130 The hot universe
Finally, let us find out when exactly the universe becomes transparent to radi-
ation. This occurs when the typical time for photon scattering begins to exceed
the cosmological time. The Rayleigh cross-section for scattering on neutral hydro-
gen is negligibly small and, therefore, despite of the low concentration of elec-
trons, the opacity is due to Thomson scattering from free electrons. Substituting
σT 6. 65 × 10 −^25 cm^2 ,cosmological timetand total number densityntfrom
(3.191) and (3.188) respectively, into
t∼1
σTntXe, (3.205)
we find that photon decoupling occurs when
Xedec∼ 6 × 103√
(^) mh^275
η 10
(
Tγ 0
Tdec) 3 / 2
. (3.206)
It follows thatTdec∼2500 K, or equivalently,zdec∼ 900 ,independent of the
cosmological parameters. For (^) mh^275 0 .3 andη 10 5 ,the ionization fraction at
this time is about 2× 10 −^2. It is interesting to note that this time coincides with
the moment whene,pand 2Slevels fall out of equilibrium and the approximate
(3.202) becomes inapplicable.
Radiation decoupling does not mean that matter and radiation lose all thermal
contact. In fact, the interaction of a small number of photons with matter keeps the
temperatures of matter and radiation equal down to redshiftsz∼ 100 .Only after
that does the temperature of baryonic matter begin to decrease faster than that of
radiation. There is no trace of this temperature in baryons seen today because most
of them are bound to galaxies where they are heated during gravitational collapse.