  Possible interactions of dark matter (DM) with Standard Model (SM) particles
can be tested with spectral distortions (SDs) of the cosmic microwave
background (CMB). In particular, a non-relativistic DM particle that scatters
elastically with photons, electrons or nuclei imprints a negative chemical
potential $\mu$ to the CMB spectrum. This article revisits the first study of
this effect, with an accurate treatment of heat exchange between DM and SM
particles. We show that the instantaneous-decoupling approximation made in the
original study systematically and significantly underestimates the amplitude of
SDs. As a consequence, we derive tighter upper bounds to the DM-SM
elastic-scattering cross section for DM masses $m_\chi \lesssim 0.1$ MeV, from
the non-detection of $\mu$-distortions by FIRAS. We also show that a future
instrument like PIXIE, sensitive to $|\mu| \sim 10^{-8}$, would be able to
probe DM-SM cross sections much smaller than first forecasted, and orders of
magnitude below current upper limits from CMB-anisotropy data, up to DM masses
of $\sim 1$ GeV. Lastly, we study the sensitivity of SDs to the electric and
magnetic dipole moments of the DM. Although SDs can place non-trivial
constraints on these models, we find that even future SD experiments are
unlikely to improve upon the best current bounds. This article is accompanied
by the public code DMDIST, which allows one to compute CMB SDs for generic
particle-DM models, specified by their cross sections for elastic scattering
with and annihilation into SM particles.
