4.4 “End of Inflation” Mechanism 75 thermal bath, v is the relative velocity between a φ particle and a thermal bath particle (which in our case is ≈ c =1 ) and 〈 〉 denotes a thermal average. The scattering cross-section σ is given by where E$_{c.m.}$ is the centre-of-mass energy, which is Substituting Eq. (4.119) into Eq. (4.118) gives This scattering cross-section is the total cross-section for all types of scat- tering (e.g. elastic) that can take place between φ and the particles in the thermal bath. For a complete Field Theory derivation of the elastic scat- tering cross-section between φ and the thermal bath, see Appendix B. The thermalization rate now becomes During the time between the end of primordial inflation and primordial in- flation reheating, T ∝ a − 3 8 and H ∝ a − 3 2 and during radiation domination T ∝ a − 1 and H ∝ a − $^{2}$. Therefore, if the constraint Γ$_{therm}$ > H is satisfied at the time of the end of primordial inflation, then it is satisfied all the way up Γ$_{therm}$ ∼ g $^{4}$T (4.121) σ ∼ g 4 T 2 (4.120) E$_{c.m.}$ ∼ T (4.119) σ ∼ g 4 E 2 c.m. (4.118)