Electroweak Interactions 125Temperature and Time Scales. Let us now derive a relation between the temper-
ature scale and the timescale. We have already found the relation (5.40) between the
size scale푅and the timescale푡during the radiation era,
푎(푡)∝√
푡, (6.42)
where we choose to omit the proportionality factor. The Hubble parameter can then
be written
퐻=푎̇
푎=^1
2 푡
. (6.43)
Note that the proportionality factor omitted in Equation (6.42) has dropped out.
In Equation (5.35) we noted that the curvature term푘푐^2 ∕푅^2 in Friedmann’s equa-
tions is negligibly small at early times during the radiation era. We then obtained the
dynamical relation
푎̇
푎=
(
8 휋퐺
3
휌
) 1 ∕ 2
. (6.44)
Inserting Equation (6.43) on the left and replacing the energy density휌on the right
by휀r∕푐^2 , we find the relation sought between photon temperature and time:
1
푡=
√
16 휋퐺푎S
3 푐^2
푔∗푇^2 = 3. 07 × 10 −^21
√
푔∗푇
2
[K^2 ][s−^1 ]. (6.45)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.
Cooling Plasma. At a temperature of 10^11 K, which corresponds to a mean energy
of about 300MeV, the particles contributing to the effective degrees of freedom푔∗
are the photon, three charged leptons, three neutrinos (not counting their three inert
right-handed components), the six quarks with three colors each, the gluon of eight
colors and two spin states, the scalar Higgs boson H^0 , and the vector gauge bosons
W±,Z^0. Thus the effective degrees of freedom is given by
푔∗= 2 + 3 ×7
2
+ 3 ×
7
4
+ 6 × 3 ×
7
2
+ 8 × 2 + 1 + 3 × 3 = 106. 75. (6.46)
Above the QCD–hadron phase transition at 200MeV the hadrons are represented
by their free quark subconstituents which contribute more degrees of freedom than