Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 140
∆
1.2
1
0.8
0.6
0.4
0.2
∆c1
∆c2
No BH formation
BH formation
∆c13
No BH formation
0 20 40 60 80 100
x
Figure 64: The curve in the (x, δ) plane indicating which parameter values lead
to collapse to a BH when δc =1/3 (full QCD Bag Model). This Figure was
obtained by joining Figures 52, 57 and 62.
7.1.4 Summary
We now compile the results obtained in Sections 7.1.1 to 7.1.3 (Bag Model).
In particular, we have joined Figures 52, 57 and 62 in a single one in order to
have a full picture of the QCD phase transition on the (x, δ) plane: Figure 64.
Figure 65 represents the same scenario but now in the (log 10 x, δ) plane: a better
representation if, for example, one wants to find the locus of the transition.
With the help of equation (217) we move from the(log 10 x, δ) plane into the
(log 10 t, δ) plane. As a result, we get Figure 66 where we have also indicated
the lines t = t− and t = t+ (which mark the location of the transition).
During the QCD transition the threshold for PBH formation experiences a
reduction. As a result, a new window for PBH formation (between δc1 and δc or
between δc1 and δc2) is opened for a brief period. On Table 32 we present some
values giving this new threshold for PBH formation during the QCD transition
according to the Bag Model when δc =1/3. We have presented the values of
δc1 and δc2 (where applicable) as a function of time and as a function of the
parameter x.
7.2 Crossover Model
During the QCD Crossover a reduction on the value of the threshold δc is
expected, due to the reduction on the sound speed. We need to determine
the analogous of function f (see condition 253) for the QCD Crossover. This
function f should account for the fact that we have a variable sound speed
value during the Crossover and that a smaller value of cs(t) contributes more
significantly to the reduction of δc than a larger one. We then introduce the
function
α(t) =1− cs(t)
cs0
(261)