Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 121
which is evaluated at the time of collapse, for the formation of the PBH. In
particular, when the fluctuation enters the horizon in a radiation–dominated
Universe, one gets (e.g. Carr, 1975; Kiefer, 2003)
δmin ≡ 1
3 ≤ δ< 1 ≡ δmax, (250)
where the lower bound comes from condition (247) and the upper bound comes
from condition (248). The extreme δmax corresponds to the situation for which
a separate Universe forms and δmin corresponds to the threshold of PBH formation.
If δ < δmin the fluctuation dissipates and there is no PBH formation
(see Section 2.4.3 of Sobrinho & Augusto, 2007).
The correct value of δmin has been a matter of discussion (see Table 3 of
Sobrinho & Augusto, 2007). We have already seen that the value δmin =1/3
is suggested by analytic arguments. However, numerical simulations considering
critical phenomena in the PBH formation (see Section 2.4.1 of Sobrinho
& Augusto, 2007) reveal a higher value, δmin ≈ 0.7, which is almost twice
the old value. Another study using peaks theory (Green et al., 2004) leads
to δmin ≈ 0.3 − 0.5, which is in good agreement with the analytic approach
(δmin =1/3). Taking into account that the threshold δmin arises from critical
behaviour, we will refer to δmin in the rest of the text as δc.
The value of the threshold δc is constant, with some exceptions, throughout
the radiation–dominated Universe. Exceptions are phase transitions (Sections
2 and 3) and annihilation processes (Section 4). During these epochs the speed
of sound vanishes or, at least, diminishes and, as a result, δc becomes smaller
(Sections 7, 8 and 9). This is very important because a smaller δc will favour
PBH production (Section 11). The condition for PBH formation is written as
δc ≤ δ< 1. (251)
For radiation domination (w =1/3), the size of the overdense region at turnaround
(i.e. at the moment when the kinetic energy of the expansion is zero),
its Schwarzschild radius, the Jeans length, and the cosmological horizon size
are all of the same order of magnitude. On the contrary, for dust domination
(w = 0), the Jeans length is much smaller than the horizon size. For an overdense
region that experiences a radiation phase for much of its evolution and a
dust–like phase for the rest we define an effective Jeans length (e.g. Cardall &
Fuller, 1998)
1
RJ,eff = (1 − f) (252)
3Gρc
where f denotes the fraction of the overdense region spent in the dust–like phase
of the transition.
6.2 The mechanism of PBH formation
The ultimate fate of an initially super–horizon density fluctuation, upon horizon
crossing, is mainly determined by a competition between dispersing pressure