  Black holes located within a dark matter cloud can create overdensity regions
known as dark matter spikes. The presence of spikes modifies the
gravitational-wave signals from binary systems through changes in the
gravitational potential or dynamical friction effects. We assess the importance
of including relativistic effects in both the dark matter distribution and the
dynamical friction. As a first step we numerically calculate the particle dark
matter spike distribution in full general relativity, using both Hernquist and
Navarro-Frenk-White profiles in a Schwarzschild background, and we produce
analytical fits to the spike profiles for a large range of scale parameters.
Then we use a post-Newtonian prescription for the gravitational-wave dephasing
to estimate the effect of relativistic corrections to the spike profile and to
the dynamical friction. Finally we include the torques generated by dynamical
friction in fast-to-generate relativistic models for circular extreme
mass-ratio inspirals around a nonspinning black hole. We find that both types
of relativistic corrections positively impact the detectability of dark matter
effects, leading to higher dephasings and mismatches between gravitational-wave
signals with and without dark matter spikes.
