Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 176
In Sections 11.1 to 11.5 we made use of a few abreviations concerning different
contributions to β(tk). We list these abreviations in Table 41. For example,
RBE, represents a case for which there are non–negligible contributions from
radiation (R), from the QCD Bag Model (B), and from the EW Bag Model
(E) but with negligible contributions from the electron–positron annihilation as
well as from the QCD Lattice Fit and from the QCD Crossover (if one chooses
one of these models instead of the Bag Model).
We might have also situations with one or more contributions exceeding the
observational limits (these are labeled with an ∗). For exeample, RB ∗ LCE ∗ ,
represents a case for which we have, besides the contribution from radiation R,
contributions from the QCD Lattice Fit (L) or from the QCD Crossover (C).
The QCD Bag Model is excluded due to observational constraints (B ∗ ). The
same happens for the EW Bag Model (E ∗ ). The contribution from the electron–
positron annihilation epoch is negligible in this case.
11.1 Radiation–dominated universe
In this section we determine the fraction of the universe going into PBHs at
different epochs for a radiation–dominated universe with a running–tilt power
spectrum (Section 10). We consider, for t+, all orders of magnitude between
10 −23 s (end of inflation) and 10 8 s. We are interested in a blue spectrum, i.e,
a spectrum for which n>1. In fact, if we want to have interesting values for
β, then we should have, at least, n 1.22. As an upper limit we consider
n ≈ 2.0. This corresponds to a cut–off at tk ∼ 10 5 s (see Figure 93) which
excludes PBHs with masses larger than ∼ 10 10 M⊙ (which is equivalent to the
mass of the present day largest Supermassive Black Hole (SMBH) candidates;
e.g. Natarajan & Treister, 2008).
Each pair of the form (t+, nmax) determines a different location and a different
value for the maximum value of β(tk). In general one of three things might
occur:
(1) β(tk) exceeds the observational constraints ⇒ the pair (t+, nmax) must
be rejected.
(2) β(tk) is negligible (< 10 −100 ) for all values of tk, in which case we take
β(tk) = 0.
(3) β(tk) is always bellow the observational constraints and, at least during
some epoch, above 10 −100 ; these are the cases of interest to us.
Let us consider, as an example, the case n =1.30. If t+ = 10 −17 s, then
β(tk) exceeds the observational constraints as it is clear from Figure 94. On the
other hand, if t+ = 10 −16 s we obtain a valid curve for β(tk) (see Figure 95)
with βmax ∼ 10 −17 . As one moves t+ to later epochs the value of βmax becomes
smaller (see Figure 95) until, for t+ = 10 −10 s, we reach βmax ∼ 10 −133 (see
Figure 96). Thus, we consider that, in the case of n =1.30, there is a window