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We consider the intermediate mass-ratio inspiral of a stellar-mass compactobject with an intermediate-mass black hole that is surrounded by a dark matterdensity spike. The interaction of the inspiraling black hole with the darkmatter particles in the spike leads to dynamical friction. This can alter thedynamics of the black hole binary, leaving an imprint on the gravitational wavesignal. Previous calculations did not include in the evaluation of thedynamical friction coefficient the contribution from particles that move fasterthan the black hole. This term is neglected in the standard Chandrasekhar'streatment where only slower moving particles contribute to the deceleratingdrag. Here, we demonstrate that dynamical friction produced by the fast movingparticles has a significant effect on the evolution of a massive binary withina dark matter spike. For a density profile $\rho\propto r^{-\gamma}$ with$\gamma\lesssim 1$, the dephasing of the gravitational waveform can be severalorders of magnitude larger than estimated using the standard treatment. As$\gamma$ approaches $0.5$ the error becomes arbitrarily large. Finally, we showthat dynamical friction tends to make the orbit more eccentric for any $\gamma< 1.8$. However, energy loss by gravitational wave radiation is expected todominate the inspiral, leading to orbital circularization in most cases.

Dynamical friction in dark matter spikes: corrections to Chandrasekhar's formula