4.4 “End of Inflation” Mechanism 70 4.4.3.5 The Field Value $^{ψ}$∗ Substituting the observed spectrum value P $_{ζ}$( k$_{0}$ ) = 2 . 142 × 10 − 9 into both Eq. (4.60) and Eq. (4.64), with α ∼ 1 , gives the same constraint, which is This constraint automatically satisfies the requirement of a suitable perturba- tive expansion, Eq. (4.61). Substituting Eq. (4.92) into Eq. (4.77), regarding the dynamics of thermal inflation, gives Rearranging this for m$_{0}$ gives the constraint We require the field value of ψ to be much larger than its perturbation, i.e. $^{ψ}$∗ ≫ $^{δψ}$∗ , so that the perturbative approach is valid. Therefore we obtain, with $^{δψ}$∗ ∼ $^{H}$∗ , and $^{ψ}$∗ ≫ $^{H}$∗ (4.95) $^{δψ}$∗ $^{ψ}$∗ ≪ 1 (4.96) m$_{0}$ ≪ ( 10 $^{$^{4}$H}$∗ ) α h M α − 1 P (4.94) h ≫ ( 10 − 4 $^{H}$∗ ) α m$_{0}$ M α − 1 P (4.93) $^{ψ}$∗ ∼ ( 10 − $^{4}$m2$_{0}$M 2 α − 2 P h $^{$^{2}$H}$∗ ) 1 2 α$_{−}$ 1 (4.92)