Exact theory of freeze-out

We show that the standard theory of thermal production and chemical decoupling of WIMPs is incomplete. The hypothesis that WIMPs are produced and decouple from a thermal bath implies that the rate equation the bath particles interacting with the WIMPs is an algebraic equation that constraints the actual WIMPs abundance to have a precise analytical form down to the temperature x_* = m_χ/T_*. The point x., which coincides with the stationary point of the equation for the quantity Δ = Y-Y₀, is where the maximum departure of the WIMPs abundance Y from the thermal value Y₀ is reached. For each mass m_χ and total annihilation cross section left angle σ_annυ_r right angle, the temperature x_* and the actual WIMPs abundance Y(x_*) are exactly known. This value provides the true initial condition for the usual differential equation that have to be integrated in the interval x ≥ x_*. The matching of the two abundances at x_* is continuous and differentiable. The dependence of the present relic abundance on the abundance at an intermediate temperature is an exact result. The exact theory suggests a new analytical approximation that furnishes the relic abundance accurate at the level of 1.2 % in the case of S-wave and P-wave scattering cross sections. We conclude the paper studying the evolution of the WIMPs chemical potential and the entropy production using methods of non-equilibrium thermodynamics. (orig.)