4.4 “End of Inflation” Mechanism 72 which is identical to Eq. (4.80) except for the difference in the limit. For the case Γ$_{ϕ}$ ≪ H$_{TI}$ , we would be saying that Substituting H$_{TI}$ , Eq. (4.10), into Eq. (4.104) gives which is identical to Eq. (4.81) except for the difference in the limit. As we are dealing with the thermal fluctuation of φ about φ =0 , we have 〈 δφ 〉 T = 〈 φ 〉 $_{T}$. The thermal fluctuation of φ is and we require A detailed derivation of this is given in Appendix A. In order to keep m$_{ψ,}$$_{eff}$ light, we require, from Section 4.4.3.4, 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, gT$_{1}$ ≫ m$_{0}$ (4.103) g ( M 2 P H$_{TI}$ Γ$_{ϕ}$ ) 1 4 ≫ m$_{0}$ (4.104) m$_{0}$ ≪ [ ( g $^{4}$Γ$_{ϕ}$ ) n +1 M 2 n +1 P √$_{λ}$ ] 1 3 n +2 (4.105) √ 〈 φ $^{2}$〉 T ∼ T (4.106) g ≪ 1 (4.107) hT$_{1}$ ( $^{ψ}$∗ M$_{P}$ ) α − 1 ≪ H$_{TI}$ (4.108)