Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 159
8 The threshold δc for PBH formation during
the EW phase transition
8.1 Crossover Model (SMPP)
During the EW Crossover it is expected a reduction on the value of the PBH
formation threshold δc due to the decrease on the sound speed. We adopt for f
expression (267) derived for the QCD Crossover (Section 7.2) but now with the
sound speed, cs(t), given by equation (180). First, however, we must determine
which values of ∆T we will use. We are particularly interested in a value of ∆T
for which the threshold δc attains a minimum value (because lower values of δc
favour PBH formation).
For a given ∆T we determine, with the help of funtion (1 − f)δc, the new
threshold δc1 as a function of the horizon crossing time tk. When tk ≪ tEW−
or, when tk ≫ tEW+, we get δc1 = δc. Between these two extremes there
is a value of tk for which δc1 attains a minimum value δc1,min. For example,
when ∆T = 0.001Tc and δc =1/3, we obtain δc1,min ≈ 0.33213 with
tk ≈ 8.32 × 10 −11 s.
We repeated this procedure for different values of ∆T (0 < ∆T ≤ Tc) and
concluded that, in the δc = 1/3 case, our best value is δc1,min ≈ 0.33186,
corresponding to having ∆T ≈ 0.013Tc and tk ≈ 8.32 × 10 −11 s. In Figure 82
we show a selection of the obtained results.
On Table 35 we show the results for different values of δc. Note that, although,
the value of the parameter ∆T remains almost constant the same does
not apply to tk. For a larger value of δc, the instant tk, for which we get the
lowest δc1,min, is closer to tEW−.
8.1.1 Examples
In this section we study the changes on the value of the threshold δc during
an EW Crossover, through examples of fluctuations that cross the horizon at
different epochs. The study is mainly made for δc = 1/3. We are mostly
interested in the case ∆T =0.013Tc (because it is the one that leads to the
lowest value of δc) but we also consider the cases ∆T =0.001Tc and ∆T =0.1Tc,
with the purpose of comparing the results.
Let us start with δc =1/3 and with a fluctuation that crosses the horizon at
tk =5.0 × 10 −11 s (i.e., before t1 =2.97 × 10 −10 s, cf. Section 3.2.1). In order
to identify the values of δ for which collapse to a BH will occur, in this case, we
plot, in Figure 83a, both (1 − f)δc and δ itself as functions of δ. We conclude
that the evolution of perturbations entering the horizon at this epoch are not
affected by the presence of a Crossover transition. That is because tk occurs
sufficiently before the transition.
As a second example, we consider a fluctuation that crosses the horizon at
tk =7.0×10 −11 s. In this case, we do have a lower threshold for PBH formation
δc1 ≈ 0.3330 if ∆T =0.1Tc but not if ∆T =0.001Tc or ∆T =0.013Tc as it is
clear from Figure 83b. In Figure 83c we have the case tk ≈ 8.32 × 10 −11 s. This