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 $\sigma v=a+bv^2$) 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.
