Primordial Black Holes and Cosmological Phase Transitions Report ...

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PBHs and Cosmological Phase Transitions 18 ¨φ +3H ˙ φ + V ′ (φ) = 0 (45) where we have considered κ = 0 and Λ = 0 in equation (44). Here, the prime denotes the derivative of the potential with respect to the inflaton field. Amongst the wide variety of inflationary scenarios, slow roll inflation provides a simple and generic description of inflation consistent with the WMAP data (e.g. Boyanovsky et al., 2006). The basic premise of slow roll inflation is that the potential is fairly flat during the inflationary stage. This flatness not only leads to a slowly varying inflaton and Hubble parameter, hence ensuring a sufficient number of e–folds, but also provides an explanation for the gaussianity of the fluctuations as well as for the almost scale invariance of their power spectrum. Departures from scale invariance and gaussianity are determined by the departures from flatness of the potential, namely by derivatives of the potential with respect to the inflaton field (e.g. Boyanovsky et al., 2006). The slow roll approximation corresponds to (e.g. Carr, 2005) ξ ≪ 1, |η| ≪ 1 (46) where ξ and η are the so called slow–roll parameters which are determined by the derivatives of the inflaton potential in the following manner (e.g. Carr, 2005) ξ = m2 pl 16π V ′ V 2 (47) η = m2pl V 8π ′′ . (48) V The inflationary era ends when ξ and |η| grow to order unity (e.g. Tsujikawa, 2003). At that time the scalar field starts to roll faster and finally to oscillate around the minimum and finally it decays producing radiation and reheating the Universe (e.g. Covi, 2003). If the conditions (46) are valid then equations (44) and (45) are approximately given by (e.g. Boyanovsky et al., 2006) H 2 = 8π 3m2 V (φ) (49) pl 3H ˙ φ + V ′ (φ) = 0. (50) Inflation is now an established part of Cosmology with several important aspects, such as the superhorizon origin of density perturbations, having been spectacularly validated by WMAP (e.g. Boyanovsky et al., 2006). The gaussian and nearly scale invariant spectrum of primordial fluctuations generically predict by most inflationary models fits with high precision the data provided by WMAP (e.g. Spergel et al., 2007). 1.4 Cosmological phase transitions The inflationary stage is followed by a radiation–dominated era after a short period of reheating during which the energy stored in the inflaton field decays
PBHs and Cosmological Phase Transitions 19 into quanta of many other fields, which, through scattering processes, reach a state of local thermodynamic equilibrium. This period is followed by deccelerated expansion and cooling, with the Universe successively visiting the different energy scales at which particle and nuclear physics predict symmetry breaking phase transitions. Those phase transitions are broadly characterized as either second or first–order (e.g. Boyanovsky et al., 2006). If a thermodynamic quantity changes discontinuously (for example as a function of temperature) we have a first–order phase transition. This happens because, at the point at which the transition occurs, there are two separate thermodynamic states in equilibrium. Any thermodynamic quantity that undergoes such a discontinuous change at the phase transition is referred to as an order parameter. Whether or not a first–order phase transition occurs often depends on other parameters that enter the theory. It is possible that, while another parameter is varied, the change in the order parameter of the phase transition decreases until they, together with all other thermodynamic quantities, become continuous at the transition point. In this case we refer to a second order phase transition at the point at which the transition becomes continuous (i.e., it shows a thermodynamic behaviour without discontinuities or singularities in the free energy or any of its derivatives), and a continuous crossover at the other points for which all physical quantities undergo no changes (e.g. Trodden, 1999). However, if the crossover is relatively sharp the situation may not be too different from a phase transition (e.g. Boyanovsky et al., 2006). Phase transitions are the most important phenomena in particle cosmology since, without them, the history of the Universe would simply be one of gradual cooling. In the absence of phase transitions, the only departure from thermal equilibrium is provided by the expansion of the Universe (e.g. Trodden, 1999). The SMPP (Section 1.8) predicts two phase transitions. The first one, at temperatures of ∼ 100 GeV, is the Electroweak phase transition (Section 3) which was responsible for the spontaneous breaking of the EW symmetry, which gives the masses to the elementary particles. This transition is also related to the EW baryon–number violating processes, which had a major influence on the observed baryon–asymmetry of the Universe (e.g. Aoki et al., 2006b). The second transition occurs at T ≈ 170 MeV. It is related to the spontaneous breaking of the chiral symmetry of the Quantum Chromodynamics (QCD) when quarks and gluons become confined in hadrons (Section 2). At high temperatures asymptotic freedom of QCD predicts the existence of a deconfined phase (according to lattice QCD simulations), the Quark–Gluon Plasma (QGP). At low temperatures quarks and gluons are confined in a Hadron Gas (HG) (e.g. Schmid et al., 1999). The QCD phase transition was pointed out, for a long time, as a prime candidate for a first–order phase transition (e.g. Jedamzik & Niemeyer, 1999). Recent results (e.g. Aoki et al., 2006b) provide strong evidence that the QCD transition is a simple Crossover instead. Here we will consider the two possibilities (see Section 2).
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Table 49 (continued). PBHs and Cosm
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Table 49 (continued). Radiation EW
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