  The dissipation of small-scale perturbations in the early universe produces a
distortion in the blackbody spectrum of cosmic microwave background photons. In
this work, we propose to use these distortions as a probe of the microphysics
of dark matter on scales $1\,\textrm{Mpc}^{-1}\lesssim k \lesssim
10^{4}\,\textrm{Mpc}^{-1}$. We consider in particular models in which the dark
matter is kinetically coupled to either neutrinos or photons until shortly
before recombination, and compute the photon heating rate and the resultant
$\mu$-distortion in both cases. We show that the $\mu$-parameter is generally
enhanced relative to $\Lambda$CDM for interactions with neutrinos, and may be
either enhanced or suppressed in the case of interactions with photons. The
deviations from the $\Lambda$CDM signal are potentially within the sensitivity
reach of a PRISM-like experiment if $\sigma_{\textrm{DM}-\gamma} \gtrsim
1.1\times10^{-30} \left(m_{\textrm{DM}}/\textrm{GeV}\right) \textrm{cm}^{2}$
and $\sigma_{\textrm{DM}-\nu} \gtrsim 4.8\times 10^{-32}
\left(m_{\textrm{DM}}/\textrm{GeV}\right) \textrm{cm}^{2}$ for time-independent
cross sections, and $\sigma^{0}_{\textrm{DM}-\gamma} \gtrsim 1.8 \times
10^{-40} \left(m_{\textrm{DM}}/\textrm{GeV}\right) \textrm{cm}^{2}$ and
$\sigma^{0}_{\textrm{DM}-\nu} \gtrsim 2.5 \times 10^{-47}
\left(m_{\textrm{DM}}/\textrm{GeV}\right) \textrm{cm}^{2}$ for cross sections
scaling as temperature squared, coinciding with the parameter regions in which
late kinetic decoupling may serve as a solution to the small-scale crisis.
Furthermore, these $\mu$-distortion signals differ from those of warm dark
matter (no deviation from $\Lambda$CDM) and a suppressed primordial power
spectrum (strongly suppressed or a negative $\mu$-parameter), demonstrating
that CMB spectral distortion can potentially be used to distinguish between
solutions to the small-scale crisis.
