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.

The line-of-sight, solid-angle integral of squared dark-matter density used for annihilation fluxes with a position-independent rate coefficient. It separates an astrophysical density model from particle yields and annihilation rates. A decay signal instead scales with a single power of density. Substructure or spatially varying velocity-dependent rates can prevent a simple smooth-density factorization.