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Substituting 〈 φ 〉 , $^{ψ}$∗ and Γ , Eqs. (4.7), (4.92) and (4.195) respectively, into Eq. (4.209) gives the constraint 

<img src='content_image/296598.jpg'>

## 4.5.2.3 Energy Density of the Spectator Field 

We require the energy density of ψ to be subdominant after thermal inflation up until it decays, in order that it does not cause any inflation by itself. The energy density of ψ after thermal inflation is 

<img src='content_image/296597.jpg'>

For simplicity, we assume that ψ decays around the same time as φ , i.e. that H does not change much between the time when φ decays and the time when ψ decays. Therefore, the energy density of the Universe at the time when ψ decays is ∼ M 2 $_{P}$Γ $^{2}$. We therefore require 

<img src='content_image/296596.jpg'>

<img src='content_image/296595.jpg'>

