Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 111
Table 26: Classification of overdense regions according to the state of matter
at the horizon crossing time and at the turnaround point for the QCD phase
transition (Cardall & Fuller, 1998).
Class Horizon Crossing phase Turnaround phase
A quark–gluon quark–gluon
B quark–gluon mixed
C quark–gluon hadron
D mixed mixed
E mixed hadron
F hadron hadron
where R(t−) is given by equation (71) and R(t) is given by: i) equation (69)
if x ≤ y −1 ; ii) equation (70) if y −1 < x < 1; iii) equation (71) if x ≥ 1.
The value of y, which defines the end of the transition, can now be determined
evaluating x(t+). It turns out that for the Bag Model case (cf. Section 2.3.1),
with gQGP = 61.75 and gHG = 21.25, one obtains
2 t−
4gQGP − gHG
t+
y −1 = x(t+) =
4gQGP − gHG
≈ 0.272. (218)
In Figure 44 we show the curve x(t). Notice that, equation (217) is to be used
only during the first–order QCD transition: more precisely, in the neighborhood
of the transition. For example, if one considers x ≪ y−1 then x will eventually
become negative which does not make sense.
It will also be useful to know the expression which gives the turnaround
point for each class of fluctuations. For classes A and F, which evolve completely
during a radiation–dominated phase (w = wk = wc =1/3), we obtain, taking
into account that Sk = Rk, that Ks/Kk = 1. In this case we have, from equation
(212) the result
1/2 1+δk
. (219)
Sc,A = Sc,F = Rk
δk
For class D, which evolves completely during the dust phase (w = wk = wc = 0)
we have, according to equation (212)
Sc,D = Rk
1+δk
. (220)
δk
For classes B, C and E the value of the adiabatic index w varies during the
fluctuation. For example, in the case of class B we have wk =1/3 and wc = 0.