Physical Foundations of Cosmology

128 The hot universe

equation (3.202)

Saha approximation

numerics

600 800 1000 1200 1400

redshift z

ionization

X

e

equation (3.201)

100

10 −^1

10 −^2

10 −^3

10 −^4

Fig. 3.10.

In this regime the rate of recombination is completely determined by the rate of
two-photon decay. Obviously, (3.201) and (3.202) are valid only after the ion-
ization fraction decreases significantly below unity and the deviation from the
equilibrium becomes significant. Compared with numerical results, they become
efficiently accurate after the concentration of neutral hydrogen has reached about
50% (Figure 3.10). According to (3.202), for realistic values of the cosmological
parameters

(

(^) mh^275  0 .3 andη 10  5

)

,this occurs atz1220 or, equivalently,
atT3400 K.Hence, the range of applicability of (3.202) is not very wide,
namely, 1200>z> 900 .During this time, however, the temperature drops only
from 3400 K to 2450 K but the ionization fraction decreases very substantially,
toXe( 900 ) 2 × 10 −^2 .It is interesting to compare this result with the prediction
of the equilibrium Saha formula (3.189), according to whichXe(2450 K)∼ 10 −^5.
Thus, atz 900 ,the actual ionization fraction is a thousand times larger than
the equilibrium one. It is also noteworthy that the equilibrium ionization frac-
tion is completely determined by the baryon density and the temperature but the
nonequilibriumXe(z),given in (3.201), also depends on the total density of nonrel-
ativistic matter. This is not surprising because nonrelativistic matter determines the