On matching conditions for cosmological perturbations

TL;DR

This work develops gauge-consistent matching conditions for cosmological perturbations across sharp transitions in the equation of state within a flat FRW universe. By enforcing matching on hypersurfaces of constant energy, it derives explicit conditions in both longitudinal and synchronous gauges and demonstrates that the commonly understood conservation of curvature perturbations on large scales remains valid when the transition physics is properly specified. The authors apply the framework to concrete scenarios, showing that scalar spectra need not be scale-invariant and that the evolution of the decaying mode in string cosmology is strongly constrained by the transition surface. The results clarify previous misconceptions, provide robust tools for analyzing transitions in inflationary and string-inspired cosmologies, and emphasize the practical advantage of gauge-invariant or longitudinal formulations for perturbation theory.

Abstract

We derive the matching conditions for cosmological perturbations in a Friedmann Universe where the equation of state undergoes a sharp jump, for instance as a result of a phase transition. The physics of the transition which is needed to follow the fate of the perturbations is clarified. We dissipate misleading statements made recently in the literature concerning the predictions of the primordial fluctuations from inflation and confirm standard results. Applications to string cosmology are considered.