Ultrafast effective multilevel atom method for primordial hydrogen recombination

Cosmological hydrogen recombination has recently been the subject of renewed attention because of its importance for predicting the power spectrum of cosmic microwave background anisotropies. It has become clear that it is necessary to account for a large number $n\gtrsim 100$ of energy shells of the hydrogen atom, separately following the angular momentum substates in order to obtain sufficiently accurate recombination histories. However, the multilevel atom codes that follow the populations of all these levels are computationally expensive, limiting recent analyses to only a few points in parameter space. In this paper, we present a new method for solving the multilevel atom recombination problem, which splits the problem into a computationally expensive atomic physics component that is independent of the cosmology and an ultrafast cosmological evolution component. The atomic physics component follows the network of bound-bound and bound-free transitions among excited states and computes the resulting effective transition rates for the small set of “interface” states radiatively connected to the ground state. The cosmological evolution component only follows the populations of the interface states. By pretabulating the effective rates, we can reduce the recurring cost of multilevel atom calculations by more than 5 orders of magnitude. The resulting code is fast enough for inclusion in Markov chain Monte Carlo parameter estimation algorithms. It does not yet include the radiative transfer or high-$n$ two-photon processes considered in some recent papers. Further work on analytic treatments for these effects will be required in order to produce a recombination code usable for Planck data analysis.

Figure 1

Schematic representation of the formulation of the recombination problem adopted in this work. Dotted arrows represent possibly numerous fast transitions within the interior.

Figure 2

Left panel: “Exact fudge factor” as a function of redshift ${\mathcal{A}}_{\mathrm{B}}(T_{\mathrm{m}},T_{\mathrm{r}})/{\alpha }_{\mathrm{B}}(T_{\mathrm{m}})$, for several values of $n_{\max }$, using $T_{\mathrm{m}}(z)$ computed by RecSparse for cosmological parameters as in Ref. 10. We use the fit of Ref. 41 for the case-B recombination coefficient ${\alpha }_{\mathrm{B}}(T_{\mathrm{m}})$. For comparison, the code Recfast uses a constant fudge factor $F=1.14$ to mimic the effect of high-$n$ shells. Right panel: Fraction of the effective recombinations to the $n=2$ shell that lead to atomic hydrogen in the $2s$ state. In both cases the effective rates were computed for $n_{*}=2$, i.e. with interface states $2s$ and $2p$ only, neglecting escape from the Lyman $\beta ,\gamma ,\dots$ lines.

Figure 3

Left panel: Relative differences between recombination histories computed with successively more accurate effective rates. Right panel: Recombination history for effective rates computed with $n_{\max }=500$, i.e. accounting explicitly for 125 250 states of the hydrogen atom.

Figure 4

A comparison of our ultrafast code to RecSparse [10], for different values of $n_{\max }$. The vertical axis is the fractional difference in free electron abundance rescaled by ${10}^5$ (positive indicating that RecSparse gives a larger $x_e$). We see that the maximum fractional deviation is $<8\times {10}^{-5}$. The feature around $z=1540$ is due to a time step change in RecSparse.