4.3 φ Decay Rate, Spectral Index and Tensor Fraction 54 (Sections 4.4.3.5 and 4.4.3.8 respectively) by the time cosmological scales exit the horizon during primordial inflation). In a similar fashion, the reason why the slow-roll parameter η depends only on ψ is because this parameter captures the dependance on the spectral index of the field(s) whose pertur- bations contribute to the observed primordial curvature perturbation ζ . In our case, this is only the ψ field. The definition of the running of the spectral index is [1] the second line coming from d ln ( k )=d ln ( aH ) ≃ H d t ≡ − d N , where k = aH . From Eq. (4.23) we have By differentiating the natural log of ϵ with respect to N , we obtain [1] d ln ( ϵ ) d N ≃ − 4 ϵ + 2 η (4.28) $_{n}$′$_{s}$ ≃ 2 d ϵ d N − 2 d η$_{ψψ}$ d N (4.26) ≃ 2 ϵ d ln ( ϵ ) d N − 2 d η$_{ψψ}$ d N (4.27) n ′ s ≡ d n$_{s}$ d ln ( k ) (4.24) ≃ − d n$_{s}$ d N (4.25)