Cosmological hydrogen recombination: influence of resonance and electron scattering

In this paper we consider the effects of resonance and electron scattering on the escape of Lyman a photons during cosmological hydrogen recombination. We pay particular attention to the influence of atomic recoil, Doppler boosting, and Doppler broadening using a Fokker-Planck approximation of the redistribution function describing the scattering of photons on the Lyman a resonance of moving hydrogen atoms. We extend the computations of our recent paper on the influence of the 3d/3s-1s two-photon channels on the dynamics of hydrogen recombination, simultaneously including the full time dependence of the problem, the thermodynamic corrections factor, leading to a frequency-dependent asymmetry between the emission and absorption profile, and the quantum-mechanical corrections related to the two-photon nature of the 3d/3s-1s emission and absorption process on the exact shape of the Lyman a emission profile. We show here that, because of the redistribution of photons over frequency hydrogen recombination is sped up by Delta N-e/N-e similar to -0.6% at z similar to 900. For the CMB temperature and polarization power spectra, this results in vertical bar Delta C-t/C-t vertical bar similar to 0.5-1% at l greater than or similar to 1500, which is in turn important for analyzing future CMB data in the context of the Planck Surveyor, SPT, and Act. The main contribution to this correction comes from the atomic recoil effect (Delta N-e/N-e similar to -1.2% at z similar to 900), while Doppler boosting and Doppler broadening partially cancel this correction, again slowing hydrogen recombination down by Delta N-e/N-e similar to 0.6% at z similar to 900. The influence of electron scattering close to the maximum of the Thomson visibility function at z similar to 1100 can be neglected. We also give the cumulative results, in addition including the time-dependent correction, the thermodynamic factor, and the correct shape of the emission profile. This amounts to Delta N-e/N-e similar to -1.8% at z similar to 1160 and vertical bar Delta C-t/C-t vertical bar similar to 1-3% at l greater than or similar to 1500.