequency-dependent residual after this fit. The NFW--FDM and SIDM--FDM populations become distinguishable with about 60 and 110 resolved image waveforms, respectively. These results show that repeated lensed LISA signals can probe the spatial form of dark matter in lens galaxies, rather than only the total lensing mass.