Intermediate Mass-Ratio Inspirals with Dark Matter Minispike

The dark matter (DM) distributed around the intermediate massive black hole (IMBH) forms an overdensity region called DM minispike. In this paper, we consider the binary system which consists of an IMBH with DM minispike and a small black hole. The factors which affect the evolution of the orbit include the gravity of the system, the dynamical friction and accretion of the small black hole caused by the DM minispike, and the radiation reaction of gravitational waves (GWs). Using the method of osculating orbit, we find that when the semi-latus rectum $p\ll 10^5 R_s$ ($R_s$ is the Schwarzschild radius of the IMBH) the dominated factors are the dynamical friction and accretion from the DM minispike, and the radiation reaction. When $p\gg 10^5 R_s$, the gravity from the DM minispike dominates the orbital evolution. The existence of DM minispike leads to the deviation from the Keplerian orbit, such as extra orbital precession, henceforth extra phase shift in the GW waveform. We show that these effects of DM minispike in GW waveforms can be detected by future space-based GW detectors such as LISA, Taiji, and Tianqin, and the observation of the IMBH binaries with DM minispikes may disclose the density profile of DM minispikes.