  The dark matter (DM) distributed around an intermediate massive black hole
(IMBH) forms an overdensity region called DM minispike. We consider the binary
system which consists of an IMBH with DM minispike and a small black hole
inspiralling around the IMBH in eccentric orbits. 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 semilatus rectum p<<10^5 Rs (Rs 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>>10^5 Rs, 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. By calculating the signal-to-noise ratio for GWs with and without DM
minispikes and the mismatch between them, we show that the effect of the DM
minispike in GW waveforms can potentially be detected by future space-based GW
detectors such as LISA, Taiji, and Tianqin.
