Circularization versus eccentrification in intermediate mass ratio inspirals inside dark matter spikes

Inspirals of an intermediate mass black hole (IMBH) and a solar mass type object will be observable by space based gravitational wave detectors such as the Laser Interferometer Space Antenna. A dark matter overdensity around an IMBH---a dark matter spike---can affect the orbital evolution of the system. We consider here such intermediate mass ratio inspirals on eccentric orbits, experiencing dynamical friction of the dark matter spike. We find that by including the relative velocities of the dark matter particles, the dynamical friction tends to circularize the orbit, in contrast to previous inquiries. We derive a general condition for circularization or eccentrification for any given dissipative force. In addition to the dephasing, we suggest using the circularization rate as another probe of the dark matter spike. Observing these effects would be an indicator for the particle nature of dark matter.