4.3 φ Decay Rate, Spectral Index and Tensor Fraction 55 where η is a slow-roll parameter given by where V ′′ ( ϕ ) is the second derivative of the inflaton potential with respect to the inflaton field ϕ . η is to be evaluated at the point where cosmological scales exit the horizon during primordial inflation. By differentiating η$_{ψψ}$ with respect to N using the quotient rule, we obtain with the second line coming from the fact that we have V$_{ψψ}$ not depending on N , as we are assuming that both ψ and φ have settled to a constant value (Sections 4.4.3.5 and 4.4.3.8 respectively) by the time cosmological scales exit the horizon during primordial inflation. By differentiating the natural log of H with respect to N , we obtain [1] d ln ( H ) d N ≃ ϵ (4.34) η ≡ M 2 P V ′′ ( ϕ ) V ( ϕ ) (4.29) d η$_{ψψ}$ d N = 3 H 2 d V$_{ψψ}$ d N − 3 V$_{ψψ}$ d H 2 d N 9 H 4 (4.30) = − V$_{ψψ}$ d H 2 d N 3 H 4 (4.31) = − η$_{ψψ}$ 1 H 2 d H 2 d N (4.32) = − 2 η$_{ψψ}$ d ln ( H ) d N (4.33)