Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 129
∆
1.4
1.2
1
0.8
0.6
0.4
0.2
x2 x15 x30 x90
∆c1
∆c2
No BH formation
BH formation
∆c13
No BH formation
20 40 60 80 100
x
Figure 52: The curve in the (x, δ) plane indicating which parameter values lead
to collapse to a BH, according to the QCD Bag Model, in the case x>1 and
δc =1/3. We show the values of x corresponding to the cases presented in
Figures 50 and 51. The intersection point δc1 = δc2 corresponds to x ≈ 54.8
and the intersection point δc2 = δc =1/3 corresponds to x ≈ 12.0. (adapted
from Cardall & Fuller, 1998).
7.1.2 During the mixed phase
When y −1 < x < 1 we are dealing with fluctuations of classes B, C or E
(cf. Figure 45). Notice that fluctuations of class A could reach also the range
y −1 < x < 1 but only if they have δ> 1 which leads to the formation of a
separate Universe. For a given x we can determine, with the help of equations
(231) and (234), the range of amplitudes which correspond to each class.
For example, for the case x =0.927, we have, from equation (231), that
δ ≈ 0.079 and from equation (234) that δ ≈ 0.94. This means that when x =
0.927 the overdensity will be of class E if δ< 0.079, of class C if 0.079 < δ < 0.94
and of class B if δ> 0.94. The division between class B and A occurs according
to equation (233) when δ ≈ 1.7.
In Figure 56 we plot both (1 − f)δc and δ itself as functions of δ for the
cases: (a) x =0.927, (b) x =0.6, and (c) x =0.308 with δc =1/3 for all
the three cases. The appropriate function f was used to each class (i.e. fB –
equation 255; fC – equation 256; fE – equation 257). In the case x =0.927
PBHs are formed from fluctuations of classes B and C. Fluctuations of class C
with δ< 0.28 dissipate without forming a PBH. This point δc1 ≈ 0.28 marks
the new threshold for PBH formation during the QCD phase transition when
x =0.927.
In the case x =0.6 PBHs can form from fluctuations of class C or class
E, but no longer from fluctuations of class B. The new threshold for PBH
formation is δc1 ≈ 0.29. Finally, in the case x =0.308 PBHs can only form
from fluctuations of class E (the separation between classes E and C occurs for
δCE ≈ 2.24). The new threshold for PBH formation is δc1 ≈ 0.30.