4.5 Modulated Decay Rate 107 Therefore we require and These two constraints are identical to Eqs. (4.171) and (4.172) that appear in Section 4.4.3.11. This is easy to understand, in that the only difference between the derivation of the two constraints in Section 4.4.3.11 and those that appear here, is that, in the latter, we start with ψ taking its frozen value $^{ψ}$∗ , whereas in the former, we start with ψ taking its average oscillation value. However, the average oscillation value is just $^{ψ}$∗ . Substituting 〈 φ 〉 , $^{ψ}$∗ and Γ , Eqs. (4.7), (4.92) and (4.195) respectively, into Eq. (4.216) gives the constraint which is identical to Eq. (4.173). In order for Eqs. (4.210) and (4.217) to both be satisfied, we require the upper bound in Eq. (4.210) to be much larger than the lower bound in Eq. (4.217). This gives 4.5.3 Results m$_{ψ}$ ≪ M$_{P}$ Γ $^{ψ}$∗ (4.215) h 〈 φ 〉 ψ α ∗ ≪ M α $_{P}$Γ (4.216) 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}$$_{P}$ ) 2 α (4.217) m$_{0}$ ≪ √ (2 n + 4) λ ( 100 g 2 $^{√$_{H}$$_{∗}$}$) n +1 √$_{M}$$_{P}$ n − 1 (4.218)