74 The hot universe
Unification scales. It is remarkable and fortunate that the most important robust predictions
of inflation do not depend substantially on unknown particle physics. Therefore, the
existence of such a stage may be observationally verified in the near future.
∼ 10 −^43 s( 1019 GeV)Near the Planckian scale, nonperturbative quantum gravity domi-
nates and general relativity can no longer be trusted. However, at energies slightly below
this scale, classical spacetime still makes sense and we expect that the universe is in a self-
reproducing phase. Nevertheless, self-reproduction does not eliminate the fundamental
issues of spacetime structure at the Planckian scale. In particular, the question of cosmic
singularities still remains. It is expected that these problems will be properly addressed
in an as yet unknown nonperturbative string/quantum gravity theory.3.3 Rudiments of thermodynamics
To properly describe the physical processes in an expanding universe we need,
strictly speaking, a full kinetic theory. Fortunately, the situation greatly simplifies
in the very early universe, when the particles are in a state oflocalequilibrium
with each other. We would like to stress that the universe cannot be treated as a
usual thermodynamical system in equilibrium with an infinite thermal bath of given
temperature: it is a nonequilibrium system. Therefore, bylocal equilibriumwe
simply mean that matter has maximal possible entropy. The entropy is well defined
for any system even if this system is far from equilibrium and never decreases.
Therefore, if within a typical cosmological time the particles scatter from each
other many times, their entropy reaches the maximal possible value before the size
of the universe changes significantly.
The reaction rate responsible for establishing equilibrium can be characterized
by thecollision time:
tc1
σnv, (3.5)
whereσis the effective cross-section,n is the number density of the particles
andvis their relative velocity. This time should be compared to the cosmic time,
tH∼ 1 /H,and if
tctH, (3.6)
local equilibrium is reached before expansion becomes relevant. Let us show that
at temperatures above a few hundred GeV condition (3.6) is satisfied for both
electroweak and strong interactions. At such high temperatures, all known particles
are ultra-relativistic and the gauge bosons are all massless. Therefore, the cross-
sections for strong and electroweak interactions have a similar energy dependence
and they can be estimated (e.g. on dimensional grounds) as
σO( 1 )α^2 λ^2 ∼α^2
T^2