Note that at the end of inflation, which is indicated by ¨ a = 0, or equivalently by ˙ H = − H $^{2}$, one obtains a constant Newtonian potential Φ$_{k}$ ≈ c$_{1}$ . This is perhaps the most significant result for a single field perturbation. In a long wavelength limit one obtains a constant of motion ζ [12, 174, 197] defined as 7 : This is also known as a comoving curvature perturbation [198] reads in the longitudinal gauge [12] for the slow-roll inflation as For CMB and structure formation we need to know the metric perturbation during the matter dominated era when the metric perturbation is Φ( t$_{f}$ ) ≈ (3 / 5) c$_{1}$ . Substituting the value of c$_{1}$ from Eq. (20), we obtain In a similar way it is also possible to show that the comoving curvature perturbations is given by where δ$_{φ}$ denotes the field perturbation on a spatially flat hypersurfaces, because on a comoving hypersurface δφ = 0, by definition. Therefore, on flat hypersurfaces where δt is the time displacement going from flat to comoving hypersurfaces [25, 158]. As a result Note that during matter dominated era the curvature perturbation and the metric pertur- bations are related to each other 7 If the equation of state for matter remains constant, there exists a simple relationship which connects the metric perturbations at two different times: Φ$_{k}$( t$_{f}$ ) = 1+ 2 $_{3}$(1+ w ( t$_{f}$ )) − 1 1+ 2 $_{3}$(1+ w ( t$_{i}$ )) − 1 Φ$_{k}$( t$_{i}$ ) [12, 174, 197]. 17 ζ = 2 3 H − 1 ˙ Φ$_{k}$ + Φ$_{k}$ 1 + w + Φ$_{k}$ , w = p ρ . (21) ζ$_{k}$ = Φ$_{k}$ − H 2 ˙ H ( $_{H}$− 1 ˙ Φ$_{k}$ + Φ$_{k}$ ) . (22) Φ$_{k}$( t$_{f}$ ) ≈ 3 5 H δφ$_{k}$ ˙ φ ∣ k = aH . (23) ζ$_{k}$ ≈ H ˙ φ δφ ∣ k = aH , (24) δφ$_{k}$ = ˙ φδt , (25) ζ$_{k}$ ≡ Hδt . (26) Φ$_{k}$ = − 3 5 ζ$_{k}$ . (27)