Two-photon transitions in primordial hydrogen recombination

The subject of cosmological hydrogen recombination has received much attention recently because of its importance to predictions for and cosmological constraints from cosmic microwave background observations. While the central role of the two-photon decay 2s -> 1s has been recognized for many decades, high-precision calculations require us to consider two-photon decays from the higher states ns, nd -> 1s (n >= 3). Simple attempts to include these processes in recombination calculations with an effective two-photon decay coefficient analogous to the 2s decay coefficient Lambda(2s)=8.22 s(-1) have suffered from physical problems associated with the existence of kinematically allowed sequences of one-photon decays, e.g. 3d -> 2p -> 1s, that technically also produce two photons. These correspond to resonances in the two-photon spectrum that are optically thick to two-photon absorption, necessitating a radiative transfer calculation. We derive the appropriate equations, develop a numerical code to solve them, and verify the results by finding agreement with analytic approximations to the radiative transfer equation. The related processes of Raman scattering and two-photon recombination are included using similar machinery. Our results show that early in recombination the two-photon decays act to speed up recombination, reducing the free electron abundance by 1.3% relative to the standard calculation at z=1300. However, we find that some photons between Ly alpha and Ly beta are produced, mainly by 3d -> 1s two-photon decay and 2s -> 1s Raman scattering. At later times, these photons redshift down to Ly alpha, excite hydrogen atoms, and act to slow recombination. Thus, the free electron abundance is increased by 1.3% relative to the standard calculation at z=900. Our calculation involves a very different physical argument than the recent studies of Wong and Scott and Chluba and Sunyaev, and produces a much larger effect on the ionization history. The implied correction to the cosmic microwave background power spectrum is negligible for the recently released WMAP and ACBAR data, but at Fisher matrix level will be 7 sigma for Planck.