Cold dark matter is one of the most compelling candidates for dark matter, and its intrinsic properties suggest that it can accumulate near the central supermassive black holes of galaxies, forming dense dark matter halos. Space-based gravitational wave (GW) detectors, with their high sensitivity, offer a promising avenue to probe the existence of dark matter through its effects on GW signals. In this work, we adopt the Navarro-Frenk-White (NFW) profile as the initial dark matter halo model and consider the adiabatic formation of a static, spherically symmetric black hole within the halo. In the final state spacetime, we took into account the gravitational perturbation effect of the dark matter halo, which resulted in the final spacetime metric being different from the Schwarzschild metric. Based on the relativistic dynamics in the resulting spacetime, we calculate the dark matter density and velocity distributions in the vicinity of the black hole. In this process, we employ the near-field approximation to derive analytical expressions for the distribution function of the NFW halo. Furthermore, we assess the dynamical friction exerted by the dark matter halo on orbiting bodies near the black hole, and investigate its impact on the orbital evolution and GW waveforms of extreme mass-ratio inspiral systems. Our results indicate that, in certain scenarios, the presence of a dark matter halo can lead to modifications in the phase of the emitted GWs.

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• Dark matter
• Gravitational wave sources
• Massive compact halo objects