3.2 Cosmic Inflation 39 in quasi-de Sitter inflation we have H ≈ constant , i.e. H varies extremely slowly with scale. We now transform from cosmic time t to conformal time and also consider instead the comoving field perturbation Eq. (3.40) now becomes where where we have assumed a ( t ) ∝ e $^{Ht}$. We have thus obtained a harmonic oscil- lator scenario. Following the usual procedure of QFT, we will now promote variables to operators and quantize this harmonic oscillator. We express the comoving perturbation in terms of Fourier components as where ˆ a † and ˆ a are creation and annihilation operators respectively, that satisfy d $^{2}$ϕ k ( η ) d η 2 + ω 2 $_{k}$( η ) ϕ k ( η ) = 0 (3.44) ω 2 $_{k}$( η ) = k 2 − 2 η 2 (3.45) ˆ ϕ k ( η ) = ϕ$_{k}$ ( η )ˆ a ( k ) + ϕ ∗ $_{k}$( η )ˆ a $^{†$_{(}$}$− k ) (3.46) [ ˆ a ( k ) , ˆ a †$_{(}$$_{k}$′$_{)}$ ] = (2 π ) $^{3}$δ$^{3}$( k − $_{k}$′$_{)}$ (3.47) η = − 1 $^{aH}$∗ (3.42) ϕ ≡ aδφ (3.43)