content_image/297669.jpg content_image/297670.jpg content_image/297668.jpg content_image/297667.jpg
to the start of thermal inflation. Therefore we have the constraint 

Taking Eq. (4.121) with T ∼ ( M 2 P $^{H}$∗ Γ$_{ϕ}$) 1 4 gives 

<img src='content_image/297669.jpg'>

<img src='content_image/297670.jpg'>

We also require Γ$_{therm}$ > H to be satisfied throughout the whole of thermal inflation. Therefore, we have the constraint 

<img src='content_image/297668.jpg'>

Substituting H$_{TI}$ and T$_{2}$ , Eqs. (4.10) and (4.13) respectively, into the above gives n +1 

## 4.4.3.8 The Field Value $^{φ}$∗ 

<img src='content_image/297667.jpg'>

We consider three possible cases for the value of the thermal waterfall field φ during primordial inflation, with m$_{φ,}$$_{inf}$ being the effective mass of φ during primordial inflation: 

A) φ heavy, i.e. | m$_{φ,}$$_{inf}$ | ≫ $^{H}$∗ , in which φ rolls down to its VEV. 

B) φ light, i.e. | m$_{φ,}$$_{inf}$ | ≪ $^{H}$∗ , in which φ is at the Bunch-Davies value (to be explained below). 

C) φ light, in which a SUGRA correction to the potential is appreciable, 

