Photons and Baryons before Atoms: Improving the Tight-Coupling Approximation

TL;DR

This paper evaluates and improves the tight-coupling approximation used in early universe cosmology, introducing a second-order method that enhances accuracy and computational efficiency in modeling photon-baryon interactions before recombination.

Contribution

The authors develop a second-order tight-coupling approximation that closely matches exact solutions and reduces computational time in CMB spectrum calculations.

Findings

Second-order approximation closely tracks full solutions

Bias on cosmological parameters is negligible with first-order

Computational time for CMB spectra can be reduced by ~17%

Abstract

Prior to recombination photons, electrons, and atomic nuclei rapidly scattered and behaved, almost, like a single tightly-coupled photon-baryon plasma. We investigate here the accuracy of the tight-coupling approximation commonly used to numerically evolve the baryon and photon perturbation equations at early times. By solving the exact perturbations equations with a stiff solver starting deep in the radiation-dominated epoch we find the level of inaccuracy introduced by resorting to the standard first-order tight-coupling approximation. We develop a new second-order approximation in the inverse Thomson opacity expansion and show that it closely tracks the full solution, at essentially no extra numerical cost. We find the bias on estimates of cosmological parameters introduced by the first-order approximation is, for most parameters, negligible. Finally, we show that our second-order approximation can be used to reduce the time needed to compute cosmic microwave background angular spectra by as much as ~17%.