In this paper we consider the effects of resonance and electron scattering on the escape of Lyman \alpha 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 \alpha 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 \alpha emission profile .
We show here that due to the redistribution of photons over frequency hydrogen recombination is sped up by \Delta N _ { e } / N _ { e } \sim - 0.6 \% at z \sim 900 .
For the CMB temperature and polarization power spectra this results in | \Delta C _ { l } / C _ { l } | \sim 0.5 \% - 1 \% at l \gtrsim 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 ( \Delta N _ { e } / N _ { e } \sim - 1.2 \% at z \sim 900 ) , while Doppler boosting and Doppler broadening partially cancel this correction , again slowing hydrogen recombination down by \Delta N _ { e } / N _ { e } \sim 0.6 \% at z \sim 900 .
The influence of electron scattering close to the maximum of the Thomson visibility function at z \sim 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 \Delta N _ { e } / N _ { e } \sim - 1.8 \% at z \sim 1160 and | \Delta C _ { l } / C _ { l } | \sim 1 \% - 3 \% at l \gtrsim 1500 .