Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 126
Table 28: The value of x corresponding to the intersections points δc2 = δc and
δc1 = δc2 as a function of δc, for the QCD phase transition according to the Bag
Model.
δc δc2 = δc δc1 = δc2
1/3 12.0 54.8
0.4 8.8 38.7
0.5 6.0 25.4
0.6 4.5 18.0
0.7 3.5 13.4
region [δc1,δc2] is much smaller. In the case x = 90 the fluctuations of classes
B and C do not lead any longer to the formation of PBHs.
Figure 52 indicates the region in the (x, δ) plane for which collapse to a BH
occurs (x >1 and δc =1/3). Without the phase transition, this would be a
straight horizontal line at δ =1/3. The intersection points δc1 = δc2 (x ≈ 54.8)
and δc2 = δc =1/3 (x ≈ 12.0) turn out to be very important for the calculation
of β (see Section 11).
In Figure 53 we consider, again, the cases x = 2, x = 30 and x = 90 but
now with δc assuming several values between 1/3 and 0.7. The new window for
PBH formation, i.e., the region between δc1 and δc or δc2, is larger for smaller
values of δc.
Figure 54 shows the region in the (x, δ) plane for which collapse to a BH
occurs with x>1 for δc =1/3 and for δ =0.7. Without the phase transition,
these would be two straight horizontal lines at δ = 1/3 and δ = 0.7. The
intersection points δc1 = δc2 (x ≈ 54.8 when δc =1/3; x ≈ 13.6 when δc =0.7)
and δc2 = δc (x ≈ 12.0 when δc =1/3; x ≈ 3.5 when δc =0.7) are relevant for
the calculus of β (see Section 11). For more examples see Table 28 where we
show the results for other values of δc between the limits 1/3 and 0.7.
We interpolated the values presented in Table 28 in order to obtain the
relation x(δc) for the special cases δc1 = δc2 and δc2 = δc. We obtained the
cubic polynomials
xδc1=δc2(δc) ≈−843.192δ 3 c + 1620.83δ2 c − 1083.54δc + 266.998 (259)
xδc2=δc(δc) ≈−164.72δ 3 c + 317.654δ 2 c − 214.03δc + 54.1305 (260)
which are represented in Figure 55. In Table 29 we have a compilation of the
values of δAB, δBC, δc1 and δc2 for different values of x and δc.