4.4 “End of Inflation” Mechanism 95 Therefore we obtain η$_{ψψ}$ as Our final result for the spectral index is therefore From Section 4.3.2, the running of the spectral index n ′ s is given by with η being given by Eq. (4.29), which is to be evaluated at the point where cosmological scales exit the horizon during primordial inflation. The potential of Eq. (4.174) gives which is identical to the value of ϵ , Eq. (4.176). Substituting Eq. (4.181) into Eq. (4.187) gives Our final result for the running of the spectral index is therefore η ≈ 1 2 $^{N}$∗ + 1 (4.188) η = 2 M 2 P ϕ 2 ∗ (4.187) n ′ s ≃ − 8 ϵ $^{2}$+ 4 ϵη + 4 ϵη$_{ψψ}$ (4.186) η$_{ψψ}$ = 1 3 H 2 ∗   m 2 ψ + ( 4 α 2 − 2 α ) h $^{2}$φ2 ∗ ( $^{ψ}$∗ M$_{P}$ ) 2 α − 2   (4.184) n$_{s}$ ≈ 1 − 2 2 $^{N}$∗ + 1 + 2 3 H 2 ∗   m 2 ψ + ( 4 α 2 − 2 α ) h $^{2}$φ2 ∗ ( $^{ψ}$∗ M$_{P}$ ) 2 α − 2   (4.185) n ′ s ≈ − 4 (2 $^{N}$∗ + 1) 2 + 4 (6 $^{N}$∗ + 3) H 2 ∗   m 2 ψ + ( 4 α 2 − 2 α ) h $^{2}$φ2 ∗ ( $^{ψ}$∗ M$_{P}$ ) 2 α − 2   (4.189)