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## 4.4.3.9 Energy Density of the Thermal Waterfall Field 

We require the energy density of φ to be subdominant at all times, in order that it does not cause any inflation by itself. During the period between the end of primordial inflation and the start of thermal inflation, the energy density of φ is 

<img src='content_image/313523.jpg'>

the second line coming from the thermal fluctuation of φ , which is ∼ T . Therefore, considering the Friedmann equation, we require 

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During the time between the end of primordial inflation and primordial in- flation reheating, T ∝ a − 3 8 and H ∝ a − 3 2 and during radiation domination T ∝ a − 1 and H ∝ a − $^{2}$. Therefore, if Eq. (4.137) is satisfied, then equiva- lent constraints for higher T and H are guaranteed to be satisfied as well. For the case Γ$_{ϕ}$ ≳ H$_{TI}$ , by substituting H$_{TI}$ and T$_{1}$ , Eqs. (4.10) and (4.11) respectively, into Eq. (4.137) we obtain 

<img src='content_image/313525.jpg'>

which is the same constraint as Eq. (4.107). For the case Γ$_{ϕ}$ ≪ H$_{TI}$ , by substituting H$_{TI}$ and T$_{1}$ , Eqs. (4.10) and (4.12) respectively, into Eq. (4.137) 

