  We analyze cosmic microwave background (CMB) data to constrain the mass and
interaction strengths of thermally-produced dark matter (DM) in a
self-consistent manner, simultaneously taking into account the cosmological
effects of its mass and interactions. The presence of a light thermal-relic
particle contributes non-negligibly to the radiation density during Big Bang
Nucleosynthesis (BBN), altering the light-element yields, as well as the the
effective number of relativistic particle species. On the other hand, DM
interactions with the Standard Model can affect distribution of matter in later
universe. Both mass and interactions alter CMB anisotropy on sub-degree scales.
To understand and quantify the interplay of these effects, we consider elastic
DM-baryon scattering with a momentum-transfer cross section that scales as a
power law of the relative velocity between the scattering particles. In the
range of thermal-relic DM masses relevant for BBN ($\lesssim$ 20 MeV), we find
that the reconstruction of the DM mass and the scattering cross section from
the CMB data features strong degeneracies; modeling the two effects
simultaneously increases the sensitivity of the CMB measurements to both
fundamental properties of DM. Additionally, we study the effects of late-time
residual annihilation of a light thermal relic and provide improved CMB
constraints on the DM mass and annihilation cross section. To examine
degeneracy between DM mass, cross section for elastic scattering with baryons,
and annihilation cross section, we consider a specific case of DM with an
electric and magnetic dipole moments. We present new, self-consistent
cosmological bounds for this model and discuss implications for future
searches.
