We have seen how the vacuum fluctuation of each light scalar field is converted to a classical perturbation at the time of horizon exit. One or more of these perturbations should in turn generate the curvature perturbation ζ, that is probed by observation when cosmological scales begin to enter the horizon.

In this chapter we are going to consider the simplest scenario, where the curvature perturbation already achieves its final value a few Hubble times after horizon exit. In other words, we are going to consider the scenario in which ζ is constant during the entire era when the smoothing scale is outside the horizon. According to Section 5.4.2, this means that the locally defined pressure is a unique function of the locally defined energy density throughout that era. This is achieved in a single-field inflation model, if at each position the field value a few Hubble times after horizon exit determines the subsequent pressure and energy density. During inflation, this is the same thing as saying that the trajectory φ(x, t) is independent of x, up to a shift in t. In other words, it is the same as saying that the inflationary trajectory is an attractor. After inflation though, it would be possible for some other light field to play a significant role. We shall assume in this chapter that such is not the case.