124 The hot universe
a sequence of steps. However, even in cascading recombination, at least one very
energetic photon is emitted corresponding to the energy difference between the 2P
and 1Sstates of the hydrogen atom. This so-called Lyman-αphoton,Lα,has energy
3 BH/ 4 =117 000 K and a rather large resonance absorption cross-section,σα
10 −^17 –10−^16 cm^2 ,at the recombination temperature. Therefore, theLαphotons are
reabsorbed in a short time,τα(σαnH)−^1 ∼ 103 –10^4 s, after emission. We have to
compare this time to the cosmological time at recombination. During the matter-
dominated epoch, the cosmological time can easily be expressed in terms of the
radiation temperature by equating the energy density of cold particles (see (3.1))
with the critical densityεcr= 1 /
(
6 πt^2)
and noting thatT=Tγ 0 ( 1 +z).We obtaintsec 2. 75 × 1017(
(^) mh^275
)− 1 / 2 (Tγ 0
T
) 3 / 2
. (3.191)
At the moment of recombination,ταt∼ 1013 s and, consequently, theLαpho-
tons are not significantly redshifted before reabsorption. To simplify our consider-
ations we will neglect the redshift effect.
The presence of a large number ofLαphotons and other energetic photons
results in a greater abundance of electrons, protons and hydrogen atoms in excited
2 Sand 2Pstates than expected according to the equilibrium Saha formula. This
delays recombination so that, for a given temperature, the actual ionization fraction
exceeds its equilibrium value. The full system of kinetic equations describing non-
equilibrium recombination is rather complicated and is usually solved numerically.
To solve them analytically, we use the method of quasi-equilibrium concentrations,
as was applied to the problem of nucleosynthesis. The results obtained by this
method are in good agreement with the numerical calculations.
We will make a series of simplifying assumptions whose validity can be checked
a posteriori.First of all, we neglect all highly excited hydrogen states and retain
only the 1S, 2 Sand 2Pstates of neutral hydrogen. The remaining ingredients are
electrons, protons and thermal photons, as well asLαand other nonthermal pho-
tons emitted during recombination. The main reactions in which these components
participate are symbolically represented in Figure 3.9. The direct recombination to
the ground state can be ignored because it results in no net change of neutral hydro-
gen, as explained above. Thermal radiation dominates in ionizing the 2Sand 2P
states. In fact, to ionize an excited atom, the energy of a photon must only be larger
thanBH/4. The number of such thermal photons is much greater than the number
of energetic nonthermal photons, and hence, when considering the ionization of
excited atoms, we can ignore the distortion of the thermal radiation spectrum.
In contrast, thermal photons play no significant role in transitions between 1S
and 2Pstates after the beginning of recombination. These transitions are mostly due