content_image/299060.jpg content_image/299059.jpg content_image/299058.jpg
(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] 

<img src='content_image/299060.jpg'>

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 

<img src='content_image/299059.jpg'>

By differentiating the natural log of ϵ with respect to N , we obtain [1] 

<img src='content_image/299058.jpg'>

