 Dark-matter annihilation radiation

For identical self-conjugate dark matter particles, the pair-annihilation rate per volume is $⟨\sigma v⟩n_{\chi }^2/2$. If each annihilation releases radiated energy $E_a$, volume emissivity is $j=E_a⟨\sigma v⟩{\rho }^2/(2m_{\chi }^2)$. Optically thin isotropic surface brightness is $(4\pi {)}^{-1}\int jdz$. The squared-density dependence assumes a spatially constant rate coefficient and a smooth density; unresolved substructure changes the average of the square.