Two-photon transitions in hydrogen and cosmological recombination

TL;DR

This paper analyzes two-photon transitions in hydrogen and their impact on cosmological recombination, providing analytic models and quantifying the resulting corrections to the ionization history.

Contribution

It introduces simple analytic formulas for two-photon emission profiles and assesses their effect on hydrogen recombination, highlighting the importance of interference effects.

Findings

Two-photon emission profiles are nearly Lorentzian near resonances.

Quantum-electrodynamical corrections cause significant deviations in the wings.

Two-photon processes lead to less than 0.4% correction in ionization history.

Abstract

We study the two-photon process for the transitions ns --> 1s and nd --> 1s in hydrogen up to large n. For n<=20 we provide simple analytic fitting formulae to describe the non-resonant part of the two-photon emission profiles. Combining these with the analytic form of the cascade-term yields a simple and accurate description of the full two-photon decay spectrum, which only involves a sum over a few intermediate states. We demonstrate that the cascade term naturally leads to a nearly Lorentzian shape of the two-photon profiles in the vicinity of the resonances. However, due to quantum-electrodynamical corrections, the two-photon emission spectra deviate significantly from the Lorentzian shape in the very distant wings of the resonances. We investigate up to which distance the two-photon profiles are close to a Lorentzian and discuss the role of the interference term. We then analyze how the deviation of the two-photon profiles from the Lorentzian shape affects the dynamics of cosmological hydrogen recombination. Since in this context the escape of photons from the Lyman-alpha resonance plays a crucial role, we concentrate on the two-photon corrections in the vicinity of the Lyman-alpha line. Our computations show that the changes in the ionization history due to the additional two-photon process from high shell (n>2) likely do not reach the percent-level. For conservative assumptions we find a correction DN_e/N_e~-0.4% at redshift z~1160. This is numerically similar to the result of another recent study; however, the physics leading to this conclusion is rather different. In particular, our calculations of the effective two-photon decay rates yield significantly different values, where the destructive interference of the resonant and non-resonant terms plays a crucial role in this context (abridged)