HyRec: A fast and highly accurate primordial hydrogen and helium recombination code

TL;DR

HyRec tackles the accurate computation of primordial hydrogen and helium recombination histories in the era of precision CMB measurements. It combines an effective multi-level atom framework with a fast, non-perturbative treatment of radiative transfer, including two-photon processes and Ly$\alpha$ frequency diffusion, to deliver histories in about $\sim 2$ seconds while achieving $\lesssim 10^{-4}$–level accuracy for hydrogen and $\lesssim 10^{-3}$ for helium. A key innovation is the effective four- to multi-level reduction (EMLA) that folds the interior high-$n$ states into a small set of interface states via precomputed effective rates, enabling efficient coupling to a radiative-transfer solver that evolves the radiation field simultaneously with level populations. The helium sector is handled with a fast, physically motivated approximation that captures the dominant line and continuum processes, and is cross-validated against detailed calculations to within a percent level. Overall, HyRec provides a robust, publicly available tool for high-precision recombination histories essential for interpreting current and upcoming CMB data without large computational costs.

Abstract

We present a state-of-the-art primordial recombination code, HyRec, including all the physical effects that have been shown to significantly affect recombination. The computation of helium recombination includes simple analytic treatments of hydrogen continuum opacity in the He I 2 1P - 1 1S line, the He I] 2 3P - 1 1S line, and treats feedback between these lines within the on-the-spot approximation. Hydrogen recombination is computed using the effective multilevel atom method, virtually accounting for an infinite number of excited states. We account for two-photon transitions from 2s and higher levels as well as frequency diffusion in Lyman-alpha with a full radiative transfer calculation. We present a new method to evolve the radiation field simultaneously with the level populations and the free electron fraction. These computations are sped up by taking advantage of the particular sparseness pattern of the equations describing the radiative transfer. The computation time for a full recombination history is ~2 seconds. This makes our code well suited for inclusion in Monte Carlo Markov chains for cosmological parameter estimation from upcoming high-precision cosmic microwave background anisotropy measurements.

Figures (10)

• Figure 1: Peebles $C$-factor [Eq. (\ref{['eq:Peebles C']})] and ratio of the population of the $n=2$ shell to its value in Saha equilibrium with the continuum, as a function of redshift.
• Figure 2: Schematic representation of the hydrogen atom, with the nomenclature used in this paper. Slow transitions from the "weak interface" states to the ground state are counted as transitions from the $n=2$ state with which they are in equilibrium.
• Figure 3: Fractional changes in the ionization history relative to the effective three-level atom model. The "RecFast" model is an effective three-level atom with the case-B recombination coefficient multiplied by a fudge factor F = 1.14. The same prescription for the evolution of the matter temperature is used in all cases, see Sec.\ref{['sec:results']}.
• Figure 4: Fractional changes in the ionization history when including higher-order Lyman transitions and feedback between them, compared to the effective multi-level atom model with $2s$ and $2p$ only.
• Figure 5: Sparsity pattern of the linear system solved for evolving simultaneously the level populations and the radiation field, in the presence of two-photon transtions and frequency diffusion.