Observability of CMB spectrum distortions from dark matter annihilation

Even after dark matter chemically freezes out in the early universe, electromagnetic cascades from dark matter annihilation can still perturb the background photon spectrum when the universe temperature cools down to 0.5 keV. We revisit the CMB spectrum distortions caused by $s$-wave dark matter annihilation under the updated Planck data and the future CMB sensitivity, concluding that $s$-wave annihilation cannot create observable distortions under forecast sensitivities of the (Super-)PIXIE missions. We further detail the case of $p$-wave dark matter annihilation, demonstrating the observability of the primordial $\mu$-distortion. Taking current constraints from primordial light elements, structure formations, cosmic electron-positron rays, and gamma rays, we find that the $\mu$-distortion reaching the observational limit as large as $\mu\simeq 3\times 10^{-8}$ can only be realized with a dark matter mass at 10--50 MeV and a kinetic decoupling temperature around 1 keV. The upper bound of the $p$-wave annihilation cross section can be strengthened by an order of magnitude if the $\mu$-distortion is not detected.