Physical Foundations of Cosmology

3.6 Recombination 123

Therefore, for now, we neglect the excited hydrogen atoms and introduce the ion-
ization fraction:

Xe≡

ne

ne+nH

. (3.187)

Sincenp=neand

ne+nH 0. 75 × 10 −^10 η 10 nγ 3. 1 × 10 −^8 η 10

(

T/Tγ 0

) 3

cm−^3 , (3.188)

(3.185) becomes

X^2 e

1 −Xe

=exp

(

37. 7 +

3

2

ln

(

BH

T

)

−

BH

T

−lnη 10

)

. (3.189)

The expression in the exponent vanishes at

Trec

BH

43. 4 −lnη 10

 3650

(

1 + 2. 3 × 10 −^2 lnη 10

)

K. (3.190)

At this time, the ionization fraction isXe 0 .6 and the temperatureTreccharacter-
izes the moment when hydrogen recombination begins. At earlier timesXe→1;
for example, 1−Xe 10 −^5 atT5000 K.As soon as the temperature decreases
belowTrec,recombination proceeds very rapidly. According to (3.189), a 10% fall
in temperature reduces the ionization fraction by a factor of 10; atT∼2500 K, we
haveXe∼ 10 −^4.
The equilibrium Saha formula tells us that the ionization fraction should con-
tinue to decrease exponentially as the temperature drops. However, this does not
occur in an expanding universe; the ionization fraction freezes out instead. More
importantly, the equilibrium description fails almost immediately after the begin-
ning of recombination. The main reason for the failure of the Saha formula is the
large number of energetic photons emitted when the nuclei and electrons combine.
These nonthermal photons significantly distort the high-energy tail of the thermal
radiation spectrum exactly at energies crucial to recombination. As a result, it be-
comes essential to take into account deviation from equilibrium and we must use
kinetic theory.

3.6.3 Hydrogen recombination: the kinetic approach

Direct recombination to the ground state accompanied by the emission of a photon
does not substantially increase the number of neutral atoms because the emitted
photon has enough energy to immediately ionize the first neutral hydrogen atom
it meets. These two competing processes occur at very high rates and result in
no net change innH. More efficient is cascading recombination, in which neutral
hydrogen is first produced in an excited state and then decays to the ground state in