content_image/321601.jpg content_image/321602.jpg content_image/321603.jpg
T$_{1}$ corresponds to the temperature when the potential energy density becomes comparable with the energy density of the thermal bath, for which the density is ρ$_{γ}$ ∼ T $^{4}$. In the case of Γ$_{ϕ}$ ≪ H$_{TI}$ , thermal inflation will begin at a temperature 

In both cases, thermal inflation ends at a temperature 

<img src='content_image/321601.jpg'>

<img src='content_image/321602.jpg'>

T$_{2}$ corresponds to the temperature when the tachyonic mass term of the thermal waterfall field becomes equal to the thermally-induced mass term in Eq. (4.5). 

## 4.3 φ Decay Rate, Spectral Index and Tensor Fraction 

## 4.3.1 φ Decay Rate 

The decay rate of φ is given by 

<img src='content_image/321603.jpg'>

