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

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 $2^1{P}^o-1^{1}S$ line, the He I] $2^3{P}^o-1^{1}S$ 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 $\sim 2$ 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.

Figure 1

Peebles $C$ factor [Eq. (11)] 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 $\mathrm{F}=1.14$. The same prescription for the evolution of the matter temperature is used in all cases, see Sec. 5e.

Figure 4

Fractional changes in the ionization history when including higher-order Lyman transitions and feedback between them, compared to the effective multilevel 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 transitions and frequency diffusion.

Figure 6

Changes in the recombination history when including two-photon decays and Raman scattering (no diffusion), compared to our “base” model. The line labeled “$2s$”shows the changes in $x_e$ when one properly accounts for stimulated $2s\to 1s$ decays, as well as absorptions of distortion photons and Raman scattering form $2s$. The other lines show the cumulative correction when adding two-photon transitions from higher levels.

Figure 7

Changes in the recombination history when including frequency diffusion in Lyman-$\alpha$, compared to a model with two-photon transitions but no diffusion.

Figure 8

Comparison of HyRec with the $\mathrm{MLA}+\mathrm{\text{two}}$-photon code of Hirata (2008) and to the results of Hirata & Forbes (2009) who account for $\mathrm{Ly}\alpha$ diffusion. See text for comments.

Figure 9

The function $\mathcal{E}({\tau }_{\mathrm{c}})$ (solid line) and our approximation (dashed line).

Figure 10

A comparison of the HyRec He $\mathrm{II}\to \mathrm{I}$ recombination history to that computed by the “full physics” code of Switzer & Hirata [33]. The maximum deviation is 0.3%.