(ii) Multilevel atom calculations, that properly account for all transitions between excited states of hydrogen and generalize the effective three-level atom model. Such modifications will mostly affect the low redshift tail of the visibility function, and the accuracy requirement is somewhat lower than for (i).
Chapter 3
Effective multilevel atom method for primordial hydrogen
recombination 1
3.1 Introduction
We saw in Section 2.2.2 that the effective three-level atom (TLA) model [31, 32] cannot fully capture the effect of the bath of blackbody photons on the recombination process: stimulatedrecombinations must also be accounted for, as well as bound-bound transitions among excited states and photoion- izations events that impede some of the “cascading” electrons to reach the first excited states. Such processes will essentially affect the low-redshift tail of the ionization history, but are still important given the sensitivity of upcoming CMB experiments.
One way to account exactly for these processes is to solve simultaneously for the free electron fraction and the populations of all (or at least, a large number of) excited states. This method of solution, which we refer to as the standard multilevel atom (MLA) method, is commonly used in the study of emission lines in the interstellar medium. In fact, it was first used in the context of primordial recombination to compute the spectrum of emitted lines [74, 75]. After it was understood that non- equilibrium of excited states and stimulated recombinations may have an effect on the recombination history itself [5], the first detailed MLA calculations were carried by Seager et al. [29, 30]. In practice, one cannot account for an infinite number of excited states and must impose a cutoff in MLA computations, i.e., discard states with principal quantum number higher than a givennmax. Refs. [29, 30] computed recombination histories including up tonmax= 300 excited states, assuming that angular momentum substates within a given energy shell are in statistical equilibrium. They
1The material in this chapter was mostly adapted from the paper Ultrafast effective multilevel atom method for primordial hydrogen recombination, Y. Ali-Ha¨ımoud & C. M. Hirata, Phys. Rev. D 82, 063521 (2010), with the exception of Section 3.4.4, which is an exerpt from the paperHyRec: A fast and highly accurate primordial hydrogen recombination code, Y. Ali-Ha¨ımoud & C. M. Hirata, Phys. Rev. D83, 043513 (2011). Reproduced here with permission, copyright (2010, 2011) by the American Physical Society.
found that the residual ionization fraction at low redshift is decreased by approximately 10% with respect to the TLA prediction. It was later shown that the desired sub-percent accuracy can only be reached when explicitly resolving the out-of-equilibrium angular momentum substates [35, 36], which requires the MLA codes to follow Nlevel = nmax(nmax+ 1)/2 individual states. Moreover, the ordinary differential equations (ODEs) describing the level populations are stiff, requiring the solution of largeNlevel×Nlevel systems of equations at each integration time step. This problem has been solved by several authors [36, 37, 38]; however, these standard MLA codes take hours to days to compute a single recombination history.
Eventually, it is necessary to be able to produce not only accurate but also fastrecombination histories, to be included in Markov Chain Monte Carlo (MCMC) codes for cosmological parameter estimation. The MCMC requires CMB power spectra (and hence recombination histories) to be gen- erated at each proposed point in cosmological parameter space, with a typical chain samplingO(105) points [76]. Furthermore, dozens of MCMCs are often run with different combinations of observa- tional constraints and different parameter spaces. This makes it impractical to include recombination codes that run for more than a few seconds in the MCMC. One solution is to precompute recombina- tion histories xe(z|H0, TCMB,Ωmh2,Ωbh2, YHe, Nν) on a grid of cosmological parameters, and then use elaborate interpolation algorithms to evaluate the recombination history for any cosmology [77], or to construct fitting functions [30, 78]. However, such procedures need to be re-trained every time additional parameters are added, and are rather unsatisfying regarding their physical significance.
In this chapter, we present a new2, effective multilevel atom method (hereafter EMLA), that is a simple generalization of Peebles’ TLA model presented in Section 2.2.1, but accounts exactly for the multilevel structure of hydrogen, and allows for the inclusion of virtually infinitely many excited states of hydrogen and all relevant bound-bound and bound-free transitions. The improved recombination equations are exactly equivalent to the standard MLA equations, but have the same computational cost as the TLA model (for an equal treatment of radiative transfer), while being much more accurate. The basic idea is that the vast majority of the excited hydrogen levels are populated and depopulated only by optically thin radiative transitions (bound-bound and bound- free) in a bath of thermal photons; we show that their effect can be “integrated out” leaving only a few functions of the matter and radiation temperaturesTm andTr(this list would include the free electron density ne if we incorporated collisions), which can be pre-tabulated. These functions are just the generalization of the case-B recombination coefficient. In an actual call to the recombination code from an MCMC, it is then only necessary to solve aneffective MLA with a smaller number of levels, which eliminates the computationally difficultNlevel×Nlevelsystem solution in the traditional MLA. [The idea is similar in spirit to the line-of-sight integral method for the computation of the
2It has recently been brought to my attention that this method was independently suggested by M. S. Burgin [79, 80]. Due to translation delays, this work only became known in the Western world several months after the publication of our paper.
CMB power spectrum [24], which eliminated a large number of independent variables from the cosmological perturbation theory system of ODEs (the high-order moments of the radiation field, Θ`for`1) in favor of pretabulated spherical Bessel functions.] Our method achieves a speed-up of the recombination calculation by 5 to 6 orders of magnitude.
This chapter is organized as follows. In Section 3.2 we review the general picture of hydrogen recombination, and the bound-bound and bound-free transition rates involved in the calculation.
In Section 3.3 we describe the standard MLA method. We present our new EMLA method in Section 3.4 and demonstrate its equivalence with the standard MLA formulation. We describe our numerical implementation and results in Section 3.5, and conclude in Section 3.6. Appendix 3.A gives mathematical proofs for various relations satisfied by the effective rates. Appendix 3.B gives the details of our numerical evaluation of the effective rates.