Primordial Black Holes and Cosmological Phase Transitions Report ...

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$Kawasaki dynamic in continuum Math. Encounters XXXV ... - CCM$

PBHs and Cosmological Phase Transitions 98 the role of the QCD’s Bag constant (see Section 2.3.1), is derived from the potential V . However we use a simpler version, based on the approximation (Enqvist et al., 1992) B(T ) ≈ l 1 − T Tc ≈ l 1 − 2 T 2 T 2 c . (183) If one wants to study how a given fluctuation behaves during the EW phase transition, then it is of crucial importance to know the duration of the transition. This means that we need to define a specific beginning t = tEW− as well as a specific end t = tEW+ to the EW transition. Here tEW− and tEW+ are the limits for the time interval during which the sound speed vanishes. Although the temperature of the Universe is not constant during the EW phase transition, is stays all the time near the critical value Tc. Thus, the value of R(tEW+) (i.e. the value of the scale factor at the end of the transition) is given, approximately, by (see equation 78) R(tEW+) ≈ T0 Tc . (184) On the other hand, for t = tEW+ equation (86) becomes Λ R(tEW+) = exp c 3 (tSN teq − t0) 2/3 1/2 tEW+ . (185) Inserting equation (184) into equation (185) we obtain, for the instant when the transition ends teq tEW+ = tSN −1/3 Λ exp −2c 3 (tSN T0 2 − t0) . (186) tSN The evolution of the scale factor while the high and low temperature phases coexist in a first–order EW phase transition, i.e., during the c2 s = 0 part, may be determined by the entropy conservation as long as we assume that the transition evolves close to equilibrium (e.g. Jedamzik & Niemeyer, 1999). Thus, we here adopt equation (148) from the QCD case, writing it in the form ∆R = R(tEW+) R(tEW−) = 1/3 s(tEW−) = 1+ s(tEW+) ∆g 1/3 . (187) tSN When t = tEW− equation (72) becomes Λ R(tEW−) = exp c 3 (tSN 2/3 1/2 teq tEW+ tEW− − t0) tSN gl teq teq Tc tEW+ 2/3 (188) where we consider new =2/3 and nqcd =1/2. From equations (185) and (188) we obtain tEW− = tEW+ √ . (189) ∆R3
PBHs and Cosmological Phase Transitions 99 cs 2 0.4 0.3 0.2 0.1 0 tEW tEW -9.7 -9.6 -9.5 -9.4 -9.3 Log 10t1s Figure 41: The sound speed c2 s (t) for the EW phase transition according to the Bag Model with Tc = 100 GeV. During the coexistence phase, which occurs between the instants tEW− = 2.3 × 10−10 s (see text for more details) and tEW+ =3.15 × 10−10 s, the sound speed drops to zero. Considering Tc = 100 GeV we obtain, with the help of equation (186), the value tEW+ = 3.15 × 10 −10 s. In order to determine the value of tEW− one must determine first the value of ∆R. This is a problem, because we need to know the value of ∆g = gh −gl (see equation 187). We consider, for gl (i.e., the number of degrees of freedom at the end of the transition), in accordance with the SMPP, the value gl = 95.25. However, we do not have any clue for the real value of gh. In the context of the MSSM it may be as large as 228.75 when all particles and superpartners are present (cf. Table 15) and, in the context of the SMPP it is gh = 96.25. Within this range of values ∆R varies between ≈ 1.0035 and ≈ 1.3392. Inserting this values into equation (189) with tEW+ =3.15 × 10 −10 s it turns out that tEW− should be between 2.03 × 10 −10 s and 3.13 × 10 −10 s. We consider here tEW− =2.3 × 10 −10 s, corresponding to ∆g ≈ 80. In Figure 41 we show the sound speed profile for the EW transition, according to these results. It must be noted, however, that the value of tEW− was introduced only with the purpose of giving an example. At this stage, we do not have any observational evidence supporting this or any other value. In fact, we do not have any concludent results on the existence of extensions of the SMPP beyond the EW cosmological phase transition. 3.3 The baryogenesis problem There is a large body of observational evidence that suggests that there is more matter than antimatter in the Universe up to scales of the order of the Hubble
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CCM-Centro de Ciências Matemática
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Contents List of Figures vii List o
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PBHs and Cosmological Phase Transit
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List of Tables 1 The Universe timel
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Table 49 (continued). PBHs and Cosm
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Table 49 (continued). Radiation EW
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