  The Planck experiment has measured the Cosmic Microwave Background (CMB)
angular spectrum with unprecedented accuracy, and these results are likely to
remain the state-of-the art for a decade or more. Since these measurements are
going to be used to forecast Euclid's ability to test alternative dark matter
models, we aim to set the final CMB constraint on the cross section of dark
matter-photon scattering in the early universe. As the formalism has not been
published in the literature yet, we present the key steps to derive the full
set of linearized Einstein and Boltzmann equations and elucidate necessary
modifications to the line of sight and the tight coupling approximation. These
approximation schemes are commonly used in Boltzmann codes to reduce
computational costs, but the tight coupling regime has been neglected in
previous works related to dark matter-photon interactions. Our computations
account for a dark matter sound speed, which was neglected in previous work. We
find that its effect is negligible on the CMB spectra but fairly important on
the linear matter power spectrum. We then analyze the Planck data, with and
without the 2015 polarization data at high multipoles. In both cases our
constraint on the ratio of the elastic scattering cross section to the dark
matter mass becomes more stringent with respect to previous constraints. The
change is about 35\% when the full Planck dataset is added (which includes
E-mode polarization at high multipoles and the lensing potential power
spectrum) and about 20\%. when we only account for the polarization data at low
multipoles and the full temperature spectrum. Our most conservative limit reads
as $\sigma_{\mathrm{DM}-\gamma} \leq 2.25\times
10^{-6}\,\sigma_\mathrm{Th}\left(m_\mathrm{DM}/\mathrm{GeV}\right)$ at 95\%
confidence. We do not expect this limit to change by adding the recently
published SPT data.

Cosmology and Nongalactic Astrophysics