  It was found that the dark matter (DM) in the
intermediate-mass-ratio-inspiral (IMRI) system has a significant enhancement
effect on the orbital eccentricity of the stellar massive compact object, such
as a black hole (BH), which may be tested by space-based gravitational wave
(GW) detectors including LISA, Taiji and Tianqin in future observations
\citep{2019PhRvD.100d3013Y}. In this paper, we will study the enhancement
effect of the eccentricity for an IMRI under different DM density profiles and
center BH masses. Our results are as follows: $(1)$ in terms of the general DM
spike distribution, the enhancement of the eccentricity is basically consistent
with the power-law profile, which indicates that it is reasonable to adopt the
power-law profile; $(2)$ in the presence of DM spike, the different masses of
the center BH will affect the eccentricity, which provides a new way for us to
detect the BH's mass; $(3)$ considering the change of the eccentricity in the
presence and absence of DM spike, we find that it is possible to distinguish DM
models by measuring the eccentricity at the scale of about $10^{5} GM/c^{2}$.
