Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 183
log 10Βtk
-15
-15.5
-16
-16.5
-17
-17.5
n1.40, log 10t1s8
-10.5 -10 -9.5 -9 -8.5 -8
log10 tk
1 s
Figure 100: The fraction of the universe going into PBHs for a running–tilt
power spectrum with n+ =1.40 and t+ = 10 −8 s. The red line corresponds
to the contribution from the EW Crossover while the black line represents the
contribution from the radiation domination. The maximum difference between
the two in β(tk) is of order unity ( 10 0.1 ).
from the EW Crossover.
11.3 Electron–positron annihilation
The cosmological electron–positron annihilation ocurred when the age of the
universe was ∼ 1 s. Thus, the additional contribution from this epoch to the
global value of β is more relevant when t+ ∼ 1 s. Integrating equation (288),
with the threshold δc1 replaced by the appropriate values (e.g. Table 38), we
find that the cases with a non–zero contribution from the electron–positron
annihilation are in the range −2 ≤ log 10(t+/1s) ≤ 2 and 1.52 ≤ n+ ≤ 1.76, as
shown on Table 43.
Let us start with t+ = 1 s. In this case the contribution from the electron–
positron annihilation epoch is almost equal in magnitude to the contribution
from radiation (although with peaks at different epochs). In Figure 101a we
show, as a first example, the case t+ = 1 s and n+ =1.56. From the radiation
contribution we have βmax ∼ 10 −29 located at tk ∼ 10 −1.14 s and from the
electron–positron annihilation contribution we have βmax ∼ 10 −34 located at
tk ∼ 10 −0.07 s. As a second example we show in Figure 101b the case t+ = 1 s
and n+ =1.62. Now, we have from the radiation contribution βmax ∼ 10 −10 located
at tk ≈ 10 −1.2 s and from the electron–positron annihilation contribution
βmax ∼ 10 −12 located at tk ≈ 10 −0.08 s.
(contribution from radiation), then move to the red line (EW Crossover contribution) and,
finally, move once again to the black line.