Ly α escape during cosmological hydrogen recombination: the 3d-1s and 3s-1s two-photon processes

We give a formulation of the radiative transfer equation for Lyman alpha photons, which allows us to include the two-photon corrections for the 3s-1s and 3d-1s decay channels during cosmological hydrogen recombination. We use this equation to compute the corrections to the Sobolev escape probability for Lyman alpha photons during hydrogen recombination, which then allow us to calculate the changes in the free electron fraction and CMB temperature and polarization power spectra. We show that the effective escape probability changes by Delta P/P similar to +11% at z similar to 1400 in comparison with the one obtained using the Sobolev approximation. This accelerates hydrogen recombination by Delta N-e/N-e similar to -1.6% at z similar to 1190, implying that vertical bar Delta C-l/C-l vertical bar similar to 1-3% at l greater than or similar to 1500 with shifts in the positions of the maxima and minima in the CMB power spectra. These corrections will be important to the analysis of future CMB data. The total correction is the result of the superposition of three independent processes, related to (i) time-dependent aspects of the problem; (ii) corrections due to quantum mechanical deviations in the shape of the emission and absorption profiles in the vicinity of the Lyman alpha line, from the normal Lorentzian; and ( iii) a thermodynamic correction factor, which is found to be very important. All of these corrections are neglected in the Sobolev-approximation, but they are important in the context of future CMB observations. All three can be naturally obtained in the two-photon formulation of the Lyman alpha absorption process. However, the corrections (i) and (iii) can also be deduced in the normal "1 + 1" photon language, without necessarily going to the two-photon picture. Therefore, only (ii) is really related to the quantum mechanical aspects of the two-photon process. We show here that (i) and (iii) represent the largest individual contributions to the result, although they partially cancel each other close to z similar to 1100. At z similar to 1100, the modification due to the shape of the line profile contributes about Delta N-e/N-e similar to -0.4%, while the sum of the other two contributions gives Delta N-e/N-e similar to -0.9%.