Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 149
Inserting expression (269) into equation (268) and replacing R(tk), as given by
equation (71), with nqcd =2/3 in Sc,A we obtain
fALat = 3
−3/2 tk 1+δ
tk
2 δ
1+δ
δ
α(t)
tk
√ tdt. (270)
In the case of a fluctuation of class B we write
with
fB = fBLat + S3 c,B − S3 1
S 3 c,B
fBLat = 1
S3 S1
α(t)dS
c,B Sk
3
(271)
(272)
where S1 and Sc,B are given by expressions (225) and (228) , respectively. Inserting
expression (269) into equation (272), replacing R(tk) in Sc,B and considering
that S1 ≡ R(t−), we obtain
fBLat = 3
2
t −1/2
k x
−3 t−
−1/4 (1 + δ)3/4
δ
tk
α(t) √ tdt (273)
where x = x(tk). In the case of a fluctuation of class C we have
with
fC = fCLat + S3 2 − S3 1
S 3 c,C
fCLat = 1
S3 S1
α(t)dS
c,C Sk
3
(274)
(275)
where S2C and Sc,C are given by expressions (226) and (229) respectively. Inserting
expression (269) into equation (275), replacing R(tk) in Sc,C and considering
that S1 ≡ R(t−), we obtain
fCLat = 3
−3/2 1+δ
tk
y
2 δ
−1/2
t−
α(t)
tk
√ tdt (276)
with y −1 ≈ 0.729 (Section 5.3.2). In the case of fluctuations of classes E and F
(tk >t−) we continue to use, respectively, for fE and fF , the expressions (257)
and (258) of the Bag Model.
In the next section we study PBH formation during the QCD phase transition,
from fluctuations of classes A, B, C, E and F, according to the Lattice Fit.
We divide the study in Before, During and After, as we did for the Bag Model.
At the end of the section we compile the results. The study is mostly done for
δc =1/3 but we also consider the effect of larger values of δc (up to 0.7).