Two-photon transitions in hydrogen and cosmological recombination

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-α resonance plays a crucial role, we concentrate on the two-photon corrections in the vicinity of the Lyman-α 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 ∆N 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. 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.

HyRec: A fast and highly accurate primordial hydrogen and helium recombination code

“…Dubrovich & Grachev [25] suggested that two-photon transitions from higher levels may have a significant effect on the recombination history. Later computations confirmed this idea [26], and provided an accurate treatement of radiative transfer in the presence of two-photon transitions, as well as a solution for the double-counting problem (which arises for resonant two-photon transitions, already included in the one-photon treatment as "1+1" transitions) [24,27].…”

“…(23)(24)(25)(26)(27)(28), are strictly equivalent to the standard MLA equations presented in the previous section, as was derived in Paper I [Eq. (31) was not derived in that paper and we give a proof in Appendix A 2].…”

Nonresonant effects and hydrogen transition line shape in cosmological recombination problems

“…Subsequently it was studied by Wong and Scott (2007). This problem was also intensively investigated by Chluba and Sunyaev (2007), who stimulated the appearance of this paper by their numerous questions. The conclusion of the abovementioned works is that the process of two-photon cascade transitions under consideration can lead to corrections of a few tenths of a percent.…”

“…The goal of this paper is, in particular, a detailed quantum-mechanical analysis of the problem and an investigation of the line wings, which, as Chluba and Sunyaev (2007) showed, play a significant role in the formation of deviations from the standard picture of recombination.…”

Theory of the multiphoton cascade transitions with two photon links: comparison of quantum electrodynamical and quantum mechanical approaches

“…This regularization is performed by the introduction of the level widths in the energy denominators which otherwise turn to zero for the photon frequencies, corresponding to the cascade resonances. In the QM phenomenological approach only the width of the intermediate state is usually employed for regularization [12], [13], [27]. We call this approach phenomenological since the width is introduced as an phenomenological parameter.…”