Physical Foundations of Cosmology

4.2 Quantum chromodynamics and quark–gluon plasma 147

but atTQCDthey constitute only a small fraction of the total energy density.
Baryon number is very small and therefore we can neglect the appropriate chemical
potentials. The contribution of the quark–gluon plasma to the total pressure is then

pqg=

κqg

3

T^4 −B(T), (4.32)

where functionB(T)represents thecorrectiondue to the soft, low-energy modes
and

κqg=

π^2

30

(

2 × 8 +

7

8

× 3 × 2 × 2 ×Nf

)

. (4.33)

The first term on the right hand side here accounts for the contribution of eight
gluons (with two polarizations each) and the second one is due toNflight quark
flavors withmqT(every flavor has three colors, two polarization states and the
extra factor 2 accounts for antiquarks).
Unfortunately the correction termB(T)cannot be calculated analytically from
first principles. To get an idea of how it may look, we can use a phenomenological
description of confinement, for instance, the MIT bag model. According to this
model, quarks and gluons are described by free fields inside bags (bounded regions
of space), identified with hadrons, and these fields vanish outside the bags. To
account for appropriate boundary conditions in a relativistically invariant way, one
adds to the Lagrangian “a cosmological constant”B 0 (called the bag constant),
which is assumed to vanish outside the bag. This “cosmological constant” induces
negative pressure and prevents quarks escaping from the bag. In a quark–gluon
plasma, where the bags “overlap”,B(T)=B 0 =const everywhere.
Given the pressure, the energy density and entropy can be derived using the
thermodynamical relations (3.33) and (3.31):

sqg=

4

3

κqgT^3 −

∂B

∂T

,εqg=κqgT^4 +B−T

∂B

∂T

. (4.34)

As soon as the temperature drops below some critical valueTc,which on quite
general grounds is expected to be aboutQCD200 MeV,most quarks and gluons
will be trapped and confined within the lightest hadrons−pions (π^0 ,π±).Their
masses are about 130 MeV and at the time of the phase transition they can still
be treated as ultra-relativistic particles. After quarks and gluons are captured, the
total number of degrees of freedom drops drastically, from 16 (for gluons)+ 12 Nf
(for quarks)to only 3 for pions. The pressure and entropy density of the ultra-rela-
tivistic pions are then

ph=

κh

3

T^4 ,sh=

4

3

κhT^3 , (4.35)

whereκh=π^2 / 10.