  Future space-based laser interferometry experiments such as LISA are expected
to detect $\cal O$(100--1000) stellar-mass compact objects (e.g., black holes,
neutron stars) falling into massive black holes in the centers of galaxies, the
so-called extreme-mass-ratio inspirals (EMRIs). If dark matter forms a "spike"
due to the growth of the massive black hole, it will induce a gravitational
drag on the inspiraling object, changing its orbit and gravitational-wave
signal. We show that detection of even a single dark matter spike from the
EMRIs will severely constrain several popular dark matter candidates, such as
ultralight bosons, keV fermions, MeV--TeV self-annihilating dark matter, and
sub-solar mass primordial black holes, as these candidates would flatten the
spikes through various mechanisms. Future space gravitational wave experiments
could thus have a significant impact on the particle identification of dark
matter.
