The subject of cosmological hydrogen recombination has received much attention recently because of its importance to predictions for and cosmological constraints from CMB observations .
While the central role of the two-photon decay 2 s \rightarrow 1 s has been recognized for many decades , high-precision calculations require us to consider two-photon decays from the higher states ns,nd \rightarrow 1 s ( n \geq 3 ) .
Simple attempts to include these processes in recombination calculations with an effective two-photon decay coefficient analogous to the 2 s decay coefficient \Lambda _ { 2 s } = 8.22 s ^ { -1 } have suffered from physical problems associated with the existence of kinematically allowed sequences of one-photon decays , e.g . 3 d \rightarrow 2 p \rightarrow 1 s , 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 3 d \rightarrow 1 s two-photon decay and 2 s \rightarrow 1 s 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 & Scott and Chluba & Sunyaev , and produces a much larger effect on the ionization history .
The implied correction to the CMB power spectrum is neligible for the recently released WMAP and ACBAR data , but at Fisher matrix level will be 7 \sigma for Planck .