Generating the curvature perturbation at the end of inflation

a. Introduction The primordial curvature perturbation ζ exists already a few Hubble times before the observable Universe enters the horizon. Its timeindependent value at that stage sets the initial condition for the subsequent evolution of all perturbations in the scalar and adiabatic mode, which in turn seem to the dominant (perhaps the only) cause of inhomogeneity in the Universe. The curvature perturbation is almost gaussian, with an almost scale-independent spectrum Pζ = (5 × 10). The curvature perturbation is supposed to originate from the vacuum fluctuations during inflation of one or more light scalar fields, which on each scale are promoted to classical perturbations around the time of horizon exit. Different proposals have been made regarding when and how this happens. In this note I explore a new possibility; that the significant, and possibly dominant, contribution to the curvature perturbation is generated at the end of inflation. b. Generating the curvature perturbation For clarity I assume Einstein gravity, slow-roll inflation and canonically normalized fields. None of these essential for the generation of the curvature perturbation. What is essential though, to make the curvature perturbation almost scale-independent, is that inflation is almost exponential while cosmological scales leave the horizon, corresponding to a Hubble parameter H which varies slowly on the Hubble timescale. The slow-roll inflationary trajectory will lie in a subspace of field space, spanned by one or more light fields which are the components of the inflaton. Any additional light fields have a negligible effect on the inflationary trajectory. We are going to be interested in Fourier modes of the curvature perturbation ζ with a given wavenumber k of cosmological interest. To evaluate them we can smooth the scalar fields on a scale of order 1/k (to be precise, on a somewhat smaller scale). This scale leaves the horizon at some epoch k = aH , where H = ȧ/a is the almostconstant inflationary Hubble parameter. At horizon exit the canonically-normalized fields can be defined so that one of them points along the trajectory; if the inflaton has more than one component I call