Physical Foundations of Cosmology

3.6 Recombination 125

thermal radiation

e,p

2 P

2 S

1 S

γ+γ

Lα

Fig. 3.9.

to the nonthermalLαphotons. When the deviation from equilibrium becomes large
and the 2Slevel is overpopulated, we can ignore the transition 1S+γ+γ→
2 Scompared to the two-photon decay 2S→ 1 S+γ+γ. (A transition with a
single photon is forbidden by angular momentum conservation.) The rate for the
two-photon decay,W 2 S→ 1 S 8 .23 s−^1 ,is very small (compare, for example, with
W 2 P→ 1 S 4 × 108 s−^1 ); nevertheless, this decay plays the dominant role in non-
equilibrium recombination. In terms of the pipe-and-reservoir picture, the two-
photon transition is the main source of irreversible leakage from thee,preservoir
to the 1Sreservoir. Because all other processes result in high-energy photons which
reionize neutral hydrogen and return electrons to thee,preservoir, the rate of net
change in the ionization fraction is

dXe

dt

=−

dX 1 S

dt

=−W 2 SX 2 S, (3.192)

whereXe≡ne/nT,X 2 S≡n 2 S/nT,andnTis the total number density of neutral
atoms plus electrons, given in (3.188). Once a substantial fraction (∼50%) of neutral
hydrogen has formed, (3.192) is a good approximation to use until nearly the end
of recombination.