Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 160
Table 35: The lowest value of δc1,min for the EW Crossover for different values
of δc. It is also shown the corresponding values of ∆T/Tc and tk.
δc δc1,min ∆T/Tc tk(s)
1/3 0.33186 0.0131 8.32 × 10 −11
0.4 0.39823 0.0091 9.33 × 10 −11
0.5 0.49778 0.0111 1.10 × 10 −10
0.6 0.59734 0.0101 1.23 × 10 −10
0.7 0.69689 0.0101 1.35 × 10 −10
corresponds to the case for which a minimal value for δc1 ≈ 0.33186 is achieved
and that occurs for ∆T =0.013Tc.
We considered also the cases tk =1.5 × 10 −10 s – Figure 83d (tk t2). For all these cases it is clear
that the effect of the EW Crossover, in terms of the reduction of the value of
δc, is not as significant as it was for fluctuations crossing the horizon a little bit
before the beginning of the transition (cf. Figures 83a and 83b). The value of
δc1 smoothly aproaches δc =1/3 as one moves to larger values of tk for all the
considered values of ∆T .
Figure 84 shows the region on the (log 10(tk/1s), δ) plane for which collapse
to a BH occurs in the case of the EW Crossover with δc =1/3 and when ∆T
assumes the values 0.001Tc, 0.013Tc and 0.1Tc. Without the phase transition,
these would be three straight horizontal lines at δ =1/3. We have also represented,
for reference, the location of the EW Crossover. It is clear that the
case ∆T =0.013Tc is the most important in the context of PBH production
(with δc =1/3). For that reason we will consider, from now on, for the EW
Crossover, only this case.
Figure 85 indicates the region on the (log 10(tk/1s), δ) plane for which collapse
to a BH occurs for δc =1/3 and for δ =0.4. Without the phase transition,
these would be two straight horizontal lines at δ =1/3 and δ =0.4.
8.1.2 Summary
Due to the EW Crossover the PBH formation threshold δc experiences a reduction
on its background value. On Table 36 we present some of these new values
δc1 as a function of time for the case δc =1/3.
8.2 Bag Model (MSSM)
In the case of an EW first–order phase transition we replace the lower limit in
the condition (251) by expression (253). The quantity f, which represents the