Wronskian formulation of the spectrum of curvature perturbations

Abstract

We present a new formulation for the evaluation of the primordial spectrum of curvature perturbations generated during inflation, using the fact that the Wronskian of the scalar field perturbation equation is constant. In the literature, there are many works on the same issue focusing on a few specific aspects or effects. Here we deal with the general multi-component scalar field, and show that our new formalism gives a method for evaluating the final amplitude of the curvature perturbation systematically and economically. The advantage of the new method is that one only has to solve a single mode of the scalar field perturbation equation backwards in time from the end of inflation to the stage at which the perturbation is within the Hubble horizon, at which the initial values of the scalar field perturbations are given. We also clarify the relation of the new method with the new, recently developed delta N formalism.