132 Thermal History of the Universe
The density parameter is훺휈=^3
11푁훾
휌cℎ^2∑
푖푚푖. (6.68)
Recombination Era. As long as there are free electrons, the primordial photons
are thermalized by Thomson scattering against them, and this prohibits the electrons
from decoupling, in contrast to neutrinos. Each scattering polarizes the photons, but
on average this is washed out. The electromagnetic reaction rate is much higher than
the weak reaction rate of the neutrinos; in fact, it is higher than the expansion rate of
the Universe, so the condition in Equation (6.57) is fulfilled.
Eventually the Universe expands and cools to such an extent, to about 1000K, that
electrons are captured into atomic orbits, primarily by protons but also by the trace
amounts of ionized helium and other light nuclei. This process is referred to as recom-
bination. Unlike the unstable particles푛,휋,휇that decay spontaneously liberating
kinetic energy in exothermic reactions, the hydrogen atom H is abound stateof a
proton and an electron. Its mass is less than the p and e−masses together,
푚H−푚p−푚e=− 13 .59 eV, (6.69)so it cannot disintegrate spontaneously into a free proton and a free electron. The
mass difference in Equation (6.69) is thebinding energyof the hydrogen atom.
The physics of recombination is somewhat subtle. Initially one might think that
recombination occurs when the photon temperature drops below 13.59eV, making
formation of neutral hydrogen energetically favorable. Two characteristics of the
physics push the recombination temperature lower. The first, and the easiest to elabo-
rate, is that there are vastly more photons than electrons and so in thermal equilibrium
even a small proportion of high-energy photons are sufficient to maintain effectively
complete ionization. Photons in thermal equilibrium have the blackbody spectrum
given by Equation (6.10). Even for photon temperatures somewhat below 13.59eV
there will be enough highly energetic photons in the Wein tail (as the high-energy sec-
tion is termed) to ionize any neutral hydrogen. The large amount of entropy in the
Universe also favors free protons and electrons.
With respect to the thermal history of the Universe, the fact that photons do not
scatter against neutral atoms is critical. As recombination proceeds and the number of
electrons falls, matter and radiation decouple. This has two results. First, with matter
and radiation no longer in thermal equilibrium the thermal history of the two proceed
independently. Perturbations in matter are no longer damped by interaction with
radiation and any such perturbations can grow into structures through gravitational
instability. Decoupling thus initiates the period of structure formation that has led to
our present Universe being populated with stars, galaxies, galaxy clusters and so on.
The second result is that, with photons no longer scattering against a sea of elec-
trons, the photons can stream freely through the Universe; upon recombination, the
Universe becomes transparent to light. Prior to recombination, the Universe was
opaque to electromagnetic radiation (although not to neutrinos) and it would have
been impossible to do astronomy if this situation had persisted until today. The freely