Profiling Dark Matter Spikes with Gravitational Waves from Accelerated Binaries

Dark matter halos can develop a density spike, e.g., around a galactic supermassive black hole, with the profile $\rho \propto r^{-\gamma_{\rm sp}}$ determined both by the galaxy's formation history and the microphysics of dark matter. We show that future LISA/DECIGO observations, of intermediate/stellar-mass binary mergers inside the spike around the supermassive black hole, can measure $\gamma_{\rm sp}$ at a few-percent--level precision. The spike induces a distinctive time-dependent acceleration along the non-circular orbit taken by the binary's center of mass, which is observable as a secular modulation of the gravitational wave signal. This method -- insensitive to confounding astrophysical effects (dynamical friction, tidal effects, etc.) and not reliant on unknown dark matter particle physics -- provides a clean diagnostic of density spikes and a new probe of dark matter.