4.5 Modulated Decay Rate 106 Substituting 〈 φ 〉 , $^{ψ}$∗ and Γ , Eqs. (4.7), (4.92) and (4.195) respectively, into Eq. (4.209) gives the constraint 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 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 ρ$_{ψ}$ ≪ M 2 $_{P}$Γ 2 (4.213) h 2 ψ 2 α ∗ M 2 α − 2 P 〈 φ 〉 2 + 1 2 m 2 $_{ψ}$ψ 2 ∗ ≪ M 2 $_{P}$Γ 2 (4.214) ρ$_{ψ}$ = h 2 ψ 2 α ∗ M 2 α − 2 P φ 2 + 1 2 m 2 $_{ψ}$ψ 2 ∗ (4.211) ∼ h 2 ψ 2 α ∗ M 2 α − 2 P 〈 φ 〉 2 + 1 2 m 2 $_{ψ}$ψ 2 ∗ (4.212) h ≪ ( g $^{2}$m$_{0}$ M α − 1 P ) 2 α − 1   √ (2 n + 4) λ m$_{0}$M n P  $^{2}$$^{α$_{−}$}$$^{1}$ n +1 ( $_{100}$√$_{H}$$_{∗}$ m$_{0}$M α − 1 P ) 2 α − 2 (4.210) 4.5.2.3 Energy Density of the Spectator Field