A New Calculation of the Recombination Epoch

TL;DR

This paper presents a highly accurate, level-by-level calculation of recombination for hydrogen and helium in the early universe, evolving ~300 atomic levels alongside electrons and the matter temperature. It reveals two key advances: a roughly 10% reduction in the residual ionization fraction $x_e$ at $z \,\lesssim\,800$ due to non-equilibrium excited-state dynamics in hydrogen, and a significantly slower HeI recombination that completes just before hydrogen recombination. To facilitate practical use, the authors derive an approximate set of equations—two ODEs for the hydrogen proton fraction and HeII fraction plus a temperature equation—that reproduce the full multi-level results with carefully chosen coefficients and a fudge factor $F$; they also provide guidance on when to employ the full model versus the approximate scheme. The results have direct implications for precise modeling of the CMB anisotropy power spectrum and the ionization history in arbitrary cosmologies, enabling fast, accurate computations without tracking hundreds of atomic levels.

Abstract

We have developed an improved recombination calculation of H, HeI, and HeII in the early Universe which involves a line-by-line treatment of each atomic level. We find two major differences compared with previous calculations. Firstly, the ionization fraction x_e is approximately 10% smaller for redshifts <~800, due to non-equilibrium processes in the excited states of H. Secondly, HeI recombination is much slower than previously thought, and is delayed until just before H recombines. We describe the basic physics behind the new results and present a simple way to reproduce our calculation. This should enable fast computation of the ionization history (and quantities such as the power spectrum of CMB anisotropies which depend on it) for arbitrary cosmologies, without the need to consider the hundreds of atomic levels used in our complete model.