124 Thermal History of the Universe
For each species of relativistic fermions participating in the thermal equilibrium
there is a specific number density. To find the total number density of particles sharing
the available energy we have to count each particle species푖weighted by the corre-
sponding degrees of freedom푔푖.
Equation (6.13) with a factor푔푖explicitly visible is
휀푖=^1
2푔푖푎S푇^4 , (6.38)
where푎Sis Stefan’s constant. It turns out that this expression gives the correct energy
density for every particle species if we insert its respective value of푔푖from Table A.5.
Equation (6.12) can be correspondingly generalized to relativistic fermions. Their
number density is
푁f=^3
4푁훾. (6.39)
In general, the primordial plasma was a mixture of particles, of which some are rel-
ativistic and some nonrelativistic at a given temperature. Since the number density
of a nonrelativistic particle [given by the Maxwell–Boltzmann distribution, Equation
(6.28)] is exponentially smaller than that of a relativistic particle, it is a good approx-
imation to ignore nonrelativistic particles. Different species푖with mass푚푖have a
number density which depends on푚푖∕푇, and they may have a thermal distribution
with a temperature푇푖different from that of the photons. Let us define theeffective
degrees of freedomof the mixture as
푔∗=
∑
bosons푖푔푖+
∑
fermions푗푔푗
(푇
푗
푇) 4
. (6.40)
As explained in the context of Equation (6.37) the sum over fermions includes a fac-
tor^78 , accounting for the difference between Fermi and Bose statistics. The factor
(푇푗∕푇)^4 applies only to neutrinos, which obtain a different temperature from the pho-
tons when they freeze out from the plasma (as we shall see later). Thus the energy
density of the radiation in the plasma is
휀r=^1
2푔∗푎S푇^4. (6.41)
The sum of degrees of freedom of a system of particles is of course the number
of particles multiplied by the degrees of freedom per particle. Independently of the
law of conservation of energy, the conservation of entropy implies that the energy is
distributed equally between all degrees of freedom present in such a way that a change
in degrees of freedom is accompanied by a change in random motion, or equivalently
in temperature.
Thus entropy is related to order: the more degrees of freedom there are present, the
more randomness or disorder the system possesses. When an assembly of particles
(such as the molecules in a gas) does not possess energy other than kinetic energy
(heat), its entropy is maximal when thermal equilibrium is reached. For a system of
gravitating bodies, entropy increases by clumping, maximal entropy corresponding
to a black hole.