n = g 2 π 2 ∫ ∞ 0 f ( p ) p $^{2}$d p (2.34) = A ζ (3) g π 2 T 3 (2.35) 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 f ( p ) ≈ 1 e p T ± 1 (2.33) f ( E ) = 1 e ( E$_{−}$µ T ) ± 1 (2.32) 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 For a collection of particles in thermal equilibrium, the distribution func- tion is 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. 15 2.2 The Big Bang