Astronomy & Astrophysics

A&A 480, 629-645 (2008)
DOI: 10.1051/0004-6361:20077921

Two-photon transitions in hydrogen and cosmological recombination

J. Chluba¹ and R. A. Sunyaev^{1, 2}

¹ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching bei München, Germany
e-mail: jchluba@mpa-garching.mpg.de
² Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia

(Received 21 May 2007 / Accepted 22 November 2007)

Abstract
We study the two-photon process for the transitions $n{\rm s}\rightarrow 1{\rm s}$ and $n{\rm d}\rightarrow 1{\rm s}$ in hydrogen up to large n. For $n\leq 20$ 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 $\Delta N_{\rm e}/N_{\rm e}\sim-0.4\%$ at redshift $z\sim 1160$ . 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. We also show that the bulk of the corrections to the ionization history is only due to the 3s and 3d-states and that the higher states do not contribute significantly.

Key words: atomic processes -- atomic data -- radiation mechanisms: general -- cosmology: theory -- cosmic microwave background