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 α 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 α 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 quantummechanical corrections related to the two-photon nature of the 3d/3s-1s emission and absorption process on the exact shape of the Lyman α emission profile. We show here that due to the redistribution of photons over frequency hydrogen recombination is sped up by ∆Ne/Ne ∼ −0.6% at z ∼ 900. For the CMB temperature and polarization power spectra this results in |∆Cl/Cl| ∼ 0.5% − 1% at l & 1500, and therefore will be important for the analysis of future CMB data in the context of the Planck Surveyor, Spt and Act. The main contribution to this correction is coming from the atomic recoil effect (∆Ne/Ne ∼ −1.2% at z ∼ 900), while Doppler boosting and Doppler broadening partially cancel this correction, again slowing hydrogen recombination down by ∆Ne/Ne ∼ 0.6% at z ∼ 900. The influence of electron scattering close to the maximum of the Thomson visibility function at z ∼ 1100 can be neglected. We also give the cumulative results when in addition including the time-dependent correction, the thermodynamic factor and the correct shape of the emission profile. This amounts in ∆Ne/Ne ∼ −1.8% at z ∼ 1160 and |∆Cl/Cl| ∼ 1% − 3% at l & 1500.