  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.
