  Recent studies have postulated that the presence of dark matter (DM) spikes
around IMBHs could lead to observable dephasing effects in gravitational wave
(GW) signals emitted by Intermediate Mass Ratio Inspirals (IMRIs). While prior
investigations primarily relied on non-self-consistent analytic methods to
estimate the influence of DM spikes on eccentric IMRIs, our work introduces the
first self-consistent treatment of this phenomenon through $N$-body
simulations. Contrary to previous studies, which suggested that dynamical
friction (DF), a cumulative effect of two-body encounters, is the primary
mechanism responsible for energy dissipation, we reveal that the slingshot
mechanism, a three-body effect, plays a more significant role in driving the
binary system's energy loss and consequent orbital shrinkage, similar to
stellar loss cone scattering in Massive Black Hole (MBH) binaries. Furthermore,
our work extends its analysis to include rotation in DM spikes, a factor often
overlooked in previous studies. We find that binaries that counter-rotate with
respect to the spike particles merge faster, while binaries that co-rotate
merge slower, in opposition to the expectation from DF theory. While our models
are idealistic, they offer findings that pave the way for a more comprehensive
understanding of the complex interactions between DM spikes, IMRIs, GW
emission, and the ability to constrain DM microphysics. Our work systematically
includes Post-Newtonian (PN) effects until 2.5 order and our results are
accurate and robust.
