An exact Jacobi map in the geodesic light-cone gauge

TL;DR

This paper derives an exact expression for the Jacobi map in the geodesic light-cone gauge, enabling precise calculations of light propagation effects in cosmology, and compares it across different gauges to improve understanding of inhomogeneities.

Contribution

It provides an explicit, exact form of the Jacobi map in the GLC gauge and demonstrates how to express it in other gauges, enhancing precision in cosmological light propagation studies.

Findings

Exact Jacobi map expression in GLC gauge

Factorization of the Jacobi map into local quantities

Consistency between different gauge calculations

Abstract

The remarkable properties of the recently proposed geodesic light-cone (GLC) gauge allow to explicitly solve the geodetic-deviation equation, and thus to derive an exact expression for the Jacobi map J^A_B(s,o) connecting a generic source s to a geodesic observer o in a generic space time. In this gauge J^A_B factorizes into the product of a local quantity at s times one at o, implying similarly factorized expressions for the area and luminosity distance. In any other coordinate system J^A_B is simply given by expressing the GLC quantities in terms of the corresponding ones in the new coordinates. This is explicitly done, at first and second order, respectively, for the synchronous and Poisson gauge-fixing of a perturbed, spatially-flat cosmological background, and the consistency of the two outcomes is checked. Our results slightly amend previous calculations of the luminosity-redshift relation and suggest a possible non-perturbative way for computing the effects of inhomogeneities on observations based on light-like signals.