Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 117
by equation (86). Thus, we have
Λ
Rk = R(tk) = exp c
3 (tSN
2/3
teq tk
− t0)
and
tSN
Λ
Sc = R(tc) = exp c
3 (tSN
2/3
teq tc
− t0)
tSN
teq
teq
1/2
(239)
1/2
. (240)
Inserting expressions (239) and (240) into equation (238) we obtain a relation
between the horizon crossing time tk and the turnaround time tc as
tc = tk
1+δk
. (241)
δk
5.4 Fluctuations during the EW phase transition
5.4.1 Crossover (SMPP)
We now describe the evolution of a fluctuation during the EW Crossover in the
same way that we did for the QCD Crossover case (cf. Section 5.3.3). Thus, Rk
and Sc are given, once again, by expressions (239) and (240). In addition, the
relation between the turnaround instant tc and the horizon crossing time tk is
given by expression (241).
5.4.2 Bag Model (MSSM)
Let us consider how a fluctuation evolves in the presence of a first–order EW
phase transition according to the Bag Model. Here we continue to follow the
model proposed by Cardall & Fuller (1998) for the QCD first–order phase transition
(see Section 5.3.1). Considering the possible locations of tk and tc, we
define six different classes of density fluctuations (Table 27).
It is very useful to have x as a function of time. We get this by adapting
equation (217) derived for the QCD case. Thus, we have
x(t) =
4g ′ EW
3 R(tEW−)
R(t) − gEW
4g ′ EW − gEW
(242)
which is valid only in the neighborhood of the EW transition. Here R(tEW−)
is given by equation (73) and R(t) is given by: i) equation (71) if x ≤ y−1 ; ii)
equation (72) if y−1 < x < 1; iii) equation (73) if x ≥ 1.
The value of y, which defines the end of the EW transition, can now be
determined evaluating x(tEW+). If one assumes ∆g ≈ 80 and gEW = 95.25 (see
Section 3.2.2), then one obtains
y −1 = x(tEW+) =
4g ′ 2 tEW−
EW − gEW
tEW+
4g ′ EW − gEW
≈ 0.460. (243)