Physical Foundations of Cosmology

148 The very early universe

Whether the transition from the quark–gluon plasma to hadronic matter is char-
acterized by truly singular behavior of the basic thermodynamical quantities or
their derivatives (first or second order phase transitions, respectively), or whether
it is merely a cross-over with rapid continuous change of these quantities, crucially
depends on the quark masses. A first order phase transition is usually related to
a discontinuous change of the symmetries characterizing the different phases. In
SU( 3 )pure gauge theory, without dynamical quarks, the expected first order phase
transition has been verified in numerical lattice calculations. In the case of two
quark flavors one expects a second order phase transition (continuous change of
symmetry). In the limit of three massless quarks, again for reasons of symmetry,
we expect a first order transition. When the quark masses do not vanish, the appro-
priate symmetries can be broken explicitly and a cross-over is expected. This is the
situation most likely realized in nature: of the three quarks relevant for dynamics,
two(u,d)are very light and one(s)is relatively heavy. However, despite more
than 20 years of efforts, the character of the cosmological quark–gluon transition
has not yet been firmly established. This is due to the great difficulty of computing
with light dynamical quarks on the lattice. The possibility of a true phase transition,
therefore, has not been ruled out.
Irrespective of the nature of the transition, there is a very sharp change in the
energy density and entropy in the narrow temperature interval aroundTc.This
result is confirmed in lattice calculations and clearly indicates the liberation of the
quark degrees of freedom. A first order phase transition is the most interesting for
cosmology and therefore we briefly discuss it, assumingB(T)=B 0 =const,as in
the bag model. This reproduces the bulk features of the equation of state obtained
in numerical lattice calculations. The pressure and entropy density as functions of
temperature are shown in Figure 4.5. AtTc,even if the phase transition is first order,

s

T^3

p

T^4 T

4

−B 0

c T

3

c

Fig. 4.5.