WMAP7 and future CMB constraints on annihilating dark matter: implications for GeV-scale WIMPs

(Context) We calculate constraints from current and future cosmic microwave background (CMB) measurements on annihilating dark matter (DM) with masses below the electroweak scale: m_{DM}=5-100 GeV. In particular, we focus our attention on the lower end of this mass range, as DM particles with masses m_{DM} ~ 10 GeV have been recently claimed to be consistent with the CoGeNT and DAMA/LIBRA results, while also providing viable DM candidates to explain the measurements of Fermi and WMAP haze. (Aims) We study the model (in)dependence of CMB spectrum on particle physics DM models, large scale structure formation and cosmological uncertainties. We attempt to find a simple and practical recipe for estimating current and future CMB bounds on a broad class of DM annihilation models. (Results) We show that in the studied DM mass range the CMB signal of DM annihilations is independent of the details of large scale structure formation, distribution and profile of DM halos and other cosmological uncertainties. All particle physics models of DM annihilation can be described with only one parameter, the fraction of energy carried away by neutrinos in DM annihilation. As the main result we provide a simple and rather generic fitting formula for calculating CMB constraints on the annihilation cross section of light WIMPs. We show that thermal relic DM in the CoGeNT, DAMA favoured mass range is in a serious conflict with present CMB data for the annihilation channels with few neutrinos, and will definitely be tested by the Planck mission for all possible DM annihilation channels. Also, our findings strongly disfavor the claim that thermal relic DM annihilations with m_{DM} ~ 10 GeV and $<sigma_av> ~ 9x10^{-25} cm^{3}s^{-1} could be a cause for Fermi and WMAP haze.