134 Thermal History of the Universe
Saha prediction. Consequently, all the times predicted by this nonequilibrium analysis
differs notably from the Saha prediction, but, interestingly, in such a way that the
times of decoupling and last scattering have practically no dependence on cosmolog-
ical parameters.
Summary. Recombination, decoupling and last scattering do not occur at the exactly
same time. It should also be noted that these terms are often used interchangeably in
the literature, so what we refer to as the LSS is sometimes called the time of recom-
bination or decoupling. Recombination can be defined as the time when 90% of the
electrons have combined into neutral atoms. This occurred at redshift
푧recombination≈ 1100. (6.70)Last scattering is defined as the time when photons start to stream freely. This
occurred at redshift
푧LSS= 1089 , (6.71)when the Universe was 379 000(훺 0 ℎ^2 )−^1 ∕^2 years old and at a temperature of 0.26eV,
thus right after the recombination time.
Decoupling is defined as the time when the reaction rate (scattering) fell below
the expansion rate of the Universe and matter fell out of thermal equilibrium with
photons. This occurred at redshift
푧decoupling≈ 890. (6.72)All three events depend on the number of free electrons (the ionization fraction) but
in slightly different ways. As a result these events do not occur at exactly the same
time, but close enough to explain the confusion in naming.
The build up of structures started right after decoupling. Reionization of the
Universe occurred at푧reionization= 20 ±5 and the matter structures were in place at
푧structures=5.
6.5 Big Bang Nucleosynthesis
Let us now turn to the fate of the remaining nucleons. Note that the charged current
reactions in Equations (6.47) and (6.48) changing a proton to a neutron areendother-
mic: they require some input energy to provide for the mass difference. In the reaction
in Equation (6.48) this difference is 0.8MeV and in reaction in Equation (6.49) it is
1.8MeV (use the masses in Table A.4!). The reversed reactions areexothermic.They
liberate energy and they can then always proceed without any threshold limitation.
The neutrons and protons are then nonrelativistic, so their number densities are
each given by Maxwell–Boltzmann distributions [Equation (6.29)]. Their ratio in equi-
librium is given by
푁n
푁p=
(
푚n
푚p) 3 ∕ 2
exp(
−
푚n−푚p
kT