Derivation of Thermally averaged cross sections In many sources discussing neutrino decoupling I find the following claim:
"The thermally averaged rate of weak interactions is given by:
$\Gamma = n \langle\sigma |v|\rangle$,
where $\langle\sigma |v|\rangle$ is a thermally averaged cross section given by:
$\langle\sigma |v|\rangle=G_F^2 T^2$.
My problem is: 
It is the first time I see this formula in my life and I can't seem to find a derivation of it. I understand that $G_F$ comes from the usual particle physics cross section. I also know that the particle physics cross section usually depends on some energy variable and that the thermal average of a energy variable will usually yield a temperature $T$ dependence. However I would like to see a careful derivation of this formula. Can someone shed some light on this?
Thank you.
Side note: I am currently studying "The early universe" by Kolb and Turner. I am quite confortable with QFT and particle physics but I am still a beginner in this field that lies in the frontier between particle physics and cosmology.
 A: I know you want a careful treatment, but if you already have Kolbe and Turner then they have the most careful treatment I've seen at a textbook level. They essentially assume the boltzmann equation is familiar, however, so I recommend something like Kardar volume 1 chapter 3 or other statistical physics book.
The actual cross section formula is complicated, however, $\sigma$ is literally the particle physics cross section. What v is is somewhat subtle, see classic paper by Gondolo and Gelmini, but is almost always defined as either the relative velocity or half that. The thermal average then is at the most basic approximation, just an integral over a non relativistic classic gas. However this part can get more complicated depending on the number of final states, if QM matters, what states have different masses, etc.
The $G_F^2 T^2$ is not at all an exact formula and only applies in the high temperature regime. What one does is basically what you said, we know the weak cross section couplings and propagators give us the $G_F^2$, and then if $T \gg M$, $T$ is the only other scale in the theory so just perform dimensional analysis. However, in many situations the temperature is low so instead you'd put in $M$.
