120 The hot universe
3.6 Recombination
The most important matter ingredients in thermodynamical processes after nucle-
osynthesis are thermal radiation, electrons, protonsp(hydrogen nuclei) and fully
ionized helium nuclei, He^2 +.The concentrations of the other light elements are very
small and we neglect them here. As the temperature decreases, the ionized helium
and hydrogen nuclei begin to capture the available free electrons and become elec-
trically neutral. In a short period of time, nearly all free electrons and nuclei have
combined to form neutral atoms and the universe becomes transparent to radiation.
Since this process occurs so quickly, we refer to this epoch as the recombination
moment.
We must, however, distinguish the helium and hydrogen recombinations, be-
cause they happen at different times. Helium has significantly larger ionization
potentials than hydrogen and therefore becomes neutral earlier. However, after he-
lium recombination, many free electrons remain and the universe is still opaque to
radiation. Only after hydrogen recombination have most photons decoupled from
matter; these are the photons that give us a “baby photo” of the universe. As a
result, hydrogen recombination is a more interesting and dramatic event from an
observational point of view.
Helium recombination, nevertheless, has some cosmological relevance. When
helium becomes neutral it decouples from the plasma thus altering the speed of
sound in the radiation–baryon fluid. We will see in Chapter 9 that this speed influ-
ences the CMB temperature fluctuations.
Recombination is not an equilibrium process. Hence, the formulae derived under
the assumption of local equilibrium can only be used to estimate when recombina-
tion occurs. This is sufficient when we consider helium recombination. However, the
subtleties of hydrogen recombination are very important for the calculation of the
CMB temperature fluctuations. Therefore, after estimating the hydrogen recombi-
nation temperature based on the equilibrium equations, we will use kinetic theory
to reveal the details of nonequilibrium recombination.
3.6.1 Helium recombination
The electric charge of the helium nucleus is 2, so it must capture two electrons to
become neutral. This occurs in two steps. First, the helium captures one electron,
becoming a singly charged, hydrogen-like ion He+. The binding energy of this ion
is four times larger than the binding energy for hydrogen:
B+=me+m 2 +−m+= 54 .4eV, (3.175)wherem 2 +andm+are the masses of He^2 +and He+respectively. This energy
corresponds to a temperature of 632 000 K.To estimate the temperature at which