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Effects of Velocity-Dependent Dark Matter Annihilation on the Energy Spectrum of the Extragalactic Gamma-ray Background

We calculate the effects of velocity-dependent dark matter annihilation cross sections on the intensity of the extragalactic gamma-ray background. Our formalism does not assume a locally thermal distribution of dark matter particles in phase space, and is valid for arbitrary velocity-dependent annihilation. As concrete examples, we calculate the effects of p-wave annihilation (with the $v$-weighted cross section of $σv=a+bv^2$) on the mean intensity of extragalactic gamma rays produced in cosmological dark matter halos. This velocity variation makes the shape of the energy spectrum harder, but this change in the shape is too small to see unless $b/a\agt 10^6$. While we find no such models in the parameter space of the Minimal Supersymmetric Standard Model (MSSM), we show that it is possible to find $b/a\agt 10^6$ in the extension MSSM$\otimes U(1)_{B-L}$. However, we find that the most dominant effect of the p-wave annihilation is the suppression of the amplitude of the gamma-ray background. A non-zero $b$ at the dark matter freeze-out epoch requires a smaller value of $a$ in order for the relic density constraint to be satisfied, suppressing the amplitude by a factor as low as $10^{-6}$ for a thermal relic. Non-thermal relics will have weaker amplitude suppression. As another velocity-dependent effect, we calculate the spectrum for s-wave annihilation into fermions enhanced by the attractive Sommerfeld effect. Resonances associated with this effect result in significantly enhanced intensities, with a slightly softer energy spectrum.