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 174 11 The fraction of the Universe going into PBHs The fraction of the Universe going into PBHs at a given epoch tk is given by (e.g. Sobrinho & Augusto, 2007) β(tk) = δmax 1 √ exp − 2πσ(tk) δc δ2 2σ2 dδ (286) (tk) where σ(tk) represents the mass variance at that epoch, δmax = 1, and δc represents the threshold for PBH formation. The value of δc is, in the case of a radiation–dominated universe, a constant somewhere between 1/3 and 0.7 (see e.g. Sobrinho & Augusto, 2007). However, if the universe experiences a phase transition, the value of δc experiences a reduction which favours PBH formation (Sections 7, 8 and 9). In this section, we consider that, during radiation domination, δc =1/3 and that, during the QCD transition, the EW transition, and the electron–positron annihilation epoch, δc assumes the values obtained in Sections 7, 8, and 9, respectively. In the presence of a Crossover–like transition, such as the QCD Crossover (Section 2.3.3), the EW Crossover (Section 3.2.1) or the electron–positron annihilation (Section 4), equation (286) must be replaced by β1(tk) = δc 1 √ exp − 2πσ(tk) δc1 δ2 2σ2 dδ (tk) 1 + √ 2πσ(tk) 1 δc exp − δ2 2σ2 dδ (tk) (287) where the additional integral accounts for the contribution from the Crossover epoch. We refer to the second integral, which is equal to the integral in expression (286), as the contribution from radiation. Denoting this integral by βRad(tk) equation (287) becomes β1(tk) = δc 1 √ exp − 2πσ(tk) δc1 δ2 2σ2 dδ + βRad(tk) (288) (tk) Naturally, if we are dealing with epochs sufficiently apart from the transition such that δc1 ≈ δc then equation (286) remains valid. On the other hand, in the presence of a Bag Model–like transition, such as the QCD Bag Model transition (Section 2.3.1) or the EW Bag Model transition (Section 3.2.2), equation (286) is valid only up to some instant after which there is an additional window [δc1,δc2] allowing PBH formation (cf. Figure 66 and Table 32 for the QCD, Figure 86 and Table 37 for the EW). For these cases equation (286) must be replaced by β2(tk) = δc2 1 √ exp − 2πσ(tk) δc1 δ2 2σ2 dδ + βRad(tk) (289) (tk)
PBHs and Cosmological Phase Transitions 175 Table 40: The different scenarios concerning the calculus of β. Scenario EW model QCD model e − e + model 1 Crossover Bag Model Crossover 2 Crossover Lattice Fit Crossover 3 Crossover Crossover Crossover 4 Bag Model Bag Model Crossover 5 Bag Model Lattice Fit Crossover 6 Bag Model Crossover Crossover Eventually, we reach some point where there is no more δc2 (see e.g. Figure 66). In that case there is a single window [δc1, 1] for PBH formation and we must use, instead, equation (288). Finally, in the case of a QCD Lattice Fit (Section 2.3.2) we must consider another extra window [δcA,δc] allowing PBH formation (cf. Figure 80, Table 34). In this case we must replace equation (286) by β3(tk) = δc 1 √ exp − 2πσ(tk) δcA δ2 2σ2 dδ + βRad(tk) (290) (tk) Over a brief period we might have to consider the window [δc1,δc2] (cf. Figure 80). For that period we must use, instead β4(tk) = δc 1 + √ 2πσ(tk) δc2 1 √ exp − 2πσ(tk) δc1 δ2 2σ2 dδ+ (tk) δcA exp − δ2 2σ2 dδ + βRad(tk) (tk) (291) Moving to later epochs we reach some point after which there is no more δc2 available (see e.g. Figure 80). In that case there is a single window [δc1, 1] for PBH formation and we use equation (288). Taking into account the considered models for the EW phase transition (Section 3.2), for the QCD transition (Section 2.3), and for the electron–positron annihilation (Section 4), there are six different possible scenarios (2 EW models × 3 QCD models × 1 e − e + model) concerning the determination of the curve β(tk). We list those scenarios in Table 40. For a given scenario, and for a given instant tk, one must choose the appropriate expression to determine the value β(tk). Proceding this way, for different values of tk, one can determine the curve β(tk) for that particular scenario.
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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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