Multi-messenger detection of black hole binaries in dark matter spikes

We investigate the inspiral of a high mass-ratio black hole binary located in the nucleus of a galaxy, where the primary central black hole is surrounded by a dense dark matter spike formed through accretion during the black hole growth phase. Within this spike, dark matter undergoes strong self-annihilation, producing a compact source of $\gamma$-ray radiation that is highly sensitive to spike density, while the binary emits gravitational waves at frequencies detectable by LISA. As the inspiralling binary interacts with the surrounding dark matter particles, it alters the density of the spike, thereby influencing the $\gamma$-ray flux from dark matter annihilation. We demonstrate that the spike self-annihilation luminosity decreases by $10\%$ to $90\%$ of its initial value, depending on the initial density profile and binary mass ratio, as the binary sweeps through the LISA band. This presents a new opportunity to indirectly probe dark matter through multi-messenger observations of galactic nuclei.