content_image/326336.jpg content_image/326335.jpg
However, for all viable parameter values in our model, Eqs. (4.149) and (4.150) are both individually never the dominant constraint when we consider them alongside all of the other constraints that are detailed in this thesis for this Thermal Inflation model. 

## $^{φ}$∗ Case C 

We do not have any additional constraints here for this case, as φ = 0 . 

## 4.4.3.10 Time of Transition from Thermal Inflation to Thermal Waterfall Field Oscillation 

In order for the equations of the δN formalism that are derived within the context of the “end of inflation” mechanism to be valid, we require the tran- sition from thermal inflation to thermal waterfall field oscillation to be suf- ficiently fast [34]. More specifically, we require 

<img src='content_image/326336.jpg'>

where Δ t ≡ t$_{2}$ − t$_{1}$ is the time taken for the transition to occur and $^{δt$_{1}$}$→ 2 is the proper time between a uniform energy density spacetime slice just before the transition at t$_{1}$ and one just after the transition at t$_{2}$ when φ starts to oscillate around its VEV. Qualitatively, we require the thickness of the transition slice to be much smaller than its warping. The primordial curvature perturbation that is generated by the “end of inflation” mechanism is given by 

<img src='content_image/326335.jpg'>

