3.4 Lepton era 93
As we found, atT<40 MeV, antibaryons can be neglected and, therefore,
np,nnp,n.The conservation law of the total baryonic charge (3.70) implies that
B
np
s(
1 +
nn
np)
(3.80)
remains constant. The factor inside the parentheses is of order unity. Using formula
(3.61) fornp,we obtain
mp−μp(t)
T(t)ln(
1
B
(mp
T) 3 / 2 )
. (3.81)
ForB 10 −^10 ,the chemical potentialμpchanges from about−115 MeV to
+967 MeV as the temperature drops from 40 MeV to 1 MeV.The number density
of protons decays asT^3 ∝a−^3 .Substituting (3.81) into (3.63), we obtain the
following estimate for the contribution of protons to the total entropy,
sp
s(
mp−μp(t)
T(t)+
5
2
)
np
sBln(
1
B
(mp
T) 3 / 2 )
. (3.82)
Thus,nonrelativisticprotons contribute only a small fraction of order 10−^8 to the
total entropy. It is interesting to note that the total entropy of protons themselves is
not conserved. It follows from (3.82) that this entropy logarithmically increases as
the temperature decreases. This has a simple physical explanation. If nonrelativistic
protons were completely decoupled from the other components, their temperature
would decrease faster than the temperature of the relativistic particles, namely, as
1 /a^2 instead of 1/a.Therefore, to maintain thermal equilibrium with the dominant
relativistic components, the protons borrow the energy and entropy needed from
them.
Problem 3.12Verify that the conservation of entropy and of the total number of
nonrelativistic particles implies that their temperature decreases in inverse propor-
tion to the second power of the scale factor. How does the chemical potential depend
on the temperature in this case?
To estimate the chemical potential of electrons,μe,we use the conservation law
for the electric charge (see (3.71)). Because the universe is electrically neutral, we
haveQ= 0 .Taking into account that electrons are still relativistic atT>1 MeV
and skipping the negligible contributions fromτ- andμ-leptons andπ−mesons in
(3.71), we find
μe
Tne
snp
s