144 Thermal History of the Universe
Second, there must be C and CP violation in the theory, as these operators change
baryons into anti-baryons and leptons into anti-leptons. If the theory were C and CP
symmetric, even the baryon-violating reactions (6.105) and (6.106) would be matched
by equally frequently occurring reactions with opposite훥퐵,sononetBB-asymmetry
would result. In fact, we want baryon production to be slightly more frequent than
anti-baryon production.
Third, we must require these processes to occur out of thermal equilibrium. In ther-
mal equilibrium there is no net production of baryon number, because the X-reactions
(6.105) and (6.106) go as frequently in the opposite direction. Hence the propitious
moment is the phase transition when the X-bosons are freezing out of thermal equi-
librium and decay. If we consult the timetable in Section 6.2, this would happen at
about 10^14 GeV: the moment for the phase transition from the GUT symmetry to its
spontaneously broken remainder.
The GUT symmetry offers a good example, which we shall make use of in this
section, but it is by no means obvious that GUT is the symmetry we need and that the
phase transition takes place at GUT temperature. It is more likely that we have the
breaking of a symmetry at a lower energy, such as supersymmetry.
Leptoquark Thermodynamics. Assuming the GUT symmetry, the scenario is there-
fore the following. At some energy퐸X=푘푇Xwhich is of the order of the rest masses
of the leptoquark bosons X,
퐸X≃푀X푐^2 , (6.107)all the X, Y vector bosons , the Higgs bosons, and the gluons are in thermal equilibrium
with the leptons and quarks. The number density of each particle species is about the
same as the photon number density, and the realations in Equation (6.100) hold.
When the age of the Universe is still young, as measured in Hubble time휏H,com-
pared with the mean life휏X=훤X−^1 of the X bosons, there are no X decays and therefore
no net baryon production. The X bosons start to decay when
훤X≲휏H−^1 =퐻. (6.108)This is just like the condition in Equation (5.62) for the decoupling of neutrinos. The
decay rate훤Xis proportional to the mass푀X,
훤X=훼푀X, (6.109)where훼is essentially the coupling strength of the GUT interaction. It depends on the
details of the GUT and the properties of the X boson.
We next take the temperature dependence of the expansion rate퐻from Equa-
tions (6.43) and (6.45). Replacing the Newtonian constant퐺by its expression in terms
of the Planck mass푀P, as given in Equation (6.2), we find