# Baryon Physics and Tight Coupling Approximation in Boltzmann Codes

## TL;DR

This paper derives covariant baryon equations for Boltzmann codes, improves the tight coupling approximation, and shows that using these equations can lead to 1% deviations in cosmological parameter estimates.

## Contribution

It provides two covariant derivations of baryon equations and implements an improved tight coupling approximation in Boltzmann codes, highlighting differences from previous methods.

## Key findings

- Covariant baryon equations satisfy Bianchi identities.
- Improved tight coupling approximation up to second order.
- 1% deviation in cosmological parameters using the new equations.

## Abstract

We provide two derivations of the baryonic equations that can be straightforwardly implemented in existing Einstein--Boltzmann solvers. One of the derivations begins with an action principle, while the other exploits the conservation of the stress-energy tensor. While our result is manifestly covariant and satisfies the Bianchi identities, we point out that this is not the case for the implementation of the seminal work by Ma and Bertschinger and in the existing Boltzmann codes. We also study the tight coupling approximation up to the second order without choosing any gauge using the covariant full baryon equations. We implement the improved baryon equations in a Boltzmann code and investigate the change in the estimate of cosmological parameters by performing an MCMC analysis. With the covariantly correct baryon equations of motion, we find 1% deviation for the best fit values of the cosmological parameters that should be taken into account. While in this paper, we study the Lambda-CDM model only, our baryon equations can be easily implemented in other models and various modified gravity theories.