Conformal Fermi Coordinates

TL;DR

This paper introduces Conformal Fermi Coordinates (CFC), a new coordinate system that extends Fermi Normal Coordinates to larger scales, enabling precise analysis of long- and short-wavelength mode interactions in cosmology.

Contribution

The paper presents CFC, a generalization of Fermi Normal Coordinates that remains valid outside the horizon, simplifying calculations of mode couplings in cosmology.

Findings

CFC allows analysis of mode coupling on all scales larger than the sound horizon.

The method clarifies the physical interpretation of long-wavelength tensor effects.

Previous results on tensor mode effects are recovered and clarified.

Abstract

Fermi Normal Coordinates (FNC) are a useful frame for isolating the locally observable, physical effects of a long-wavelength spacetime perturbation. Their cosmological application, however, is hampered by the fact that they are only valid on scales much smaller than the horizon. We introduce a generalization that we call Conformal Fermi Coordinates (CFC). CFC preserve all the advantages of FNC, but in addition are valid outside the horizon. They allow us to calculate the coupling of long- and short-wavelength modes on all scales larger than the sound horizon of the cosmological fluid, starting from the epoch of inflation until today, by removing the complications of the second order Einstein equations to a large extent, and eliminating all gauge ambiguities. As an application, we present a calculation of the effect of long-wavelength tensor modes on small scale density fluctuations. We recover previous results, but clarify the physical content of the individual contributions in terms of locally measurable effects and "projection" terms.