Cosmological Perturbations and Invariant Observables in Geodesic Lightcone Coordinates

TL;DR

This paper develops gauge-invariant observables in cosmological perturbation theory using geodesic lightcone coordinates, providing explicit expressions and demonstrating their gauge independence in the full non-linear regime.

Contribution

It introduces a method to construct gauge-invariant observables in gravity with geodesic lightcone coordinates, showing their gauge independence and deriving explicit second-order expressions.

Findings

Gauge-invariant observables are fully gauge-independent in non-linear gravity.

Explicit linear and second-order expressions for key cosmological variables.

Linearized equations of motion follow from gauge-invariant Einstein's equations.

Abstract

We consider a recent approach to the construction of gauge-invariant relational observables in gravity in the context of cosmological perturbation theory. These observables are constructed using a field-dependent coordinate system, which we take to be geodesic lightcone coordinates. We show that the observables are gauge-independent in the fully non-linear theory, and that they have the expected form when one adopts the geodesic lightcone gauge for the metric. We give explicit expressions for the Sasaki-Mukhanov variable at linear order, and the Hubble rate -- as measured both by geodesic observers and by observers co-moving with the inflaton -- to second order. Moreover, we show that the well-known linearised equations of motion for the Sasaki-Mukhanov variable and the scalar constraint variables follow from the gauge-invariant Einstein's equations.