# Gauge-Transformation Properties of Cosmological Observables and its Application to the Light-Cone Average

## TL;DR

This paper investigates the gauge-transformation properties of cosmological observables beyond linear order and derives a second-order expression for the light-cone average that remains invariant under diffeomorphisms.

## Contribution

It provides a general law for gauge transformations of cosmological observables beyond linear order and applies it to derive a gauge-invariant second-order light-cone average.

## Key findings

- Derived the second-order expression for light-cone averages in cosmology.
- Demonstrated the gauge-invariance of the second-order light-cone average.
- Provided a general law for gauge transformations of observables beyond linear order.

## Abstract

Theoretical descriptions of observable quantities in cosmological perturbation theory should be independent of coordinate systems. This statement is often referred to as gauge-invariance of observable quantities, and the sanity of their theoretical description is verified by checking its gauge-invariance. We argue that cosmological observables are invariant scalars under diffeomorphisms and as a consequence their theoretical description is gauge-invariant, only at linear order in perturbations. Beyond linear order, they are usually not gauge-invariant, and we provide the general law for the gauge-transformation that the perturbation part of an observable does obey. We apply this finding to derive the second-order expression for the observational light-cone average in cosmology and demonstrate that our expression is indeed invariant under diffeomorphisms.