Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 125
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Figure 50: PBH formation during the QCD transition according to the Bag
Model for the case x = 2 and δc =1/3. The solid curve corresponds to the
function (1 − f)δc and the dashed curve to the identity δ. The pink region
corresponds to fluctuations of class B (see text). To the left of this region we
have fluctuations of class C (green) and to the right fluctuations of class A
(yellow). The borders between the different classes are given by δAB = 1 and
δBC ≈ 0.58. Collapse to a BH occurs for values of δ for which the dashed line is
above the solid curve (while δ< 1). The intersection point at δ ≈ 0.25 marks a
new threshold δc1 for PBH formation (adapted from Cardall & Fuller, 1998).
overdensity will be of class C if 0 < δ < 0.58, of class B if 0.58 < δ < 1 and of
class A if δ> 1.
In order to identify the values of δ for which collapse to a BH occurs (when
x = 2 and δc =1/3), we plot in Figure 50 both (1−f)δc and δ itself as functions
of δ. Notice that one should use the function f appropriate to each class (i.e.
fA – equation 254; fB – equation 255; fC – equation 256). We can, then, have
PBHs formed from fluctuations of classes B and C only, since fluctuations of
class A would lead to the formation of a separate Universe (since for them we
always have δ> 1)–Section 6.1.
Fluctuations of class C with δ< 0.25 dissipate before forming a PBH. This
point δc1 ≈ 0.25 marks a new and lower threshold for PBH formation during
the QCD phase transition when x = 2.
In Figure 51 we plot the cases: (a) x = 15, (b) x = 30, and (c) x = 90 with
δc =1/3 for all the three cases. In the case x = 15 there are two regions for
which PBH formation is allowed: i) a region for δ ≥ 1/3, which corresponds
to PBH formation from fluctuations of class A during the radiation–dominated
Universe; ii) a region between δc1 ≈ 0.15 and δc2 ≈ 0.27 corresponding to the
formation of PBHs from fluctuations of classes B and C. The gap between
δ =0.27 and δ =1/3 corresponds to: i) fluctuations of class A which dissipate
because they have δ< 1/3; ii) fluctuations of class B which dissipate because
they do not spend enough time on the dust–like phase, allowing collapse to
begin.
The case x = 30 is similar to the case x = 15. Notice, however, that now the
B
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A