Data from Planck measurements of the cosmic microwave background (CMB) place important constraints on models with light dark matter (DM) and light mediators especially when both lie in the mass range below $sim 1 $ GeV. In models involving kinetic mixing where the dark photon acts as the mediator, these constraints are easily satisfied and the appropriate DM relic density achievable if the DM is, e.g., a complex scalar, where $p$-wave annihilation occurs, or is the lighter component of a split pseudo-Dirac state where co-annihilation dominates. In both of these cases, although higher order in the dark gauge coupling, $g_D$, the corresponding annihilation processes including dark photon initial state radiation (ISR) will be dominantly $s$-wave with essentially temperature independent cross sections. The rates for these dark ISR associated processes, though not yielding cross sections large enough to contribute to the relic density, can still run into possible conflicts with the bounds arising from the CMB. In this paper we perform a preliminary study of the present and potential future constraints that the CMB imposes on the parameter spaces for both of these scenarios due to the existence of this dark ISR. Further analyses of the effects of dark ISR in DM annihilation is clearly warranted.
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