content_image/306508.jpg content_image/306507.jpg content_image/306504.jpg content_image/306505.jpg
at the time of neutrino decoupling, when the temperature T ∼ 1MeV, with all the standard model particles present at that time being in thermal equi- librium. For a particular particle process to be in thermal equilibrium, the interaction rate between the particles has to be much larger than the Hubble parameter (expansion rate), Γ ≫ H , so that the process has the “time” to occur, before the expansion of the Universe stifles the process. 

For a collection of particles in thermal equilibrium, the distribution func- tion is 

<img src='content_image/306508.jpg'>

with + for bosons and − for fermions and where E is the particle energy and µ is the Chemical Potential . For the case where the temperature T is much larger than the mass and chemical potential of the particle species, which was applicable in the very early Universe, the distribution function simplifies to 

<img src='content_image/306507.jpg'>

where E ∼ T ≃ p , with p being the momentum of the particle. This distri- bution function yields the blackbody distribution of photons. The number density is given by 

<img src='content_image/306504.jpg'>

<img src='content_image/306505.jpg'>

