4.4 “End of Inflation” Mechanism 94 We obtain an expression for $^{ϕ}$∗ in terms of $^{N}$∗ by using the equation We define the end of primordial inflation to be when ϵ =1 . This gives Therefore we have We also need to calculate η$_{ψψ}$ . Using our potential, Eq. (4.2), we obtain V$_{ψψ}$ at the time cosmological scales exit the horizon as Substituting Eq. (4.181) into Eq. (4.176) gives ϕ$_{end}$ = √$_{2}$$_{M}$$_{P}$ (4.178) $^{N}$∗ ≈ 1 M 2 P ∫ ϕ$_{∗}$ ϕ$_{end}$ V ( ϕ ) V ′ ( ϕ ) d ϕ (4.177) $^{N}$∗ ≈ 1 M 2 P ∫ ϕ$_{∗}$ √$_{2}$$_{M}$$_{P}$ V ( ϕ ) V ′ ( ϕ ) d ϕ (4.179) ≈ 1 2 M 2 P ∫ ϕ$_{∗}$ √$_{2}$$_{M}$$_{P}$ ϕ d ϕ (4.180) $^{ϕ}$∗ ≈ √ 4 $^{N}$∗ + 2 M$_{P}$ (4.181) ϵ ≈ 1 2 $^{N}$∗ + 1 (4.182) V$_{ψψ}$ | ∗ = m 2 ψ + ( 4 α 2 − 2 α ) h $^{2}$φ2 ∗ ( $^{ψ}$∗ M$_{P}$ ) 2 α − 2 (4.183)