  We propose a new method to constrain elastic scattering between dark matter
(DM) and standard model particles in the early Universe. Direct or indirect
thermal coupling of non-relativistic DM with photons leads to a heat sink for
the latter. This results in spectral distortions of the cosmic microwave
background (CMB), the amplitude of which can be as large as a few times the
DM-to-photon number ratio. We compute CMB spectral distortions due to
DM-proton, DM-electron and DM-photon scattering for generic energy-dependent
cross sections and DM mass m_DM >~ 1 keV. Using FIRAS measurements we set
constraints on the cross sections for m_DM <~ 0.1 MeV. In particular, for
energy-independent scattering we obtain sigma[DM-proton] <~ 10^(-24) cm^2
(keV/m_DM)^(1/2), sigma[DM-electron] <~ 10^(-27) cm^2 (keV/m_DM)^(1/2) and
sigma[DM-photon] <~ 10^(-39) cm^2 (m_DM/keV). An experiment with the
characteristics of PIXIE would extend the regime of sensitivity up to masses
m_DM ~ 1 GeV.
