Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 71
Taking into account the pressure coexistence condition (cf. equation 116)
we obtain, from equations (123) and (125), the following expression for the bag
constant (e.g. Cardall & Fuller, 1998; Schmid et al., 1999)
B = π2
90 (gQGP − gHG) T 4 c . (126)
The energy density ρ and entropy density s for the Bag Model follow from
equations (11), (12) and (123). In the case of the energy density we have, for
the QGP phase (e.g. Schmid et al., 1999)
with
ρQGP (T )=ρ ideal
QGP (T )+B (127)
ρ ideal π2
QGP (T )=
30 gQGP T 4
and for the HG phase we get (e.g. Boyanovsky et al., 2006)
(128)
ρHG(T )= π2
30 gHGT 4 . (129)
The evolution of the average energy density ρ as a function of time during a
first–order QCD transition is given by (Jedamzik, 1997)
3
R(t−)
ρ(t) =
ρQGP (Tc)+
R(t)
1
3 ρHG(Tc)
− 1
3 ρHG(Tc). (130)
With the help of equations (126), (127), (128), and (129) this becomes
ρ(t) = 1 π
3
2
30 T 4
3
R(t−)
c 4gQGP
− gHG . (131)
R(t)
In the case of the entropy density, we have, for the QGP (e.g. Schmid et al.,
1999; Jedamzik, 1997)
sQGP (T )=s ideal
QGP (T )= 2π2
45 gQGP T 3
and for the HG
(132)
sHG(T )=s ideal
HG (T )= 2π2
45 gHGT 3 . (133)
In this model, the entropy, jumps at the critical temperature Tc. This is due
to the fact that on the coexistence line both, pressure and temperature, are
constant. This jump in the entropy (which is depicted in Figure 22), means
that the Bag Model leads to a first–order phase transition with a latent heat
(equation 112, e.g. Boyanovsky et al., 2006; Schmid et al., 1999)
l = Tc∆s = 2π2
45 (gQGP − gHG)T 4 c =4B. (134)