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 Λ CDM model only, our baryon equations can be easily implemented in other models and various modified gravity theories.
Highlights
Studies in modern cosmology heavily rely on cosmological linear perturbation theory [1,2].To understand the evolution of these linear perturbations, Einstein’s equations are solved numerically at the linear order in perturbations around a homogeneous and isotropic background
We find that the new equations, solved by Cosmic Linear Anisotropy Solving System (CLASS), give some deviations from the previous results, but for the parameter estimation of the ΛCDM model, such deviations are inside the statistical error bars
Since the baryon equations of motion MB66–67 are not compatible with MB29–30, they end up breaking general covariance
Summary
Studies in modern cosmology heavily rely on cosmological linear perturbation theory [1,2]. This fix solves all three problems mentioned above We use these new baryon equations of motion in order to derive the tight coupling approximation equations up to the second order. We implement these corrections for the baryon evolution, in the CLASS code. Since the baryon equations of motion MB66–67 (the ones used in Boltzmann codes) are not compatible with MB29–30 (which instead follow from conservation of energy-momentum), they end up breaking general covariance. The fact that the equations of motion implemented for the baryon in existing Boltzmann codes do not satisfy the stress-energy-momentum conservation (and general covariance) is due to an approximation (which modified equations MB29–30 into MB66–67) that we want to discuss here.
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