Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 81
cs 2
0.4
0.3
0.2
0.1
0
t
-4.4 -4.2 -4 -3.8 -3.6
Log 10t1s
Figure 29: The sound speed c2 s (t) for the QCD phase transition according to the
Bag Model with Tc = 170 MeV. During the coexistence phase, which occurs
between the instants t− =6.25 × 10−5 s and t+ =1.08 × 10−4 s, the sound
speed drops to zero.
On the Bag Model and Lattice Fit cases, the temperature remains constant
(T = Tc) for a while. The same does not occur during a Crossover where the
temperature continues to decrease with time. Inserting expression (151) into
equation (143) we obtain, for the speed of sound during the QCD Crossover,
the following expression
⎡
⎤
c 2 s(t) =
⎢
⎣3+
∆T
∆gT(t)sech
t
gHG + gQGP +∆gtanh
2 T (t)−Tc
∆T
T (t)−Tc
∆T
⎥
⎦
−1
. (152)
We are now able to present the sound speed profile for the QCD phase transition
for a given temperature Tc as a function of time. In Figure 29, 30 and 31 we
show the curve c2 s(t) for, respectively, the Bag Model, the Lattice Fit and the
Crossover for a QCD temperature of Tc = 170 MeV.
According to the Lattice Fit the sound speed decreases until it vanishes at
some instant t− (cf. Figure 30). It is useful to know not only the interval during
which cs = 0, but also, the interval during which c2 s makes its way down from
1/3 to zero.
Therefore, we define T1 >Tc as the temperature for which c2 s equals 95%
of its ‘background’ value: c2 s,0 = 1/3. This corresponds to some instant of
time t1 < t−. From equation (139), with Tc = 170 MeV and γ = 1/3,
one obtains T1 ≈ 2296 MeV ≈ 13.5Tc. Similarly, from equation (141), with
t− = 9.37 × 10 −5 s and γ =1/3 one obtains t1 ≈ 5.1 × 10 −7 s.