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## 4.3.2.1 The Case Γ$_{ϕ}$ ≳ H TI 

We have 

<img src='content_image/305784.jpg'>

<img src='content_image/305785.jpg'>

where N$_{TI}$ is the number of e-folds of thermal inflation and the scale factors are the following: a$_{end,inf}$ is at the end of primordial inflation, a$_{reh,inf}$ is at pri- mordial inflation reheating, a$_{start,TI}$ is at the start of thermal inflation, a$_{end,TI}$ is at the end of thermal inflation, a$_{reh,TI}$ is at thermal inflation reheating and a$_{eq}$ is at the time of matter-radiation equality. For the period between the end of primordial/thermal inflation and primordial/thermal inflation reheat- ing, a goes as T − 8 3 . The proof is as follows. During this time, T ∼ ( M 2 $_{P}$H Γ) 1 4 . As H goes as t − 1 we have T ∝ t − 1 4 . During the field oscillations, the Universe is matter dominated and so we have a ∝ t 2 3 . Therefore t ∝ a 3 2 . Putting this all together we find T ∝ t − 1 4 ∝ a − 3 8 and therefore a ∝ T − 8 3 . For all other times, a goes as T − 1 and so we have 

<img src='content_image/305783.jpg'>

We need to calculate T$_{piv}$ . We consider the period between when the pivot scale reenters the horizon and the present. Throughout this period the Uni- 

