90 The hot universe
Theτ-lepton also decays, for example intoe−,ν ̄e,ντ,therefore
μτ=μe−−μνe+μντ. (3.67)Finally, from the reactionsπ−→ν ̄μμ−andpe−nνe,we deduce that
μπ−=μe−−μνe (3.68)and
μp=μn+μνe−μe−. (3.69)All other possible reactions lead to relations that are consistent with the ones above.
We recall that the chemical potentials for antiparticles are equal in magnitude to
the chemical potentials for their corresponding particles but have opposite sign.
The five independent chemical potentials can be expressed through the five
conserved quantities. The conservation of the total baryon number means that the
number density of baryons minus antibaryons decreases in inverse proportion to
the third power of the scale factora. If matter is in equilibrium, the total entropy
is conserved and, as the universe expands, the entropy densitysalso scales asa−^3.
Therefore, the baryon-to-entropy ratio
B≡
np+nn+n
s, (3.70)
remains constant. We denote here byn≡n−n ̄the excess of the corresponding
particles over their antiparticles. Similarly, the conservation law for total electric
charge can be written as
Q≡np−ne−nμ−nτ−nπ−
s
=const, (3.71)and for each type of lepton number we have
Li≡ni+nνi
s=const, (3.72)wherei≡e,μorτ.Because allncan be expressed through the temperatureT
and the corresponding chemical potentials, the system of equations (3.70), (3.71)
and (3.72), together with the conservation law for the total entropy,
d(
sa^3)
dt= 0 , (3.73)
allow us to determine the six unknown functions of time:T(t),μe−(t),μn(t),μνe(t),
μνμ(t),μντ(t).
What is known about the numerical values ofB,QandLi? The universe appears
to be electrically neutral and henceQ= 0 .The baryon-to-entropy ratio is rather well