92 The hot universe
baryon–antibaryon pairs becomes small compared to the baryon excess at temper-
atures below 40 MeV while positrons can be neglected atT<20 keV.
At low temperatures, the conserved charge is mostly carried by the light-
est particles possessing the given charge. For example, taking into account that
μ=μn,from (3.60) we obtain
n
nn(
m
mn) 3 / 2
exp(
−
m−mn
T)
exp(
−
176 MeV
T)
. (3.78)
Thus, atT<176 MeV, the contribution ofparticles to the total baryon number
can be discarded and the baryon asymmetry is due to the lightest baryons−protons
and neutrons. Similarly, at temperatures below 100 MeV, the electric charge excess
carried by leptons and mesons is mostly due to the overabundance of electrons, since
μ- andτ-leptons and the lightlest-charged mesons have relatively large masses,
namely,mμ106 MeV,mτ 1 .78 GeV andmπ±140 MeV.
3.4.1 Chemical potentials
At temperatures higher than a few MeV, the weak and electromagnetic interactions
are efficient and baryons, leptons and photons are in localthermalandchemical
equilibrium. Note that in general, thermal and chemical equilibria are distinct. For
example, while strong and electromagnetic interactions keep neutrons, protons and
radiation at the same temperature, if the weak interaction rate is smaller than the
expansion rate the chemical potentials of protons and neutrons do not need to satisfy
a chemical equilibrium condition.
At temperatures below 100 MeV, we can neglect all heavy baryons and leptons.
Let us estimate the chemical potentials of various matter components at these
temperatures, beginning with neutrinos. Assuming that the lepton numbersLiare
much smaller than unity, we find from (3.72) and (3.54) that
μντ,μ
T∼Lτ,μ, (3.79)where the entropy density is estimated ass∼T^3 and we have taken into account
that the main contribution toLτ,μcomes fromντ,μbecause theτ- andμ-leptons
have large masses.The electrons are the lightest leptons which carry the electric
charge needed to compensate the electric charge of the baryons. Therefore, their
contribution toLeis not negligible and the estimate, analogous to (3.79), applies to
thesumμe+μνerather than to the chemical potential of electron neutrinos alone.
We see that the chemical potentials of relativistic particles decrease in proportion
to the temperature as the universe expands.