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

¹ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching bei München, Germany
² Canadian Institute for Theoretical Astrophysics, 60 St. George Street, Toronto, ON M5S 3H8, Canada e-mail: jchluba@cita.utoronto.ca
³ Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia

Received: 2 April 2009
Accepted: 27 August 2009

Abstract

We give a formulation of the radiative transfer equation for Lyman α 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 α 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 ΔP/P ~ +11% at z ~ 1400 in comparison with the one obtained using the Sobolev approximation. This accelerates hydrogen recombination by ΔN_e/N_e ~ -1.6% at z ~ 1190, implying that |ΔC_l/C_l| ~ 1-3% at l 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 α 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 α 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 ~ 1100. At z ~ 1100, the modification due to the shape of the line profile contributes about ΔN_e/N_e ~ -0.4%, while the sum of the other two contributions gives ΔN_e/N_e ~ -0.9%.