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 16 Figure 3: The schematic illustration of the potential of the chaotic inflation model. This belongs to the class of the large field model (Tsujikawa, 2003; Guth, 2000). Figure 4: The schematic illustration of the potential of the natural inflation model which belongs to the class of the small field model. Here f represents the width of the inflaton potential (tipically f ∼ mP ∼ 10 19 GeV). When φ = πf the potential vanishes (Tsujikawa, 2003).
PBHs and Cosmological Phase Transitions 17 Figure 5: The schematic illustration of the potential of the hybrid (double) inflation model. This model is characterized by multiple scalar fields. In this example we have two scalar fields: φ and χ. For large φ we have a situation similar to the other models with a single field. However, when φ reaches the critical value φc a different behaviour takes place. In particular, the energy density during inflation can be much lower than normal while still giving suitably large density perturbations, and secondly the field φ can be rolling extremely slowly which is of benefit to particle physics model building (Liddle, 1999; Tsujikawa, 2003). The inflaton φ is an homogeneous scalar field, whose potential energy V (φ) leads to the exponential expansion of the Universe. The number of e–folds that elapsed during inflation (equation 40) can be written also as a function of φ (e.g. Boyanovsky et al., 2006) N(φ(t)) = − 1 m 2 pl φe φ(t) V (φ) V ′ dφ. (41) (φ) The energy density and the pressure density of the inflaton can be described, respectively, as (e.g. Liddle & Lyth, 1993) ρ = 1 2 ˙ φ 2 + V (φ) (42) p = 1 2 ˙ φ 2 − V (φ). (43) Substituting equations (42) and (43) for equations (2) and (4) we get (e.g. Liddle & Lyth, 1993) 7 H 2 = 8πG 3 1 2 ˙ φ 2 + V (φ) = 8π 3m2 1 pl 2 ˙ φ 2 + V (φ) (44) 7 We have written equation (44) also in terms of the Planck mass mpl =(c/G) 1/2 with = c = 1.
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Table 49 (continued). PBHs and Cosm
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Table 49 (continued). Radiation EW
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