Physical Foundations of Cosmology

3.3 Rudiments of thermodynamics 75

whereλ∼ 1 /pis the de Broglie wavelength andp=E∼Tis the typical momen-
tum of the colliding ultra-relativistic particles. The correspondingdimensionless
running coupling constantsαvary only logarithmically with energy and are of or-
der 10−^1 –10−^2. Taking into account that the number density of the ultra-relativistic
species isn∼T^3 ,we find that

tc∼

1

α^2 T

. (3.8)

Comparing this time to the Hubble time,

tH∼

1

H

∼

1

√

ε

∼

1

T^2

, (3.9)

we find that at temperatures belowT∼O( 1 )α^2  1015 –10^17 GeV, but above a
few hundred GeV (where (3.7) is applicable), (3.6) is satisfied and the electroweak
as well as the strong interactions are efficient in establishing equilibrium between
quarks, leptons and intermediate bosons.
The discerning reader might question whether one can apply the formulae for
cross-sections derived in empty space to interactions which occur in extremely
dense “plasma”. To get an idea of the strength of the plasma effects, we have to
compare the typical distance between the particles 1/n^1 /^3 ∼ 1 /T to the “size”
of the particles

√

σ∼α/T.If the coupling constantαis smaller than unity, the
plasma effects are not very relevant.
Primordial gravitons and, possibly, other hypothetical particles that interact
through thedimensionful gravitational constant already decouple from the rest
of matter at Planckian times and propagate, subsequently, freely.
Below 100 GeV,the Z andW± bosons acquire mass (MW 80 ,4 GeV,
MZ 91 ,2 GeV) and, thereafter, the cross-sections of the weak interactions begin
to decrease as the temperature drops. As a result, the neutrinos decouple from the
rest of matter. Finally the electromagnetic interactions also become inefficient and
photons propagate freely. All these processes will be analyzed in detail later in the
chapter, but first we would like to concentrate on the very early stages when known
particles were in equilibrium with radiation and with each other. In this case, matter
can be described in a very simple way: all particles are completely characterized
by their temperature and corresponding chemical potential.

3.3.1 Maximal entropy state, thermal spectrum, conservation

laws and chemical potentials

In this section, we outline an elegant derivation of the main formulae describing the
maximal entropy state. This derivation is based entirely on the notion of entropy for