content_image/337809.jpg content_image/337810.jpg content_image/337811.jpg content_image/337812.jpg content_image/337813.jpg content_image/337814.jpg content_image/337815.jpg
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 : 

<img src='content_image/337809.jpg'>

This is also known as a comoving curvature perturbation [198] reads in the longitudinal gauge [12] for the slow-roll inflation as 

<img src='content_image/337810.jpg'>

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 

<img src='content_image/337811.jpg'>

<img src='content_image/337812.jpg'>

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 

<img src='content_image/337813.jpg'>

where δt is the time displacement going from flat to comoving hypersurfaces [25, 158]. As a result 

<img src='content_image/337814.jpg'>

Note that during matter dominated era the curvature perturbation and the metric pertur- bations are related to each other 

<img src='content_image/337815.jpg'>

