4.4 “End of Inflation” Mechanism 61 culate distinct observational signatures (such as non-Gaussianity) that may test this scenario in the near future. 4.4.1 Generating ζ As φ is coupled to ψ , the “end of inflation” mechanism will generate a con- tribution to the primordial curvature perturbation ζ [31]. We will use the δN formalism to calculate this contribution. In this formalism, the final coordinate slice is the transition slice, going from thermal inflation to field oscillation. The δN formalism allows us to calculate the primordial curvature perturbation as The number of e-folds between the start and end of thermal inflation is given by where a$_{1}$ = a$_{start,TI}$ and a$_{2}$ = a$_{end,TI}$ . Substituting T$_{1}$ and T$_{2}$ , Eqs. (4.11) and (4.13) respectively, into Eq. (4.57) gives N$_{TI}$ ≃ ln  $^{√$_{2}$$_{gV}$}$ 1 4 0 m   (4.58) N$_{TI}$ = ln ( a$_{2}$ a$_{1}$ ) = ln ( T$_{1}$ T$_{2}$ ) (4.57) ζ = δN$_{TI}$ = d N$_{TI}$ d m δm + 1 2! d $^{2}$N$_{TI}$ d m 2 δm $^{2}$+ 1 3! d $^{3}$N$_{TI}$ d m 3 δm $^{3}$+ ... (4.56) 4.4.1.1 The Case Γ$_{ϕ}$ ≳ H TI