2.2 Hydrogen recombination: overview
2.2.3 Validity of the assumptions made
The simple yet insightful effective three-level atom model presented in Section 2.2.1 provides a good approximation for the recombination problem and remained essentially unaltered for several decades.
However, this simple theory relies on many simplifying assumptions, the validity of which we assess now.
Steady-state approximation for the excited states X
The formal way to assess the validity of the steady-state approximation is to compute the eigenvalues of the transition matrix in a multilevel calculation (to be described soon). This was done in Ref. [37],
where it was found that the minimum eigenvalue of the rate matrix is ∼ 1 sec−1. This could be expected as the rate of Ly-αdecays per atom in the 2pstate isRLyα, which has a minimum of∼1 sec−1 at z ≈1100 (see Fig. 2.1). This minimum rate is∼12 orders of magnitude larger than the recombination rate. The steady-state approximation was also checked explicitly in Ref. [38] where the solution of the time-dependent problem was computed (i.e., solving coupled ordinary differential equations forxe and the populations of the excited states). There again, it was found to be very accurate. Note that this approximation also underlies the use of the case-B recombination coefficient, as electrons captured in excited states are assumed to “cascade down” instantaneously to the first excited shell.
Case-B recombination X
In the model presented above, we have simply neglected altogether recombinations to the ground state. In reality, there is a small net rate of recombinations to the ground state due to redshifting of photons below the Lyman-continuum threshold, similar to what we described for the Lyman-αline:
˙
xe|1s,direct=− 8πH nHλ3cx1s
x2e
ge −x1se−EI/T
= (4/3)3 C
3RLyα
4βB
eE2/Tx˙e, (2.30) where λc = 43λLyα is the wavelength of Lyman-continuum photons at the ionization threshold and we have used Eq. (2.24) for ˙xe on the right-hand-side. We can see already that escape of Lyman- continuum photons is a small correction, due to the term eE2/T ≈exph
−13
1100 1+z
i. We checked that adding the rate (2.30) to the recombination rate leads to an acceleration of recombination by a very small amount |∆xe/xe| <7×10−6, in agreement with the results of Ref. [41]. Primordial hydrogen recombination is therefore indeed a case-B recombination, to the level of accuracy required.
Case-B recombination coefficient ×
For the first 20 years or so after the first works on primordial recombination, the main improvement that was made (besides using more up-to-date cosmological parameters) was using a more accurate case-B recombination coefficient (see Refs. [71, 5] and references therein). The most accurate fitting formula is given in Ref. [72]. However, the case-B coefficient as defined in Eq. (2.10) does not account for two important aspects:
(i) The bath of blackbody photons causestimulated recombinations[73], which are not accounted for in Eq. (2.10). Stimulated recombinations will speed-up recombination. One cannot simply re- place each coefficientαnl(Tm) by the spontaneous + stimulated recombination coefficientαnl(Tm, Tr) because the sum would be divergent, as the photon occupation number diverges at zero energies (see for example Fig. 4 of Ref. [38]).
(ii) The blackbody photons also alter the “cascading down” of captured electrons: an excited
atom may be photoionized before decaying to the first excited state. The sum in Eq. (2.10) should therefore be weighted by the probabilities to actually reach the first excited state.
(iii) At late times, when the intensity of the blackbody radiation field decreases, excited states cannot be maintained in Boltzmann equilibrium with each other. This is especially true for 2sand 2p, and one should split the case-B recombination coefficient appropriately between the two states.
From the discussion in Section 2.2.2, we can anticipate that these considerations may be impor- tant at late times.
Lyman-α escape rate ×
The treatment of the Lyman-α presented in Section 2.2.1 is very simplistic: the radiation field is just assumed to be a step function. Since ∼43% of recombinations proceed through a Lyman-α decay [70], a more sophisticated radiative transfer calculation is required. Moreover, higher-order Lyman lines also need to be considered. We can anticipate that more accurate calculations of the Ly-α decay rate and decays in the higher-order Lyman lines will affect the peak of the visibility function (see Section 2.2.2).
Two-photon decay rate ×
The majority of decays to the ground state proceed through the two-photon channel. The simple expression for decay rate given in Eq. (2.21) does not account for two processes:
(i)Stimulated two-photon decays, particularly important when one of the two photons has energy of the order of or less thanTr [47].
(ii)Absorption of non-thermal photons emitted in the Lyman-αline [48].
In addition, two-photon decays from higher-order excited states are also important [50]. Since the bulk of these decays are near resonance with a Lyman line (i.e., the higher-energy photon is close to a Lyman frequency), this also requires a radiative transfer treatment. Again, we expect such corrections to be mostly important at early timesz∼>900 and affect the peak of the visibility function.
From the above discussion, we see that corrections to the recombination model can be subdivided into two distinct categories:
(i) Radiative transfer calculations, mainly in the Lyman-α line, and also in higher-order lines.
These calculations must also properly account for two-photon decays from 2sand the higher-order states. Corrections to the simple decay rates Eqs. (2.19) and (2.21) are expected to mostly affect the early time recombination historyz∼>900. Since the visibility function peaks atz≈1100, small corrections to the net decay rates may have important consequences on theC`’s and even a priori small effects should be considered.
(ii) Multilevel atom calculations, that properly account for all transitions between excited states of hydrogen and generalize the effective three-level atom model. Such modifications will mostly affect the low redshift tail of the visibility function, and the accuracy requirement is somewhat lower than for (i).