Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 185
When t+ ≈ 10 s the contribution from the electron–positron annihilation
epoch exceeds the contribution from radiation. As a first example let us consider
the case t+ = 10 s and n+ = 1.54 (Figure 101c). Now we have a modest
contribution from the electron–positron annihilation epoch with βmax ∼ 10 −67
located at tk ∼ 10 −0.02 s and an even smaller contribution from radiation with
βmax ∼ 10 −80 located at tk ∼ 10 −0.045 s. As a second example we consider the
case t+ = 10 s and n+ =1.66 (Figure 101d). We now have more interesting
values with βmax ∼ 10 −8 located at tk ∼ 10 −0.025 s from the electron–positron
annihilation epoch contribution and βmax ∼ 10 −10 located at tk ∼ 10 −0.09 s
from the radiation contribution (the peaks nearly overlap).
When t+ ≈ 100 s the contribution from the electron–positron annihilation
epoch appears as an extension to the left on the curve of β(tk) as can be seen
in Figures 101e and 101f.
In the case t+ = 10 −1 s we might simultaneously have contributions from
the electron–positron annihilation epoch and from the QCD phase transition
(RBLCea in Table 43). However, in these cases the main contribution always
comes from radiation. As an example we have the case t+ = 10 −1 s and
n+ = 1.56 represented on Figure 35 102.
When t+ = 10 −2 s and n+ =1.54 or n+ =1.56 we also simultaneously have
contributions from the electron–positron annihilation epoch and from the QCD.
However, in these cases the main contribution to β(tk) comes from radiation
and from the QCD Crossover (if one adopts for the QCD the Bag Model or the
Lattice Fit, then these two cases are excluded due to observational constraints,
see Section 11.4). In Figure 103 we show, as an example, the case t+ = 10 −2 s
and n+ =1.54, some situations of which are excluded.
11.4 QCD phase transition
The contribution from the QCD phase transition to the global value of β depends
on the model one adopts. There are some cases which are allowed when one
considers only the contribution from radiation but which must be excluded when
one takes into account the QCD phase transition because of the observational
limits (cf. Tables 44, 45 and 46).
Consider, for example, the case t+ = 10 −4 s and n+ =1.48, represented
in Figure 104. It is clear that if one adopts a Bag Model or a Lattice Fit for
the QCD transition, this case must be excluded. However, if one adopts the
Crossover model, then it remains valid.
As a peculiar example we show the case t+ = 10 −3 s and n+ =1.52, which
is allowed when one takes into account only the contribution from radiation
domination but must be excluded whatever the model one adopts for the QCD
phase transition (Figure 105).
35 In order to interpret correctly the curves on Figure 102, assuming a QCD Bag Model,
start on the left over the blue line, then move to the black line (contribution from radiation)
and, finally, move to the cyan line (contribution from the electron–positron annihilation). In
the case of a QCD Lattice Fit start, instead, with the line in magenta and in the case of a
QCD Crossover start with the green line.