Solutions of gauge invariant cosmological perturbations in long-wavelength limit

TL;DR

The paper addresses gauge-invariant perturbations in a flat FRW universe with scalar fields in the long-wavelength limit, introducing a method to obtain the 0-mode solution by differentiating the background with respect to parameters in a mini-superspace/Hamilton-Jacobi framework. This approach yields a Mukhanov-type equation for the perturbations and provides explicit 0-mode solutions for both single- and multi-field cases, relating the curvature perturbation ${\cal R}$ to the background derivatives and the Mukhanov variables. It further demonstrates that, in inflationary scenarios, these 0-mode solutions reproduce slow-roll results and separate into adiabatic and isocurvature components, with clear expressions for specific potential forms. The method avoids solving the perturbation equation directly, offering a simpler and gauge-invariant route to assess long-wavelength perturbations and their implications for reheating and beyond.

Abstract

We investigate gauge invariant cosmological perturbations in a spatially flat Friedman-Robertson-Walker universe with scalar fields. It is well known that the evolution equation for the gauge invariant quantities has exact solutions in the long-wavelength limit. We find that these gauge invariant solutions can be obtained by differentiating the background solution with respect to parameters contained in the background system. This method is very useful when we analyze the long-wavelength behavior of cosmological perturbation with multiple scalar fields.