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 $x_\ast=m_\chi /T_\ast$. 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.