Wave-Optics Imprints of Warm Dark Matter Subhalos with Prompt Cusps on Strongly Lensed Gravitational Waves
Dark matter halos are expected to form with a prompt $ρ\propto r^{-3/2}$ density cusp at their centres, and in warm dark matter (WDM) cosmologies these cusps can dominate the inner structure of the low-mass subhalos that survive free-streaming suppression. We investigate whether that inner structure is visible to gravitational waves lensed in the wave-optics (WO) regime. Extending the strong lensing WO diffraction-integral framework to WDM, we generate subhalo populations with the {\sc sashimi-w} semi-analytic model, assign each subhalo a prompt cusp through a cusp-halo relation, and compute the frequency-dependent amplification factor $F(f)$ across the LISA band over 500 independent realisations for $m_χ\in \{6, 10, 20, 40, 80, 100\}$~keV. The signal is governed by the overlap between the WDM-suppressed subhalo mass function and the $m_{\rm sub} \sim 10^{4}$ to $10^{7}\,M_\odot$ range to which the LISA band is sensitive. Modulations are negligible for $m_χ\lesssim 10$~keV, reach the percent level by 40~keV, and then saturate to a CDM-like plateau, so the observable reliably probes $m_χ$ only over the intermediate range $\sim 20$ to 40~keV. Repeating the 40~keV ensemble on identical subhalo catalogs with the cusp amplitude set to zero, we find that prompt cusps enhance the amplitude modulation only mildly. Where cusps make up a large fraction of the subhalo mass, free-streaming has already depleted the signal-carrying mass range, and where that range is populated, the cusp lies well inside the Fresnel radius, to which the diffraction integral responds through projected enclosed mass rather than central density. A percent-level detection in a strongly lensed massive black-hole binary would therefore establish the presence of substructure on Fresnel scales without diagnosing its inner profile.