4.4 “End of Inflation” Mechanism 86 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 Therefore we require and Substituting 〈 φ 〉 , Γ and $^{ψ}$∗ , Eqs. (4.7), (4.18) and (4.92) respectively, into Eq. (4.172) gives the constraint ρ$_{ψ,}$$_{osc}$ = h 2 ψ 2 α M 2 α − 2 P φ 2 + 1 2 m 2 $_{ψ}$ψ 2 (4.167) ∼ h 2 ψ 2 α ∗ M 2 α − 2 P 〈 φ 〉 2 + 1 2 m 2 $_{ψ}$ψ 2 ∗ (4.168) ρ$_{ψ,}$$_{osc}$ ≪ M 2 $_{P}$Γ 2 (4.169) h 2 ψ 2 α ∗ M 2 α − 2 P 〈 φ 〉 2 + 1 2 m 2 $_{ψ}$ψ 2 ∗ ≪ M 2 $_{P}$Γ 2 (4.170) m$_{ψ}$ ≪ M$_{P}$ Γ $^{ψ}$∗ (4.171) h 〈 φ 〉 ψ α ∗ ≪ M α $_{P}$Γ (4.172) h ≫ 1 ( g $^{2}$m$_{0}$ ) 2 α − 1   m$_{0}$M n P √ (2 n + 4) $_{λ}$ $^{2}$$^{α$_{−}$}$$^{1}$ n +1 ( m$_{0}$ $_{100}$√$_{H}$$_{∗}$$_{M}$$_{P}$ ) 2 α (4.173)