Towards a complete treatment of recombination

Towards a complete treatment of the cosmological recombination problem


A new approach to the cosmological recombination problem is presented, which completes our previous analysis on the effects of two-photon processes during the epoch of cosmological hydrogen recombination, accounting for s-1s and d-1s Raman events and two-photon transitions from levels with . The recombination problem for hydrogen is described using an effective 400-shell multi-level approach, to which we subsequently add all important recombination corrections discussed in the literature thus far. We explicitly solve the radiative transfer equation of the Lyman-series photon field to obtain the required modifications to the rate equations of the resolved levels. In agreement with earlier computations we find that 2s-1s Raman scattering leads to a delay in recombination by at . Two-photon decay and Raman scattering from higher levels () result in small additional modifications, and precise results can be obtained when including their effect for the first shells. This work is a major step towards a new cosmological recombination code (CosmoRec) that supersedes the physical model included in Recfast, and which, owing to its short runtime, can be used in the analysis of future CMB data from the Planck Surveyor.

Cosmic Microwave Background: cosmological recombination, temperature anisotropies, radiative transfer

1 Introduction

The Planck Surveyor1 is currently observing the temperature and polarization anisotropies of the cosmic microwave background (CMB), and scientists all over the world eagerly await its first data release, scheduled for early 2011. With Planck data sets cosmologists will be able to determine key cosmological parameters with unprecedented precision, making it possible to distinguish between the various models of inflation (e.g. see Komatsu et al., 2010, for recent constraints).

Over the past 5 years, various groups (e.g. see Dubrovich & Grachev, 2005; Chluba & Sunyaev, 2006b; Kholupenko & Ivanchik, 2006; Switzer & Hirata, 2008; Wong & Scott, 2007; Rubiño-Martín et al., 2008; Karshenboim & Ivanov, 2008; Hirata, 2008; Chluba & Sunyaev, 2008; Jentschura, 2009; Labzowsky et al., 2009; Grin & Hirata, 2010; Ali-Haïmoud & Hirata, 2010) have realized that the uncertainties in the theoretical treatment of the cosmological recombination process could have important consequences for the analysis of the CMB data from the Planck Surveyor. It was shown that in particular our ability to measure the precise value of the spectral index of scalar perturbations, , and the baryon content of our Universe will be compromised if modifications to the recombination model by Recfast (Seager et al., 1999, 2000) are neglected (Rubiño-Martín et al., 2010).

To ensure that uncertainties in the cosmological recombination model do not undermine the science return of the Planck satellite, it is crucial to incorporate all important processes leading to changes in the free electron fraction close to the maxima of the Thomson visibility function (Sunyaev & Zeldovich, 1970) by more than into one recombination module. The main obstacle towards accomplishing this so far was that detailed recombination calculations took too long to allow accounting for the full cosmological dependence of the recombination corrections on a model-by-model basis. This led to the introduction of improved fudge factors to Recfast (Wong & Scott, 2007; Wong et al., 2008), or multi-dimensional interpolation schemes (Fendt et al., 2009; Rubiño-Martín et al., 2010), that allow fast but approximate representation of the full recombination code.

Although it was already argued that for the stringent error-bars of today’s cosmological parameters such approaches should be sufficient (Rubiño-Martín et al., 2010), from a physical stand point it would be much more satisfying to have a full representation of the recombination problem, that does not suffer from the limitations mentioned above, while capturing all the important physical processes simultaneously. Furthermore, such a recombination module increases the flexibility, and allows us to provide extensions, e.g., to account for the effect of dark matter annihilation, energy injection by decaying particles (e.g. see Chen & Kamionkowski, 2004; Padmanabhan & Finkbeiner, 2005; Chluba, 2010), or the variation of fundamental constants (Kaplinghat et al., 1999; Galli et al., 2009; Scóccola et al., 2009), while treating all processes simultaneously.

In this paper, we describe our new approach to the recombination problem, which enables us to fulfill this ambition by overcoming the problems mentioned above. Our code, called CosmoRec2, runs in minutes for a given set of cosmological parameters as it stands and can be optimized further to run well below a minute, eliminating the need for fudge factors to solve the recombination problem. One of the key ingredients that facilitates this increase in speed is the effective multi-level approach, which was proposed recently by Ali-Haïmoud & Hirata (2010).

We also extend our previous analysis on the effects of two-photon processes during the cosmological recombination epoch of hydrogen (Chluba & Sunyaev, 2008, 2010b) to account for s-1s and d-1s Raman scattering and two-photon transitions from levels with . The radiative transfer equation for the Lyman-series photons during hydrogen recombination is solved in detail using a PDE solver that we developed for this purpose and can accommodate non-uniform grids (see Appendix B for more details). Our results for the effect of Raman scattering on the recombination dynamics are in good agreement with earlier computations (Hirata, 2008). Furthermore, we show that two-photon decays from levels with can be neglected and Raman scattering is only important for the first few shells.

The main difficulty with two-photon and Raman processes during the recombination epoch is the presence of resonances in the interaction cross-sections related to normal ’1+1’ photon transitions that are already included into the multi-level recombination code (Hirata & Switzer, 2008; Chluba & Sunyaev, 2008; Hirata, 2008; Chluba & Sunyaev, 2010b). Unlike for the 2s-1s two-photon decay, all the higher s-1s and d-1s two-photon channels include ’’ photon sequences via energetically lower Lyman-series resonances, i.e., with . Similarly, for s-1s and d-1s Raman-scattering events all higher Lyman-series resonances, i.e., with , appear. Therefore, special care has to be taken to avoid double-counting of these resonances in the rate equations of the multi-level atom, as we explain in § 3.4.3, § 3.4.4, § 3.5.3, and § 3.5.4.

In § 2 we outline our principle approach to the recombination problem. The terms for the radiative transfer equation that allow to take all important recombination corrections into account are derived in § 3. We then solve the evolution equation for the high frequency photon field during the recombination epoch, and illustrate the different changes in § 4. In § 5, we discuss the different corrections to the ionization history, and we present our conclusions and outlook in § 6.

2 Perturbative approach to solving the full recombination problem

2.1 General aspects of the standard recombination problem

The cosmological recombination problem consists of determining an accurate estimate of the free electron fraction, , as a function of redshift. Because of particle conservation, and the number of electrons in excited states of Hi and Hei being negligible, one may write3


where denotes the number density of hydrogen nuclei, and is the fraction of helium nuclei. The populations of the different levels are given by , where ’a’ indicates the atomic species. Furthermore, for convenience.

Equation  (1) implies that the recombination problem reduces to finding solutions to . For hydrogen, the standard rate equation describing the evolution of the ground-state population has the form (see also; Seager et al., 1999, 2000)


Here is the mean photon occupation number over the Lyman- line profile, the atomic rate coefficients for spontaneous emission, and is the occupation number of the CMB blackbody photons at the Lyman- transition frequency .

The solution of Eq. (2), depends on the level populations of the 2s- and p-states. In addition, the photon distribution in the vicinity of every Lyman-resonance has to be known, to define . is often estimated by the Sobolev approximation, which, however, breaks down during recombination, leading to non-negligible corrections to the recombination dynamics (e.g. see Chluba & Sunyaev, 2009b, 2010b, a). The rate equations for the 2s- and p-states themselves can, in principle, be explicitly given. But here it is only important to realize that these lead to a large network of rate equations which depends on the populations of all other excited levels. To complicate matters further the electron temperature, , enters the whole problem via recombination coefficients, , to each level , where is the photon temperature.

The evolution of is described by one simple differential equation, which accounts for the cooling of electrons caused by the Hubble-expansion, and the energy exchange with CMB photons via Compton scattering. Other processes, e.g., such as Bremsstrahlung cooling, are subdominant (Seager et al., 2000).

The effective multi-level approach

Recently, Ali-Haïmoud & Hirata (2010) suggested to simplify the recombination problem to a subset of levels that need to be followed explicitly. Here we shall call the members of this subset ’resolved’ levels. This approach enables us to account for the effect of recombinations due to highly excited states (), without actually solving for all these level populations explicitly. The rationale being that except for the optically thick Lyman-series transitions, all other rates are mediated by the CMB blackbody photons, and hence only depend on the photon and electron temperatures.

The downside of this simplification entails the need to tabulate effective rate coefficients as a function of and prior to the computation. This however needs to be done only once, and given that the number of resolved states necessary for converging solutions is small, this effective multi-level approach results in tremendous speed-ups for recombination calculations (see Ali-Haïmoud & Hirata, 2010, for more details). For this work we also implemented such an effective rate approach. The rate coefficients for an effective 400-shell hydrogen atom were computed using our most recent recombination code (Chluba et al., 2010), while helium is described with a multi-level treatment (Chluba & Sunyaev, 2010a).

Within this framework the choice of the number of resolved states depends on the extra physics that one intends to include. For example, in a minimal model for the hydrogen recombination problem one should explicitly solve for the4 1s, 2s, and 2p level populations in tandem with the electron temperature, .

This minimal choice already allows us to include processes that affect the net rates in the 2s-1s two-photon channel and the 2p-1s Lyman- resonance, e.g., the effect of stimulated 2s-1s two-photon decay (Chluba & Sunyaev, 2006b), or the feedback of Lyman- photons on the 1s-2s rate (Kholupenko & Ivanchik, 2006). However, since we restricted ourselves to the 1s, 2s and 2p states, corrections due to Lyman- or higher resonance feedback cannot be modelled as these would require resolving p states with (Chluba & Sunyaev, 2007). We will return to these points in § 2.2.1.

Sobolev approximation for

In a multi-level approach the Sobolev approximation is invoked to obtain a solution for the photon-field around every resonance appearing in Eq. (2c). The photon occupation number around each line is then given by5


where , is the Sobolev optical depth in the Lyman- resonance, and . Here is the Voigt profile corresponding to a resonance, and the line occupation number, , is defined as:


Consequently, a simple approximation for the mean occupation number is


with being the Sobolev escape probability.

For the Lyman- resonance Eq. (3) results in a photon distribution that is rather unphysical (e.g. see discussion in Chluba & Sunyaev, 2009b). This is primarily due to the assumption that every interaction with the resonance leads to a complete redistribution of photons over the whole line profile, which for typical values of during recombination couples the photon distribution from the line center up to frequencies in the Lyman-continuum. For conditions present in our Universe, photon redistribution over frequency is much less effective, most notably in the distant wings. Thus, it is important to distinguish between scattering, real emission and absorption events, as we will discuss in § 3.

2.2 Beyond the standard rate equation for 1s

As mentioned in § 2.1.1, within the effective multi-level approach the choice for the resolved states depends on the physics to be modelled in detail. For example, in order to include the full effect of Lyman-series feedback, say up to , the minimal model that follows 1s, 2s, and 2p would at least have to be extended by all p-states up to 8p.

Also, the inclusion of two-photon processes from higher levels and Raman-scattering, requires us to re-write equation (2) in a more generalized form as,


where are the rates between the levels and . These rates depend on atomic physics, the CMB blackbody, the electron temperature, and the solution for the Lyman-series spectral distortion introduced by the recombination process.

To include two-photon corrections to the Lyman-series up to , the important levels to follow are all the d and s-states with . The corresponding partial rates to the p-states drop out of the equations, and the Lyman-series emission and absorption profiles, usually given by a Voigt function, will be replaced by the two-photon profiles for the and processes, and similarly for the Raman process. We will specify these corrections more precisely in the following sections.

Accounting for corrections from radiative transfer effects

Changes in the level populations, electron temperature and free electron fraction remain small (), when different physical processes, which were neglected in earlier treatments (e.g. see Rubiño-Martín et al., 2010, for overview) are included. This justifies treating corrections to and the populations of resolved levels, , as small perturbations. On the other hand, it has been shown that the changes in the photon field caused by time-dependence (Chluba & Sunyaev, 2009b), line scattering (Chluba & Sunyaev, 2009a; Hirata & Forbes, 2009), or two-photon corrections (Hirata, 2008; Chluba & Sunyaev, 2010b), are non-perturbative.

In §3 we derive in detail the different correction terms for the photon diffusion equation and provide modifications to the net rates of the effective multi-level atom. The idea is to first solve the recombination history using the effective multi-level approach in the ’’ photon description, i.e. equate in Eq. (6), and then compute the solution to the photon field using the radiative transfer equation. This then leads to corrections in the net rates, which are used in computing changes to the recombination dynamics, and hence modify Eq. (6). These corrections being small, demand only one iteration to converge. Detailed descriptions to the notations in the following sections and part of the methods used can be also found in Chluba & Sunyaev (2009b, 2010b, a).

3 Equation for the photon field evolution and corrections to the effective multi-level atom

To account for all corrections to the cosmological recombination problem, it is important to follow the evolution of non-thermal photons in the Lyman-series, which are produced during recombination. These photons interact strongly with neutral hydrogen atoms throughout the entire epoch of recombination, and their rate of escape from the Lyman-resonances is one of the key ingredients in accurately solving the recombination problem.

The partial differential equation governing the evolution of the photon field has the form (see Chluba & Sunyaev, 2009b, for a detailed discussion)


where is the distortion in the photon occupation number, and a distinction is made between the collision terms leading to emission and absorption, , and scattering, . As an example, the first term on the right hand side of the equation can account for two-photon corrections to the line profiles, while the second term, electron and/or resonance scattering. The second term on the left hand side describes the redshifting of photons due to Hubble expansion, and plays a crucial role in the escape of photons from the optically thick Lyman-series resonances.

In Eq. (7) the CMB blackbody has been subtracted, i.e., , where is the blackbody occupation number, because the left hand side directly vanishes for a blackbody with temperature . Also, spectral distortions created by Compton scattering off electrons with will be extremely small for conditions in our Universe6, at least if there is no additional energy release.

By changing the time-variable to redshift using , and scaling to dimensionless frequency , Eq. (7) reads


where and . We will now discuss the terms describing the resonance and electron scattering. In § 3.1 we specify the different emission and absorption terms, which then in § 4 and §5 are used to compute the corrections to the Lyman-series distortion and ionization history.

3.1 Inclusion of partial redistribution by resonance and electron scattering

Here we provide the terms for the Boltzmann equation describing the effect of (partial) photon redistribution by resonance and electron scattering. The form of the collision term for these cases within a Fokker-Planck formulation was discussed earlier (e.g. Zeldovich & Sunyaev, 1969; Basko, 1978b, a; Rybicki & dell’Antonio, 1994; Sazonov & Sunyaev, 2000; Rybicki, 2006; Chluba & Sunyaev, 2009a). Since we are only following the evolution of the distortion from a blackbody, and since it is clear that induced effects are negligible7, one can readily write


where is the dimensionless frequency and . The first term on the right hand side describes photon diffusion and the second accounts for the recoil effect.

Electron scattering

The diffusion coefficient in the case of electron scattering is (e.g. see Zeldovich & Sunyaev, 1969; Sazonov & Sunyaev, 2000)


where is the Thomson cross section. Chluba & Sunyaev (2009a) pointed out that electron scattering has an effect only at the early stages of recombination (). However it is easy to include, and also has the advantage of stabilizing the numerical treatment by damping small scale fluctuations of the photon occupation number caused by numerical errors, even in places where line scattering is already negligible.

As can be seen from the form of the diffusion coefficient in Eq. (10), the efficiency of electron scattering to a large extent is achromatic. This is in stark contrast to the case of resonance scattering, which is most efficient only in a very narrow range around the line center (see next paragraph). Furthermore, the number of free electrons drops rapidly towards the end of recombination, such that the Fokker-Planck approximation is expected to break down (Chluba & Sunyaev, 2009a). Nevertheless, the diffusion approximation remains sufficient for computations of the free electron fraction (see Ali-Haïmoud et al., 2010).

Resonance scattering

For resonance scattering by a Lyman- line the diffusion coefficient is (e.g. see Basko, 1978b, a; Rybicki, 2006; Chluba & Sunyaev, 2009a, and reference therein)


where and denote the resonant-scattering cross section and the Doppler width of the Lyman- resonance, respectively. For the Lyman- line and at . The Voigt profile , is normalized as . Where is the distance to the line center in units of the Doppler width.

The scattering probability of the Lyman- resonance, , is determined by a weighted count of all possible ways out of the -state, , excluding the Lyman-series resonance being considered, and then writing the branching ratio as8


yielding the probability for re-injection into the Lyman- resonance. The rates and the scattering probabilities, , can be pre-computed, independent of the solutions obtained from the multi-level code. We detail the procedure below.

Following Rybicki & dell’Antonio (1994), the diffusion coefficient is . We neglect corrections due to non-resonant contributions (e.g. see Lee, 2005) in calculating the scattering cross section, which would lead to a different frequency dependence far away from the resonance (e.g. Rayleigh scattering in the distant red wing, Jackson 1998). However, because it turns out that resonance scattering is only important in the vicinity of the Lyman- resonance, this approximation suffices.

It is also worth mentioning that Eq. (11) together with Eq. (9), in the limit of large optical depth9, provides a viable description for the redistribution of photon over frequency. Unlike the case of complete redistribution (in which the reemission after each scattering event occurs over the whole Voigt profile), in the recombination epoch photons are only partially redistributed as a result of the Doppler motions of the hydrogen atoms, so-called type-II redistribution as defined in Hummer (1962).

Equilibrium solution for the scattering term

Independent of the type of scattering, the equilibrium distribution with respect to the scattering term Eq. (9) is given by


This is the expected Wien spectrum with the temperature defined by the electrons. The normalization is determined by the emission and absorption process.

The optical depth to line scattering being extremely large inside the Doppler cores of the Lyman-resonances ( during Hi recombination) causes the photon distribution within the Doppler core to remain extremely close to equilibrium, .

3.2 Normal Lyman- emission and absorption terms

In the normal ’’ photon picture, the emission profile for each Lyman-series resonance is given by a Voigt-profile, , with Voigt-parameter . Given the rate, , at which fresh10 electrons reach the p-state, and the probability of photon injection into the Lyman- resonance, , the Lyman- line-emission and absorption term are (e.g. see Chluba & Sunyaev, 2010b);


The factor accounts for the translation from photon number to the occupation number because , for which the Voigt-profile is defined. Also, is the death or the real absorption probability in the Lyman-series resonance, and and are given by,


where is the ratio of the statistical weights of the initial and final states. The function can in principle be pre-computed using the solution for the populations of the levels from the initial run of the effective multi-level recombination code. However, the simplest way to define the ratio is to use the quasi-stationary approximation for the p-population (see details below). We note that in full thermodynamic equilibrium , so that no distortion is created ().

Physically, Eq. (14) includes two important aspects, which are not considered in the standard recombination calculation. Firstly, it allows for a distinction between scattering events on one side, and real emission and absorption events on the other. Secondly, it ensures conservation of blackbody spectrum in full thermodynamic equilibrium, even in the very distant wings of the lines. Refer Chluba & Sunyaev (2010b) for a detailed explanation of the latter point, and on how this leads to one of the largest corrections in the case of Lyman- transport.


To solve the evolution of the photon field, one has to know at which rate photons are produced by the Lyman-resonance. This rate depends on as defined in Eq. (15).

The rate equation for the evolution of the population in the p-level has the form (see Appendix B Chluba & Sunyaev, 2010b),


In this picture the emission, absorption and resonance scattering terms are all treated simultaneously. In addition, the asymmetry between the emission and absorption profile in the Lyman- resonance, as required by detailed balance, has been incorporated.

Under quasi-stationarity, and using the definition of the death probability, , Eq. (16) yields


such that with Eq. (15)


In the second step we used the normal Sobolev approximation, for which (for the case of Lyman- compare also with Eq. (41) in Chluba & Sunyaev, 2009b).

From Eq. (18b) we have , since for all Lyman-series resonances the second term in brackets is very small. Nevertheless, for the total normalization of the line intensity close to the line center, this small correction is important (Chluba & Sunyaev, 2009b), in particular for the Lyman- resonance.

Also we would like to mention that for the Voigt parameter of the Lyman- profiles, , the total width of the line is used, where transitions induced by the CMB blackbody (e.g. to higher levels) are included. Numerically, it is possible to compute the total width for the Lyman- resonance with .

3.3 The 2s-1s two-photon channel

The 2s-1s two-photon channel provides the pathway for about 60% of all electrons in hydrogen to settle into the ground state (Chluba & Sunyaev, 2006a). It therefore provides the most important channel in the cosmological recombination process. Here we treat the case of 2s-1s separately to illustrate the important approximations in the two-photon picture. The derivation outlined in this section is then used to obtain the corresponding terms for the two-photon processes from excited states with (see § 3.4).

The net change of the number density of electrons in the 2s level via the 2s-1s two-photon channel is given by


where (Labzowsky et al., 2005) is the vacuum 2s-1s two-photon decay rate, and denotes the 2s-1s two-photon decay profile normalized as . Including all possible ways in and out of the 2s-level the net change of the number density of electrons in the 2s-state can be written as


Here and include the effect of all transitions to bound states with and the continuum.

In order to simplify the notation we now introduce


where is some arbitrary function of frequency and and with .

Then, under quasi-stationarity the solution for the population of the 2s-state is given by


In the multi-level approach the effect of stimulated two-photon emission is neglected leading to . Also any CMB spectral distortion that is introduced by the recombination process (e.g. because of Lyman- emission) is omitted, implying . In this approximation, the result from Eq. (22) becomes identical to the one obtained using Eq. (2b) and Eq. (20), in the standard multi-level approach.

However, in the recombination problem corrections to both and are important. For the stimulated two-photon emission only the occupation number given by the undistorted CMB blackbody has to be considered and thus,


which can be precomputed as a function of temperature. Typically, exceeds unity by a few percent (Chluba & Sunyaev, 2006b).

For one can make use of the fact that the distortions at either or are very small, so that


Hence Eq. (3.3) can be re-written as,


where we defined the stimulated 2s-1s two-photon decay rate within the CMB ambient radiation field as (cf. Chluba & Sunyaev, 2006b). Also Eq. (25b) reflects the symmetry of the two-photon profile around .

Note that for only the CMB blackbody spectrum is important and therefore can, in principle, be precomputed as a function of photon temperature, . This also emphasizes the difference in the origin of the two terms of Eq. (25a), being the thermal contribution, while arises solely because of non-thermal photons created in the recombination process.

By comparing Eq. (25) with Eq. (2b) one can write down the correction to the 2s-1s net two-photon rate


Here we introduced , which during recombination is of order . In Equation (25) the integral depends on the spectral distortion introduced by the recombination process in the Wien’s tail of the CMB blackbody. Including only the Lyman- distortion provides a manner in which to take its feedback effect into account (cf. Kholupenko & Ivanchik, 2006).

The 2s-1s two-photon emission and absorption term

In contrast to the Lyman-series channels, the terms for the photon radiative transfer equation in the case of the 2s-1s channel can be directly obtained from the net rate between the 2s and 1s state as in Eq. (3.3), resulting in


Here we defined , where the factor of two results from two photons being added to the photon field, and the converts the units to per steradian.

The reason for this simple connection to the net rate equation is related to the fact that every transition from the 1s state to the 2s level is expected to lead to a complete redistribution over the 2s-1s two-photon profile. The main reason behind this assumption of redistribution is that the probability of coherent 1s-2s scattering event is tiny because the 2s-1s decay rate is extremely small compared to the time it takes to excite a 2s-electron to higher levels or the continuum.

However, some additional simplifications are possible. First, we can again replace the factors, , accounting for stimulated two-photon emission with those from the undistorted CMB blackbody. Furthermore, from Eq. (24),


Also, since the spectral distortions at very low frequencies are never important, one of the two terms in Eq. (28) (say the one related to ) can always be omitted. Therefore we can rewrite Eq. (3.3.1) as


where and


If the term is non-negligible, as might be the case at very low redshifts (), where the Lyman- photons emitted at redshifts into the 2s-1s absorption channel, one in addition has to subtract the term within the brackets of Eq. (29). In terms of , and the photon occupation number now evolves as,


Here the 2s-1s cross section is given by , and .

Eq. (31) bears a striking resemblance to the emission and absorption in the Lyman-series channels as in Eq. (14) because one of the two photons that are involved in the 2s-1s two-photon process is drawn from the undistorted CMB blackbody spectrum, so that the evolution equation essentially becomes a one-photon equation. The difference is the absence of a death probability since practically every electron that is excited to the 2s state will take a detour to higher levels or the continuum as .

3.4 Two-photon emission and absorption terms from excited levels with

One of the most interesting modifications to the solution for the photon field is related to the deviations of the profiles for the different two-photon emission and absorption channels from the Lorentzian shape (Chluba & Sunyaev, 2008). For the recombination problem only those one-photon sequences involving a Lyman-series resonance (e.g. ) are important11. In this section we shall replace the standard ’’-photon terms for these channels with the full two-photon description that takes into account the coherent nature of the process12.

We generalize the approach detailed in Chluba & Sunyaev (2010b) for emission of photons close to the Lyman- line to include corrections around the Lyman- and higher resonances.

Net rates for two-photon transitions from excited s- and d-states

The net change of the number density of electrons in the level via the -1s two-photon channel is given by


Here denotes the profile for the -1s two-photon decay, which can be computed as explained in Appendix A, and is normalized13 as . The (vacuum) two-photon decay rate is given by


The ratio of the statistical weights is for the s-states, and for d-states. Equation (33) simply reflects the one-photon decay rates and branching ratios of all the ’’ photon routes via intermediate p-states with . Stimulated emission induced by the CMB photons is not included in the definition of , since it is taken into account differentially by the integrals in Eq. (3.4.1).

With notations defined in Eq. (21), and following the procedure to derive Eq. (25), we can re-write Eq. (3.4.1) as


where we defined the stimulated -1s two-photon decay rate within the CMB ambient radiation field as .

The depends crucially only on the CMB blackbody spectrum and thus can be precomputed as a function of photon temperature, . On the other hand, like for the 2s-1s two-photon process (see Eq. (25)), arises due to non-thermal photons, and hence depends directly on the solution for the photon field.

In the normal ’’ photon picture, the two-photon profiles can be considered as a sum of -functions and therefore


Here , and the stimulated effect close to the Lyman-series resonances has been neglected, i.e. .

Two-photon emission and absorption for excited s- and d-states

The two-photon emission and absorption terms are obtained following the steps in the derivation of Eq. (31). For the -1s two-photon channel one therefore obtains


The -1s two-photon cross section is given by , and , where, because of energy conservation, . Also,


Again we emphasize the resemblance of the equation above to that of the one-photon equation for the Lyman-series emission and absorption channels as in Eq. (14).

Correcting the Lyman-series emission and absorption terms in the radiative transfer equation

Two-photon decays from a given initial state involve Lyman-series resonances with . For example, a 4d-1s two-photon emission event includes the effect of the Lyman- and resonance. In the Lyman-series emission and absorption terms as in Eq. (14), these are already accounted for as ’’ photon terms, when the profile is given by the normal Voigt function.

To avoid the double counting of these transitions in the radiative transfer equation, two modifications are necessary: (i) all death probabilities, , have to be reduced to account only for those channels that are not included in the two-photon description, and (ii) the Lyman-series emission rates have to be reduced for the same reason. This approach was also explained in Chluba & Sunyaev (2010b) for the 3s-1s and 3d-1s two-photon process. Including only the -1s two-photon process (say for 3d-1s), the modified death probability and of the Lyman- resonance becomes,

where the partial death probability, , is given by