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.