Some remarks about the old problem of recombination

TL;DR

This paper derives exact analytical expressions for spectral densities during hydrogen recombination, improving the accuracy of recombination models and quantifying effects on the ionization history.

Contribution

It provides a simple, rigorous derivation of the recombination spectral density and exact formulas, enhancing previous approximate methods.

Findings

The relative change in electron fraction due to two-photon absorption is 0.67%.

The effect of photons absorbed by the 2s->1s transition is a 0.05% variation.

The results differ by 12% from earlier approximate estimates.

Abstract

Context: In the seminal works of Zeldovich et al. (1968) and Peebles (1968), a procedure was outlined to obtain the equation of evolution of the hydrogen fraction without an explicit use of the radiative transfer equation. This procedure is based, explicitly or implicitly, on the concept of escape probability and, using the Sobolev approximation for this problem Sobolev (1960), has extensively been used since then in developing refined approximations. Aims: To derive in a simple, rigorous and general manner the above mentioned procedure and to obtain exact analytical expressions for the spectral density of radiation generated at one photon and two photon recombination transitions. These expressions are used to estimate the implications of several interesting effects. Methods: Some slight re-elaborations of basic principles of transport theory. Results: We have obtained the expressions searched for and used them in several explicit computations. We have found that the relative change in the electronic fraction due to the absorption by the two photon line, 1s->2s, of photons that have escaped from the line 2p-> 1s is 0.67%, that is a result 12% higher than the previous ones obtained using some approximations (Kholupenko & Ivanchik 2006; Hirata 2008). The photons generated by the transition 2s->1s and later absorbed by the same transition (in combination with a photon more energetic than its original partner) imply a 0.05% maximum variation of the electronic fraction. This problem has not been treated previously analytically, although a numerical estimate have been recently carried out in Chluba & Thomas 2011.