Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 209
δc reaching values as low as ≈ 0.091 for a background value δc =1/3 (Figure 66).
In Section 7.2 we considered the variation of δc during the QCD Crossover.
We introduced a new function f (see equation 262) which takes into account the
fact that, during the Crossover, the sound speed decreases but does not vanish.
We have done this through an adimensional function α(t) (equation 261) which
gives the fraction of the sound speed with respect to the background value
(1/ √ 3) at a given moment. We found that, in the case of a Crossover, the
reduction on the value of δc is much less pronunced than in the Bag Model case
with δc,min ≈ 0.274 for a background value δc =1/3 (Figure 70).
In Section 7.3 we considered the variation of δc during the QCD Lattice Fit.
In this case we have a period with a vanishing sound speed which is similar to
the Bag Model case and also a period during which the sound speed decreases
down to zero, resembling the Crossover situation (cf. Figure 30). Thus, we interpret
the Lattice Fit as a mixture of both situations and derive an appropriate
expression for the function f (see equations 268 to 276). The study was divided,
as in the Bag Model case, in before, during and after. As a result, we obtained
a reduction of δc from 1/3 to ≈ 0.12 (Figure 80).
Tipically, we have curves for β with two peaks: one from the radiation
contribution and another from the QCD contribution (e.g. Figures 106f and
109c). Contributions from the QCD Bag Model or from the QCD Lattice Fit
are, naturally, more visible than those from the QCD Crossover, since, in the
latter, the sound speed never reaches zero. However, in the case of the QCD
Crossover we might also reach high values for β (e.g. Figure 104).
There are many cases for which the contribution from the QCD (in particular
in the case of a Bag Model or a Lattice Fit) exceeds the observational constraints
(cf. Tables 44, 45 and 46).
Cut from Table 49, in Table 50 we present a list with the ten largest contributions
from the QCD Crossover. In each case we have also indicated the
contribution from radiation. For the cases shown, the contribution from the EW
phase transition is negligible and the contribution from the electron–positron
annihilation appears only in two cases (labeled ‘ea’). If one considers, for the
QCD phase transition, the Bag Model instead of a Crossover, then the ten cases
are excluded due to observational constraints. On the other hand, if one considers
the Lattice Fit model, then only the case n+ =1.52 and t+ = 10 −2 s is
allowed (labeled ‘L’).
Cut from Table 49, in Table 51 we present a list with the ten largest contributions
from the QCD Bag Model. In each case we have also indicated the
contribution from radiation. Notice that for all ten cases the contribution from
the QCD is, by far, much greater than the contribution from radiation. A choice
of a Lattice Fit model for the QCD gives similar results for all cases. On the
contrary, the Crossover model gives, for these cases, very modest results (see
also Table 49).
As an interesting situation we mention the case n+ =1.40 and t+ = 10 −7 s,
for which we have, besides the contribution from the QCD, an important contribution
from the EW phase transition as well as from radiation (Figure 109c).
The curve β(tk) spans from ∼ 10 −11 s to ∼ 10 −5 s, showing three noticeable