Lyαescape during cosmological hydrogen recombination: the 3d-1s and 3s-1s two-photon processes

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%.

HyRec: A fast and highly accurate primordial hydrogen and helium recombination code

“…Dubrovich & Grachev [25] suggested that two-photon transitions from higher levels may have a significant effect on the recombination history. Later computations confirmed this idea [26], and provided an accurate treatement of radiative transfer in the presence of two-photon transitions, as well as a solution for the double-counting problem (which arises for resonant two-photon transitions, already included in the one-photon treatment as "1+1" transitions) [24,27].…”

“…(23)(24)(25)(26)(27)(28), are strictly equivalent to the standard MLA equations presented in the previous section, as was derived in Paper I [Eq. (31) was not derived in that paper and we give a proof in Appendix A 2].…”

“…In addition, when considering two-photon transitions from higher levels, one should in principle add the ns and nd states as "interface" states. However, only two-photon transitions from 2s, 3s and 3d, are important (and 4s, 4d at the level of a few 10 −4 ) [24,27]. This means that one needs to pretabulate only a few functions of temperature that fully account for the multilevel structure of hydrogen.…”

“…Indeed, the probabilities P i K and P e K , where i = 2s, 2p, are still defined by Eqs. (26) and (27), but the inverse lifetime of the state K, Eq. (28), should now account for the net downward transition rate to the ground state, and one should make the replacement:…”

Ultrafast effective multilevel atom method for primordial hydrogen recombination

“…We note that our method only eliminates the computational complexity associated with the high-n excited states and does not include continuum processes and radiative transfer effects that have been studied by previous authors [16][17][18][19][20][21][22][23][24][25][26]. However, we note that there has been much progress in analytic treatments of these effects [21][22][23]; ultimately, we expect to improve these analytic treatments and graft them (and an analytic treatment of helium recombination [27][28][29][30][31]) on to the ultrafast EMLA code described herein to yield a recombination code that is accurate enough for Planck data analysis.…”