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## 4.4.3.5 The Field Value $^{ψ}$∗ 

Substituting the observed spectrum value P $_{ζ}$( k$_{0}$ ) = 2 . 142 × 10 − 9 into both Eq. (4.60) and Eq. (4.64), with α ∼ 1 , gives the same constraint, which is 

<img src='content_image/311522.jpg'>

This constraint automatically satisfies the requirement of a suitable perturba- tive expansion, Eq. (4.61). Substituting Eq. (4.92) into Eq. (4.77), regarding the dynamics of thermal inflation, gives 

Rearranging this for m$_{0}$ gives the constraint 

<img src='content_image/311521.jpg'>

<img src='content_image/311520.jpg'>

We require the field value of ψ to be much larger than its perturbation, i.e. $^{ψ}$∗ ≫ $^{δψ}$∗ , so that the perturbative approach is valid. Therefore we obtain, with $^{δψ}$∗ ∼ $^{H}$∗ , 

and 

<img src='content_image/311518.jpg'>

<img src='content_image/311519.jpg'>

