4.2 Quantum chromodynamics and quark–gluon plasma 149the pressure should be continuous, allowing both phases (hadrons and quark–gluon
plasma) to coexist. Hence, equating ( 4.32) and (4.35) atT=Tc,we can express
the critical temperatureTcthrough the bag constantB 0 :
Tc=(
3 B 0
κqg−κh) 1 / 4
=
(
180
(26+ 21 Nf)π^2) 1 / 4
B 01 /^4. (4.36)
ForB 01 /^4 220 MeV and forNf=3 light quark flavors,Tc150 MeV.
Problem 4.9How should (4.32) be modified if the baryon number is different from
zero? Using the conditionpqg(Tc,μB) 0 ,whereμBis the baryon’s chemical
potential, as an approximate criterion for the phase transition, draw in theTc–μB
plane the shape of the transition line separating the hadron and quark–gluon phases.
Why does the above criterion give us a good estimate?
In the case of a first order phase transition, the entropy density is discontinuous
at the transition and its jump,sqg=( 4 / 3 )(κqg−κh)Tc^3 ,is directly proportional to
the change in the number of active degrees of freedom. A first order phase transition
occurs via the formation of bubbles of hadronic phase in the quark–gluon plasma.
As the universe expands, these bubbles take up more and more space and when what
is left is mainly the hadronic phase the transition is over. During a first order phase
transition the temperature is strictly constant and is equal toTc.The released latent
heat,ε=Tcsqg,keeps the temperature of the radiation and leptons unchanged
in spite of expansion. To estimate the duration of the transition one can use the
conservation law for the total entropy.
Problem 4.10Taking into account that in the quantum chromodynamics epoch, in
addition to quarks and gluons, there are photons, three flavors of neutrinos, electrons
and muons, verify that the scale factor increases by a factor of about 1.5 during the
phase transition.
If the transition is of second order or a cross-over, the entropy is a continuous
function of temperature, which changes very sharply in the vicinity ofTc.As the
universe expands the temperature always drops, but during the transition it remains
nearly constant. For the case of a cross-over transition, the notion of phase is not
defined during the transition.
As we have already mentioned, only a first order quantum chromodynamics phase
transition has interesting cosmological consequences. This is due to its “violent
nature.” In particular, it could lead to inhomogeneities in the baryon distribution
and hence influence nucleosynthesis. However, calculations show that this effect
is too small to be relevant. There could be other, more speculative consequences,