Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 171
2004). Here, we consider a running–tilt power–law spectrum n(k) in the form
(see Sobrinho & Augusto, 2007)
n(k) =n0 + n1
2
k
ln +
kc
n2
ln
6
k
2 +
kc
n3
ln
24
k
3 kc
(283)
which is an expansion of equation (278) up to i = 4.
The simplest models of inflation suggest that the coefficients ni scale as powers
ɛi of some slow–roll parameter ɛ ≈ 0.1. This means that the expansion (283)
can be expected to be accurate to 10% for about 16 e–foldings around the pivot
scale (e.g. Düchting, 2004). This implies sensitivity down to horizon masses of
∼ 10M⊙. For scales probing the QCD epoch, the accuracy of expression (283)
is reduced to 20−30% and in the case of the EW epoch the case is by far worse.
In that cases we will regard expression (283) as a phenomenological one.
If one wants to have a significant number of PBHs produced at some epoch,
then one needs to have a spectrum with more power on that particular epoch.
In practice this is done introducing some fine–tunning into the spectrum (e.g.
Sobrinho & Augusto, 2007). In our case, this fine–tunning is done by guessing
which set of values for n2 and n3 would lead to interesting results in terms of
PBH production.
Given a location k+ (or, in terms of time t+) to the maximum of n(k) it
turns out that the corresponding value of n3 is given by equation (Sobrinho &
Augusto, 2007).
4 3n1 +2n2ln n3 = −
k+
kc
(284)
3
ln k+
kc
2
where kc is a pivot scale. Inserting this expression of n3 into the expression of
n(k) (Sobrinho & Augusto, 2007, equation 150) with k = k+ and n(k+) =nmax
we obtain for n2 the expression
6 3n0 − 3nmax + n1 ln
n2 = −
k+
kc
2 . (285)
ln k+
kc
On Sobrinho & Augusto (2007, Table 7) we have already presented the values of
n2 and n3 for different locations of the maximum k+ and for different values of
nmax. We have also presented the corresponding graphics with the curves n(k)
(Figures 69 to 74 from Sobrinho & Augusto, 2007). However, those values were
determined without taking into account the influence of a positive cosmological
constant Λ (cf. Section 1.5), which might be used when one converts an instant
of time tk to the corresponding wavenumber k by means of equation (203).
On Table 39 we present, as an example, the values for n2 and n3 for the case
nmax =1.4. Notice that these set of values are model independent in the case
of phase transitions. On Figure 91 we show the curves n2(t+) and n3(t+) for