Exact theory of freeze-out

Abstract

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 /T∗. The point x∗, which coincides with the stationary point of the equation for the quantity = Y −Y0, is where the maximum departure of theWIMPs abundance Y from the thermal value Y0 is reached. For each mass m and total annihilation cross section annvr , 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.