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 \(x_\ast=m_\chi /T_\ast\). The point \(x_\ast\), which coincides with the stationary point of the equation for the quantity \(\Delta= Y-Y_0\), is where the maximum departure of the WIMPs abundance \(Y\) from the thermal value \(Y_0\) is reached. For each mass \(m_\chi\) and total annihilation cross section \(\langle \sigma_\text{ann}v_\text{r}\rangle\), the temperature \(x_\ast\) and the actual WIMPs abundance \(Y(x_\ast)\) are exactly known. This value provides the true initial condition for the usual differential equation that have to be integrated in the interval \(x\ge x_\ast\). The matching of the two abundances at \(x_\ast\) 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.