content_image/312156.jpg content_image/312155.jpg content_image/312157.jpg content_image/312154.jpg
Therefore the δN formalism to third order gives 

<img src='content_image/312156.jpg'>

By substituting our mass definition and its differential, Eqs. (4.3) and (4.4), into Eq. (4.59) we obtain the power spectrum of the primordial curvature perturbation, which to first order is 

<img src='content_image/312155.jpg'>

It must be noted that although there will be perturbations in ψ that are generated during thermal inflation that will become classical due to the in- flation, the scales to which these correspond are much smaller than cosmo- logical scales, as thermal inflation lasts for only about 10 e-folds. Therefore we do not consider them here. 

A required condition for the perturbative expansion in Eq. (4.59) to be suitable is that each term is much smaller than the preceding one. This requirement gives 

## 4.4.1.2 The Case Γ$_{ϕ}$ ≪ H TI 

Substituting T$_{1}$ and T$_{2}$ , Eqs. (4.12) and (4.13) respectively, into Eq. (4.57) gives 1 

<img src='content_image/312157.jpg'>

<img src='content_image/312154.jpg'>

