Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 167
9 The threshold δc for PBH formation during
the cosmological e + e − annihilation
In this section we study the changes on the value of the threshold δc during
the cosmological electron–positron annihilation. We do that by considering
examples of fluctuations that cross the horizon at different epochs. The study is
mostly done for δc =1/3. We consider here the cases ∆T =0.115Tc (reduction
of the sound speed value up to 20%, cf. Table 23) and ∆T =0.276Tc (reduction
of the sound speed value up to 10%).
Let us start with δc =1/3 and with a fluctuation that crosses the horizon at
tk =0.20 s. In order to identify the values of δ for which collapse to a BH will
occur in this case, we plot in Figure 88a both (1 − f)δc and δ itself as functions
of δ. Now, f is given by expression (267) with the sound speed cs(t) given
by equation (195). We conclude that the evolution of perturbations entering
the horizon at this epoch is not affected by the annihilation process. That is
because tk occurs suficiently before t1 (t1 ≈ 1.7 s if ∆T =0.115Tc and t1 ≈ 1.1 s
if ∆T =0.276Tc – cf. Table 25).
As a second example we show in Figure 88b the case of a fluctuation that
crosses the horizon at tk =0.9 s. In this case we have a lower threshold for PBH
formation δc1 ≈ 0.3041 if ∆T =0.115Tc and δc1 ≈ 0.3125 if ∆T =0.276Tc.
In Figure 88c we show the case tk =2.3 s for which δc1 ≈ 0.3221 if ∆T =
0.115Tcand δc1 ≈ 0.3193 if ∆T =0.276Tc. As a final example we show in Figure
88d the case tk =8.0 s for which the effects are much less significative.
Figure 89 shows the region on the (log 10 tk,δ) plane for which collapse to a BH
occurs during the cosmological electron–positron annihilation with δc =1/3 and
with ∆T assuming the values 0.115Tc and 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. It is clear that the case ∆T =0.115Tc is the most important in
the context of PBH production because it leads to a more significative reduction
on the value of δc.
Figure 90 indicates the region on the (log 10 tk,δ) plane for which collapse
to a BH occurs for ∆T =0.115Tc with δc =1/3 and δc =0.7. Without the
annihilation process these would be two straight horizontal lines at δ =1/3 and
δ =0.7.
9.1 Summary
As a consequence of the cosmological electron–positron annihilation, the PBH
formation threshold δc experiences a reduction on its background value. On
Table 38 we present these new values, δc1, in function of time for the case
δc =1/3 with ∆T =0.115Tc.