Photon and Lepton Decoupling 129The condition for a given species of particle to remain in thermal equilibrium is
then that the reaction rate훤is larger than the expansion rate퐻, or equivalently that
훤−^1 does not exceed the Hubble distance퐻−^1 ,
훤
퐻≳ 1. (6.57)
Inserting the푇^5 dependence of the weak interaction rate훤wiand the푇^2 dependence
of the expansion rate퐻from Equation (6.45), we obtain
훤wi
퐻∝푇^3. (6.58)
Thus there may be a temperature small enough that the condition in Equation (5.62)
is no longer fulfilled.
Photon Reheating. Photons with energies below the electron mass can no longer
produce e+–e−pairs, but the energy exchange between photons and electrons still
continues by Compton scattering, reaction in Equation (6.30), orThomson scattering,
as it is called at very low energies. Electromagnetic cross-sections (subscript ‘em’) are
proportional to푇−^2 , and the reaction rate is then proportional to푇,so
훤em
퐻∝
1
푇
.
Contrary to the weak interaction case in Equation (6.58), the condition in Equa-
tion (6.57) is then satisfied for all temperatures, so electromagnetic interactions never
freeze out. Electrons only decouple when they form neutral atoms during theRecom-
bination Eraand cease to scatter photons. The term recombination is slightly mis-
leading, because the electrons have never been combined into atoms before. The term
comes from laboratory physics, where free electrons andionizedatoms are created by
heating matter (and upon subsequent cooling the electrons and ions recombine into
atoms) or from so-called HII regions, where interstellar plasma is ionized by ultra-
violet radiation and characteristicrecombination radiationis emitted when electrons
and ions re-form.
The exothermic electron–positron annihilation, reaction in Equation (6.31), is now
of mounting importance, creating new photons with energy 0.51MeV. This is higher
than the ambient photon temperature at that time, so the photon population gets
reheated. To see just how important this reheating is, let us turn to the law of conser-
vation of entropy.
Making use of the equation of state for relativistic particles (6.15), the entropy can
be written
푆=^4 푉
3 kT휀plasma.Substituting the expression for휀plasmafrom Equation (6.41) one obtains
푆=
2 푔∗
3
푉푎S푇^4
kT, (6.59)
which is valid where we can ignore nonrelativistic particles. Now푎S푇^4 is the energy
density, so푉푎S푇^4 is energy, just likekT, and thus푉푎S푇^4 ∕kTis a constant.푔∗is also