Primordial Black Holes and Cosmological Phase Transitions Report ...
Primordial Black Holes and Cosmological Phase Transitions Report ... Primordial Black Holes and Cosmological Phase Transitions Report ...
PBHs and Cosmological Phase Transitions 94 cs 2 0.3 0.25 0.2 0.15 c b a 85 90 95 100 105 110 115 120 T GeV Figure 37: The sound speed c 2 s (T ) for the EW Crossover with Tc = 100 GeV and: (a) ∆T =0.001Tc; (b) ∆T =0.005Tc; (c) ∆T =0.1Tc. Notice that the sound speed decreases around Tc but does not reaches zero (with the exception of the limiting case ∆T −→ 0). In Figure 38 we show the curve for c 2 s,min as a function of the ∆T parameter for the EW Crossover (Tc = 100 GeV) and, for comparison purposes, the corresponding curve for the QCD Crossover (Tc = 170 MeV). It is clear that during the EW Crossover the effects due to the reduction on the sound speed are less obvious than for the QCD case. For the EW Crossover case, we consider that the sound speed minimum value is attained for t ≈ tEW+ (corresponding to T ≈ Tc). During the EW Crossover the Universe continues to be radiation–dominated with the scale factor given by equation (86). Recalling equation (151) that gives the temperature as a function of time during the EW Crossover and inserting it into equation (178), we obtain, for the speed of sound during the EW Crossover, the following expression c 2 ⎡ 2 T (t)−Tc ⎢ ∆gT(t)sech ∆T s (t) = ⎣3+ ∆T gEW + g ′ EW +∆gtanh ⎤−1 ⎥ ⎦ . (180) T (t)−Tc ∆T Taking into account that Tc = 100 GeV ≈ 7.7×10 14 K, we obtain, from equation (151), that tc ≈ 3.12 × 10 −10 s. This corresponds to the instant of time when the sound speed reaches its minimum value. In Figure 39 we represent, again, the curves of Figure 37 but now as a function of time. We consider the effective duration of the QCD Crossover the interval for which the sound speed stays below 28 99% of its ‘background’ value c 2 s,0 =1/3. 28 We do not consider 95%, as we did in the QCD case, because the reduction of the sound
PBHs and Cosmological Phase Transitions 95 cs,min 2 cs,0 2 1 0.8 0.6 0.4 0.2 a b 0.02 0.04 0.06 0.08 0.1 TTc Figure 38: The minimum value attained by the sound speed, c2 s,min , as a function of the parameter ∆T during: (a) the EW Crossover (see equation 179); (b) the QCD Crossover (see equation 144, Figure 27). cs 2 0.3 0.25 0.2 0.15 c b a -9.7 -9.6 -9.5 -9.4 -9.3 -9.2 Log 10t1s Figure 39: The sound speed c 2 s (t) for the EW phase transition in the case of a Crossover with a reference temperature Tc = 100 GeV and: (a) ∆T =0.001Tc, (b) ∆T =0.005Tc, (c) ∆T =0.1Tc.
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PBHs <strong>and</strong> <strong>Cosmological</strong> <strong>Phase</strong> <strong>Transitions</strong> 95<br />
cs,min 2 cs,0 2<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
a<br />
b<br />
0.02 0.04 0.06 0.08 0.1<br />
TTc<br />
Figure 38: The minimum value attained by the sound speed, c2 s,min , as a function<br />
of the parameter ∆T during: (a) the EW Crossover (see equation 179); (b) the<br />
QCD Crossover (see equation 144, Figure 27).<br />
cs 2<br />
0.3<br />
0.25<br />
0.2<br />
0.15<br />
c<br />
b<br />
a<br />
-9.7 -9.6 -9.5 -9.4 -9.3 -9.2<br />
Log 10t1s<br />
Figure 39: The sound speed c 2 s (t) for the EW phase transition in the case of a<br />
Crossover with a reference temperature Tc = 100 GeV <strong>and</strong>: (a) ∆T =0.001Tc,<br />
(b) ∆T =0.005Tc, (c) ∆T =0.1Tc.
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