Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 80
t10 5 s
16
14
12
10
8
6
a
b
c
140 150 160 170 180 190 200
Tc MeV
Figure 28: The beginning (t−) and the end (t+) of the QCD phase transition
as a function of the transition temperature Tc: (a) t+, valid for both the Bag
Model and the Lattice Fit; (b) t− for the Lattice Fit and (c) t− for the Bag
Model.
When t = t− equation (70) becomes
Λ
R(t−) = exp c
3 (tSN
2/3 1/2
teq t+ t−
− t0)
tSN
teq
t+
2/3
(149)
where we have considered nqcd =2/3. From equations (146) and (149) we obtain
t− = t+
√ . (150)
∆R3 For example, when Tc = 170 MeV we obtain, from equation (147), the value
t+ ≈ 1.08 × 10−4 s which is valid (according to the assumptions made in the
preceding paragraphs) for both the Bag Model and the Lattice Fit. Inserting
this value into equation (150) one obtains t− ≈ 6.25 × 10−5 s in the case of the
Bag Model and t− ≈ 9.37 × 10−5 s in the case of the Lattice Fit. In Figure 28
we present the curves for t+ and t− as functions of the critical temperature Tc.
For the Crossover case we consider that the sound speed minimum value is
attained for t ≈ t+ (corresponding to T ≈ Tc). During the QCD Crossover
the Universe continues to be radiation–dominated with the scale factor given
by equation (86). Inserting equation (86) into equation (78) we obtain an expression
for the temperature T as a function of the time t valid for the QCD
Crossover:
Λ
T (t) =T0 exp c
3 (tSN
2/3
1/2
−1
teq t
− t0)
. (151)
tSN
teq