content_image/307188.jpg content_image/307186.jpg content_image/307187.jpg
Then, from our mass definition m , Eq. (4.3), we obtain 

## 4.5.2 Constraining the Free Parameters 

<img src='content_image/307188.jpg'>

Regarding the constraints that appear in Section 4.4.3, all are relevant to this modulated decay scenario except for those that appear in Section 4.4.3.10, as we do not require that the transition from thermal inflation to thermal waterfall field oscillation be sufficiently fast here, as well as those appearing in Section 4.4.3.11, as the spectator field will not be oscillating in this scenario (see Section 4.5.2.2). 

## 4.5.2.1 Time for φ Decay 

After the end of thermal inflation, we require there to exist an amount of time, i.e. an amount of Universe expansion, prior to the decay of the φ field. This is so that the modulated decay rate mechanism can have an effect. If the φ field decayed immediately after the end of thermal inflation, the decay rate would not be effectively modulated. Therefore, we require 

Substituting Eqs. (4.10) and (4.195) into here gives 

<img src='content_image/307186.jpg'>

<img src='content_image/307187.jpg'>

