Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 75
et al. (1999) had considered to fit the entropy density with
s(T )= 2π2
3
T gHG +∆gΘ(T − Tc) RL + (1 − RL) 1 −
45 Tc
γ T
(137)
which is valid for T > Tc. Here Θ and RL are given by equations (136) and (114)
respectively, ∆g = gQGP − gHG and a good fit is obtained for 0.3 < γ < 0.4.
We consider, for the rest of the text, γ =1/3. In Figure 22 we show the curve
of s(T ) (labeled ‘lattice fit’).
The other thermodynamic quantities (for T > Tc) can be derived from equation
(137). Below Tc again is valid the equation for an ideal HG as in the case
of the Bag Model (Schmid et al., 1999).
Inserting the entropy fit given by equation (137) into equation (14) we obtain
(Schmid et al., 1999)
c 2 s ∝
1 − Tc
1−γ T
valid for T ≥ Tc. In order to recover c 2 s =1/3 when T ≫ Tc we consider
c 2 s
(138)
1
= 1 −
3
Tc
1−γ . (139)
T
For T = Tc we have c2 s = 0 and for T < Tc we get, once again, c2 s =1/3. In
Figure 25 we show expression (139), as well as the results obtained for quenched
QCD (Figure 24), for T ≥ Tc with Tc = 170 MeV. The analytic approach
given by equation (139) and the numerical results obtained from quenched QCD
(Figure 24) both show a similar behaviour, in particular, when T gets below
∼ 2Tc.
It is useful to have also an expression for c2 s as a function of time t. Inserting
expression (78) into expression (139) we obtain
c 2
s (t) =1 1 −
3
R(t)
1−γ R(t−)
(140)
valid for t ≤ t− where t− corresponds to the beginning of the phase transition.
Considering equation (86) or, alternatively, equation (71), we may write
equation (140) in the form
c 2 s (t) =1
3
2.3.3 Crossover
1/2
1−γ
t
1 −
. (141)
t−
Recent results provide strong evidence that the QCD transition is a Crossover,
at least using staggered fermions, i.e., including fermionic fields in LGT (Aoki
et al., 2006a,b).