Several recent studies have considered modifications to the standard weakly-interacting massive particle (WIMP) scenario in which the cross section (times relative velocity v) for pair annihilation is enhanced by a factor 1/v. Since v~10^{-3} in the Galactic halo, this may boost the annihilation rate into photons and/or electron-positron pairs enough to explain several puzzling Galactic radiation signals. Here we show that if the annihilation cross section scales as 1/v, then there is a burst of WIMP annihilation in the first dark-matter halos that form at redshifts z ~ 100-200. If the annihilation is to gamma rays in the energy range 100 keV - 300 GeV, or to electron-positron pairs in the energy range GeV - 2 TeV, then there remains a contribution to the diffuse extragalactic gamma-ray background today. Upper limits to this background provide constraints to the annihilation cross section. If the photon or electron-positron energies fall outside these energy ranges, then the radiation is absorbed by the intergalactic medium (IGM) and thus ionizes and heats the IGM. In this case, cosmic microwave background constraints to the ionization history also put limits on the annihilation cross section.
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