4.3 φ Decay Rate, Spectral Index and Tensor Fraction 57 4.3.2.1 The Case Γ$_{ϕ}$ ≳ H TI We have 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 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- e N$_{∗}$ = $^{H}$∗ k ( T$_{reh,inf}$ T$_{end,inf}$ ) 8 3 T$_{start,TI}$ T$_{reh,inf}$ ( T$_{reh,TI}$ T$_{end,TI}$ ) 8 3 T$_{piv}$ T$_{reh,TI}$ e − N$_{TI}$ (4.41) H − 1 ∗ = $^{a}$∗ a$_{end,inf}$ a$_{end,inf}$ a$_{reh,inf}$ a$_{reh,inf}$ a$_{start,TI}$ a$_{start,TI}$ a$_{end,TI}$ a$_{end,TI}$ a$_{reh,TI}$ a$_{reh,TI}$ a$_{eq}$ a$_{eq}$ a$_{piv}$ H − 1 piv (4.39) = e − $_{N$_{∗}$}$a$_{end,inf}$ a$_{reh,inf}$ a$_{reh,inf}$ a$_{start,TI}$ e − $_{N$_{TI}$}$a$_{end,TI}$ a$_{reh,TI}$ a$_{reh,TI}$ a$_{eq}$ a$_{eq}$ a$_{piv}$ H − 1 piv (4.40)