3.6 Recombination 127the temperature drops belowT2450 K,thermal radiation no longer plays an
essential role, and the quasi-equilibrium concentration of the 2Sstates is determined
by equating the rates for recombination to the 2Slevel and two-photon decay (see
(3.193), where the second term can be neglected). AtT<2450 K, the second term
in the square brackets in (3.196) can be neglected and (3.196) simplifies to
dXe
dt−〈σv〉ep→ 2 SnTXe^2. (3.198)Thus, in this regime, the rate of recombination is entirely determined by the rate
of recombination to the 2Slevel and does not depend onW 2 S. At this time, the
number density ofLαphotons is almost completely depleted due to two-photon
decays and the 2Pstates also drop out of equilibrium with electrons, protons and
thermal radiation. Consequently, nearly every recombination event to the 2Pstate
or any other excited state succeeds in producing a neutral hydrogen atom. This
effect becomes relevant only at late stages and can be incorporated in (3.198) and
(3.196) by replacing〈σv〉ep→γ 2 Swith the cross-section for recombination to all
excited states. The latter is well approximated by the fitting formula
〈σv〉rec 8. 7 × 10 −^14(
BH
4 T
) 0. 8
cm^3 s−^1. (3.199)It is convenient to rewrite the corrected (3.196) using the redshift parameterz=
T/Tγ 0 −1, instead of cosmological time (see (3.191)). After some elementary
algebra, we obtain
dXe
dz0. 1
η 10
√(^) mh^275
[
0. 72
(
z
14400) 0. 3
+ 104 zexp(
−
14400
z)]−^1
X^2 e. (3.200)This equation is readily integrated:
Xe(z) 6. 9 × 10 −^4√
(^) mh^275
η 10
⎡
⎢
⎣
∫
14400 zdy
0. 72 y^0.^3 + 1. 44 × 108 yexp(− 1 /y)⎤
⎥
⎦
− 1.(3.201)
The solutionXe(z)is not very sensitive to the initial conditions whenXe(zin)
Xe(z)because the main contribution to the integral comes from z<zin.For
z> 900 ,or equivalently, atT>2450 K,the first term in the denominator of
the integrand can be neglected and expression (3.201) is well approximated by
Xe(z) 1. 4 × 109√
(^) mh^275
η 10
z−^1 exp
(
−
14400
z