Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 207
12 Conclusions and Future work
12.1 Results achieved and conclusions
The Universe is a well developed structure on the scale of galaxies and smaller
formations. This requires that at the beginning of the expansion of the Universe
(Section 1) there should have existed fluctuations (Section 5) which lead to the
formation of such structures. We now have a successful cosmological paradigm
based on the existence of an inflationary stage (Section 1.3) which allows us to
consider the quantum origin of the fluctuations. These quantum fluctuations,
produced during inflation, are stretched to scales much larger than the Hubble
radius (at the time when they were produced) and, as the expansion of the
universe goes on, each fluctuation will reenter inside the Hubble radius at some
later epoch, depending on its wavelength. With this mechanism we can explain
not only all the inhomogeneities we see today, even on the largest cosmological
scales, but also the production of PBHs.
If a perturbation crossing the horizon at time tk is large enough, then it
will begin to collapse at some later instant tc called the turnaround point. The
location of tk and tc with respect to the transition epoch allows us to identify, in
the case of a first–order phase transition, six different classes of fluctuations (cf.
Tables 26 and 27) – A, B, C, D, E, and F . In the presence of a first–order phase
transition, the PBH formation threshold δc is affected by some factor (1 − f)
where f is a function which gives the fraction of the overdense region spent in
the dust–like phase of the transition. In the approach considered, f relates the
sizes of the overdense region at tk and tc (Section 7.1).
The inflationary stage is followed by a radiation–dominated era during which
the Universe successively visits the different scales at which particle physics
predicts symmetry–breaking phase transitions. The SMPP (Section 1.8) predicts
two phase transitions: the EW phase transition (Section 3), at an energy
∼ 100 GeV, and the QCD phase transition (Section 2), at an energy ∼ 170 MeV.
The occurence of a phase transition turns out to be very important in the
context of PBH formation. In fact, during such epochs, the sound speed vanishes
for some instants (first–order phase transition) or, at least, it suffers, depending
on the strength of the transition, a more or less relevant reduction (Crossover)
and, as a consequence, the effect of pressure in stopping gravitational collapse
becomes less important, favouring PBH formation (Section 6.2).
Only the fluctuations with amplitude δ above some threshold δc can lead to
the formation of PBHs. If δ < δc the fluctuation dissipates and there is no PBH
formation at all. In the case of a radiation–dominated universe we have, from
analytical considerations, that δc =1/3 (although recent numerical simulations
revealed different values for δc, all in the range 1/3 – 0.7 (e.g. Sobrinho &
Augusto, 2007). During a phase transition, this constant background value δc,
valid for radiation domination, becomes smaller and, as a consequence, the value
of β(tk) (equation 286), which is very sensitive to the threshold δc, could show
a peak located near the phase transition epoch.
In order to determine the probability of PBH formation at a given epoch or,