# Sachs equations for light bundles in a cold plasma

## TL;DR

This paper generalizes Sachs equations to describe light bundle propagation in a cold plasma-filled universe, revealing modifications to cosmological distances and redshift relations without relying on Einstein's equations.

## Contribution

It introduces a new formulation of Sachs equations accounting for cold plasma effects, extending previous models to non-vacuum cosmological contexts.

## Key findings

- Modified reciprocity law in plasma environments
- Small changes in cosmological redshift predictions
- Altered Hubble law in plasma-filled cosmology

## Abstract

We study the propagation of light bundles in non-empty spacetime, as most of the Universe is filled by baryonic matter in the form of a (dilute) plasma. Here we restrict to the case of a cold (i.e., pressureless) and non-magnetised plasma. Then the influence of the medium on the light rays is encoded in the spacetime dependent plasma frequency. Our result for a general spacetime generalises the Sachs equations to the case of a cold plasma Universe. We find that the reciprocity law (Etherington theorem), the relation that connects area distance with luminosity distance, is modified. Einstein's field equation is not used, i.e., our results apply independently of whether or not the plasma is self-gravitating. As an example, our findings are applied to a homogeneous plasma in a Robertson-Walker spacetime. We find small modifications of the cosmological redshift of frequencies and of the Hubble law.