Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 168
∆
∆
0.34
0.33
0.32
0.31
0.34
0.33
0.32
0.31
2
1
a
∆1
0.2 0.4 0.6 0.8 1 1.2 1.4
∆
2
c
1
∆1
0.2 0.4 0.6 0.8 1 1.2 1.4
∆
∆
∆
0.34
0.33
0.32
0.31
0.34
0.33
0.32
0.31
1
b
2
∆1
0.2 0.4 0.6 0.8 1 1.2 1.4
∆
d
1
2
∆1
0.2 0.4 0.6 0.8 1 1.2 1.4
∆
Figure 88: PBH formation during the cosmological electron–positron annihilation
when δc = 1/3 and: (a) tk = 0.2 s; (b) tk = 0.9 s; (c) tk = 2.3 s;
(d) tk =8.0 s. The solid curve corresponds to the function (1 − f)δc when:
(1) ∆T =0.115Tc and (2) ∆T =0.276Tc. The dashed curve on the left corresponds
to the identity δ. Collapse to a BH occurs for values of δ for which the
dashed line is above the solid curve (while δ< 1). The new thresholds for PBH
formation are: (b)–(1) δc1 ≈ 0.3041 and (2) δc1 ≈ 0.3125; (c)–(1) δc1 ≈ 0.3221
and (2) δc1 ≈ 0.3193; (d)–(1) δc1 ≈ 0.3315 and (2) δc1 ≈ 0.3301. In case (a) the
threshold remains δc =1/3 for both curves.
∆
0.34
0.33
0.32
0.31
BH formation
No BH formation
t1
1
2
-1.5 -1 -0.5 0 0.5 1 1.5 2
Log 10tk1s
Figure 89: The curve on the (log 10 tk, δ) plane indicating which parameter values
lead to collapse to a BH in the case δc =1/3 for the cosmological electron–
positron annihilation with (a) ∆T =0.115Tc and (b) ∆T =0.276Tc. Without
the annihilation process, these would be straight horizontal lines at δ =1/3.
We have also represented, for reference, the location of the annihilation epoch
for the case ∆T =0.276Tc (vertical dashed lines).
t2