Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 78
cs,min 2 cs,0 2
0.5
0.4
0.3
0.2
0.1
0.02 0.04 0.06 0.08 0.1
TTc
Figure 27: The minimum value attained by the sound speed c 2 s,min
of the parameter ∆T for the QCD Crossover (see equation 144).
as a function
equation (143) we obtain the following expression giving an approximate value
for the minimum sound speed during a QCD Crossover
c 2 s,min ≈
3+ gQGP − gHG
∆T
Tc (gQGP + gHG)
−1
In Figure 27 we have the curve for c 2 s,min
. (144)
as a function of the ∆T parameter
when Tc = 170 MeV. Notice that when ∆T = 0 we have c2 s,min = 0 and when
∆T =0.1Tc we have c2 s,min ≈ 0.38c2s,0 ≈ 0.13 (c2s,0 =1/3 is the sound speed for
an ideal gas).
2.4 The duration of the QCD transition
If one wants to study how a given fluctuation behaves during the QCD phase
transition then it is of crucial importance to know the duration of the transition.
This means that, if we want to perform numerical integrations then we need
to define a specific beginning t = t− and a specific end t = t+ to the QCD
transition. Here t− and t+ are the limits for the time interval during which the
speed of sound vanishes. This is aplicable in the case of a first–order transition
(Bag Model and Lattice Fit). In the case of a Crossover we will define an
effective duration instead.
Taking into account that the temperature of the Universe during the QCD
phase transition is Tc we can obtain, with the help of equation (78) a numerical