Distinguishing Strongly Interacting Dark Matter Spikes via EMRI Gravitational Waves

We investigate the potential of extreme mass ratio inspirals (EMRIs) as probes of strongly interacting dark matter (SIDM) spikes around supermassive black holes. Based on Lagrangian formulations of dark matter self-interactions, we analyze the effects of number changing processes, such as 2→0 annihilation and 3→2 scattering, on the inner density profiles, and propose a generalized definition of the dissolution radius, including its dependence on the central black hole mass. Building on this framework, we construct dark matter spike models suitable for phase shift calculations and study the gravitational waves of EMRIs under the circular orbit adiabatic approximation. Our analysis shows that different interaction mechanisms produce distinct dark matter density distributions, resulting in characteristic phase differences in the gravitational waveforms. The magnitudes of these phase shifts are well above the resolution threshold of space-based detectors. We conclude that future missions, such as LISA, will be able to distinguish different SIDM interaction channels through precise waveform measurements, thereby opening a new avenue for probing the microphysics of dark matter in extreme astrophysical environments.