Primordial Black Holes and Cosmological Phase Transitions Report ...

More documents

Recommendations

Info

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

PBHs and Cosmological Phase Transitions 86 3 The EW phase transition The first phase transition predicted by the SMPP is the EW phase transition which occurs at a temperature TEW ∼ 100 GeV and at a time scale tEW ∼ 10 −10 s (e.g. Unsöld & Bascheck, 2002). At this temperature, which corresponds to an energy scale of the order of the masses of the Z 0 and W ± vector bosons (cf. Table 6), the weak interactions become short ranged after a symmetry breaking phase transition. For T < TEW the Z 0 and W ± vector bosons acquire masses through the Higgs mechanism while the photon remains massless, corresponding to the unbroken symmetry of the electromagnetic interactions (e.g. Boyanovsky et al., 2006). The value of TEW was estimated considering that the restoration of symmetry would happen when T ∼ G −1/2 F where GF ≈ 1.16637 × 10 −5 GeV −2 (e.g. Stuart, 1999) is the Fermi coupling constant (e.g. Gynther, 2006). In the EW standard model (Glashow–Salam–Weinberg model) the Higgs field is responsible for the dynamical mass generation via spontaneous symmetry breaking. At sufficiently high temperatures, T > TEW, the expectation value of the Higgs field is zero, i.e., the symmetry is restored and particles are massless. At T < TEW the symmetry breakes and particle masses become finite (e.g. Kämpfer, 2000). During this transition, according to the SMPP, all particles except the Higgs acquire their mass by the mechanism of spontaneous symmetry breaking (e.g. Schwarz, 2003). Csikor et al. (1998) obtained, using a nonperturbative analysis, that the phase transition is of first–order for Higgs masses less than 66.5±1.4 GeV while for larger Higgs masses only a rapid crossover is expected (see Figure 34). This value must be perturbatively transformed to the full Standard Model yielding 72.4 ± 1.7 GeV (Csikor et al., 1998). The exact determination of this critical Higgs–mass value, mH,c, at which the first–order EW phase transition changes to a crossover is important given its implications for the standard model (e.g. Karsch et al., 1996). The location of the endpoint of the first–order phase transition line is seen to move to smaller values of the Higgs mass as the chemical potentials µ are increased, indicating that the chemical potentials make the transition weaker. At the same time, the critical temperature is slightly increased. The value mH,c ≈ 72 GeV corresponds to the case µ = 0. If, for example, µ ≈ 30 GeV then we have mH,c ≈ 66 GeV (e.g. Gynther, 2006). Both the QCD and the EW theories contain a phase transition. The exact properties and critical temperatures of the transitions depend on the chemical potentials and the values of parameters of the theories. The EW phase diagram is considered in terms of the leptonic chemical µL potentials and the theory is parametrized by the Higgs mass, while the QCD phase diagram is considered in terms of the baryonic chemical potential µB and the theory is parametrized in terms of the strange quark mass. When the masses parameterizing the theories are small we have for both cases a first–order phase transition. However, as the chemical potentials are increased, the critical temperature of the EW phase transition increases, while the critical temperature of the QCD phase transition
PBHs and Cosmological Phase Transitions 87 Figure 34: Phase diagram of the SU(2)–Higgs model in the Tc/mH −RHW plane with RHW = mH/mW . The continuous line, representing the phase–boundary, is a quadratic fit to the data points. Above the line we are in the symmetric phase and below we are in the symmetry broken phase. However, this line, as an endpoint at RHW ≈ 0.82 (Csikor et al., 1998). decreases (see Figure 35). Thus, the responses of the systems on introducing the chemical potentials are opposite (e.g. Gynther, 2006). Perhaps the main difference between the EW and the QCD transitions is that only during the latter was the Universe reheated back to the critical temperature (see Section 2.1). This is due to the much larger value of latent heat in the QCD transition (e.g. Ignatius, 1993). The current mass limit for the Higgs is 114.3 GeV at 95% confidence level (see Yao et al., 2006) suggesting that the standard model does not feature a sharp EW phase transition (either first or second order) but it is rather a smooth Crossover (e.g. Boyanovsky et al., 2006). Since the change in relativistic degrees of freedom is tiny (∆g = 1, cf. Section 1.10) this is also a very boring event from the thermodynamical perspective (e.g. Schwarz, 2003). Since the Higgs sector of the theory carries only four of the total of 106.75 degrees of freedom (see Section 1.10), the contribution of the Higgs to the pressure is not easily visible (e.g. Gynther, 2006). Nevertheless, a first–order phase transition is still allowed in several extensions of the SMPP, including the MSSM (e.g. Kajantie et al., 1998, Section 1.9). 3.1 The critical temperature In the minimal standard model, EW symmetry breaking is induced by the ground state of a single doublet scalar field. We can write the potential for the real scalar component of the doublet which acquires a vacuum expectation
Page 1:
CCM-Centro de Ciências Matemática
Page 4 and 5:
Contents List of Figures vii List o
Page 6 and 7:
PBHs and Cosmological Phase Transit
Page 8 and 9:
Page 10 and 11:
Page 12 and 13:
List of Tables 1 The Universe timel
Page 14 and 15:
Page 16 and 17:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54: PBHs and Cosmological Phase Transit
Page 103: PBHs and Cosmological Phase Transit
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Page 215 and 216:
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
Table 49 (continued). PBHs and Cosm
Page 223 and 224:
Table 49 (continued). Radiation EW
Page 225 and 226:
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
Page 233 and 234:
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
Page 241 and 242:
show all

Primordial Black Holes and Cosmological Phase Transitions Report ...

Create successful ePaper yourself

Delete template?

Save as template?