88 The hot universe
3.3.5 Nonrelativistic particles
If the temperature is smaller than the rest mass and in addition
m−μ
T1 ,
spin-statistics do not play an essential role and the formulae for bosons and fermions
coincide to leading order. Substituting (3.48) into (3.37), we find that in this case
n−n ̄ 2 g(
Tm
2 π) 3 / 2
exp(
−
m
T)
sinh(μ
T)[
1 +
15
8
T
m]
. (3.60)
It follows that the number density of particles is
ng(
Tm
2 π) 3 / 2
exp(
−
m−μ
T)[
1 +
15
8
T
m]
, (3.61)
and the number density of antiparticles,n ̄,is suppressed by a factor of exp(− 2 μ/T)
compared tonand ifμ/T1 the antiparticles can be neglected. In the early
universe the number density of any type of nonrelativistic species never exceeds
the number density of photons, that is,nnγ∼T^3 ,and hence the inequality
(m−μ)/T1 is fulfilled. The energy density of particles is obtained by substi-
tuting (3.47) and (3.48) into (3.35), and can be expressed in terms of the particle
number density as
εmn+3
2
nT. (3.62)The pressurepnTis much smaller than the energy density and can be neglected
in the Einstein equations. The entropy density of the nonrelativistic particles can
easily be calculated from (3.31) and is equal to
s(
m−μ
T+
5
2
)
n. (3.63)Problem 3.11Ifm/T1but|m−μ|/T 1 ,one cannot ignore spin-statistics.
In this limit, however, the antiparticles are suppressed by a factor of exp(− 2 m/T)
and hence can be neglected. Calculate the corresponding energy density, pressure
and entropy for bosons and fermions in this case. Given a number densityn,verify
that at temperatures belowTB=O( 1 )n^2 /^3 /ma Bose condensate is formed.
The chemical potential of fermions can be arbitrarily large and may signif-
icantly exceed the mass. If (μf−m)/T 1 ,most fermions are degenerate.
Whenμfm, fermions near the Fermi surface have momenta of orderμf
and are therefore relativistic, so we can use the results in (3.53)−(3.55). Other-
wise, if (μf−m)m,the gas of degenerate fermions is nonrelativistic and the