3.6 Recombination 129cosmological expansion rate, which is an important factor in the kinetic description
of nonequilibrium recombination.
Problem 3.26Compare nonequilibrium recombination with the predictions of the
Saha formula for various values of the cosmological parameters (^) mh^275 andη 10.
In which cases is the deviation from the Saha result large immediately after the
beginning of recombination?
Atz<900, when the temperature drops below 2450 K,the approximate for-
mula (3.202) is no longer valid and we should use (3.201). The ionization frac-
tion continues to drop at first and then freezes out. For example, (3.201) pre-
dictsXe(z= 800 ) 5 × 10 −^3 ,Xe( 400 ) 7 × 10 −^4 andXe( 100 ) 4 × 10 −^4 ,for
(^) mh^275 0 .3 andη 10 5 .To calculate the freeze-out concentration, we note that
the integral in (3.201) converges to 0.27 aszgoes to zero; hence,
Xef 2. 5 × 10 −^3
√
(^) mh^275
η 10
1. 6 × 10 −^5
√
(^) m
(^) bh 75
. (3.203)
After the ionization fraction drops below unity, the approximate results given in
(3.201) and (3.202) are in excellent agreement with the numerical solutions of the
kinetic equations, while the Saha approximation fails completely (see Figure 3.10).
Problem 3.27Freeze-out of the electron concentration occurs roughly when the
rate of the reactionep→Hγbecomes comparable to the cosmological expansion
rate. Using this simple criterion, estimate the freeze-out concentration.
At the beginning of recombination, most of the neutral hydrogen atoms are
formed as a result of cascading transitions, and the number ofLαphotons is about
the same as the number of hydrogen atoms. What happens to all theseLαphotons
afterwards? Do they survive and, if so, can we observe them today as a (redshifted)
narrow line in the spectrum of the CMB? During recombination, the number density
ofLαphotons,nα,satisfies the quasi-equilibrium condition for theLαreservoir:
W 2 P→ 1 Sn 2 P=〈σα〉nαn 1 S. (3.204)Sincen 1 S→nT andn 2 P∝X^2 e,the number ofLα photons decreases with the
ionization fraction and nearly all of them disappear by the end of recombination.
Their number density is depleted due to two-photon decays of the 2Sstates. Hence,
there will be no sharp line in the primordial radiation spectrum. Nevertheless, as a
result of recombination, the CMB is warped in this part of the Wien region. This
region is significantly obscured, however, by radiation from other astrophysical
sources.