4.4 “End of Inflation” Mechanism 62 Therefore the δN formalism to third order gives By substituting our mass definition and its differential, Eqs. (4.3) and (4.4), into Eq. (4.59) we obtain the power spectrum of the primordial curvature perturbation, which to first order is It must be noted that although there will be perturbations in ψ that are generated during thermal inflation that will become classical due to the in- flation, the scales to which these correspond are much smaller than cosmo- logical scales, as thermal inflation lasts for only about 10 e-folds. Therefore we do not consider them here. A required condition for the perturbative expansion in Eq. (4.59) to be suitable is that each term is much smaller than the preceding one. This requirement gives Substituting T$_{1}$ and T$_{2}$ , Eqs. (4.12) and (4.13) respectively, into Eq. (4.57) gives 4.4.1.2 The Case Γ$_{ϕ}$ ≪ H TI h $^{$^{2}$H}$∗ ψ 2 α − 1 m $^{2}$M 2 α − 2 P ≪ 1 (4.61) P 1 2 ζ = αh $^{$^{2}$H}$∗ ψ 2 α − 1 πm $^{2}$M 2 α − 2 P (4.60) ζ = δN$_{TI}$ = − δm m + 1 2 δm 2 m 2 − 1 3 δm 3 m 3 (4.59) N$_{TI}$ ≃ 8 3 ln  $^{√$_{2}$$_{g}$$_{(}$$_{M}$}$$^{2}$ P H$_{TI}$ Γ$_{ϕ}$) 1 4 m   (4.62)