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 150 7.3.1 Before the mixed phase When x ≥ 1(tk ≤ t−) we are dealing with fluctuations of classes A, B or C. For a given x we can determine the range of amplitudes corresponding to each class. More precisely, the solution of the equation fB(d) =fC(d) gives the boundary between classes B and C and the solution of the equation fA(d) =fB(d) the boundary between classes B and A. Let us start with the case x = 15 (tk ≈ 1.6 × 10 −5 s) and δc =1/3. The boundaries between the different classes are δAB ≈ 0.25 and δBC ≈ 0.21. In order to identify the values of δ for which collapse to a BH occurs we plot in Figure 71 both (1−f)δc and δ itself as functions of δ. Notice that one should use the function f appropriate to each class (i.e. fA – equation 270; fB – equation 271; fC – equation 274). As it is clear we could have, in this case, PBHs from fluctuations of classes A (provided that δ< 1), B and C (provided that δ> 0.18 because otherwise the fluctuation will dissipate without forming a PBH). This point δc1 ≈ 0.18 marks a new and lower threshold for PBH formation during the QCD phase transition when x = 15 and δc =1/3. We have also represented on Figure 71 the curve (1 − fBag)δc where fBag regards only to the contribution within the Bag Model (i.e., we are considering fALat = fBLat = fCLat = 0 ) of the period t−
PBHs and Cosmological Phase Transitions 151 ∆ 1.4 1.2 1 0.8 0.6 0.4 0.2 0 C B A ∆1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ∆ Figure 71: PBH formation during the QCD transition according to the Lattice Fit for the case x = 15 (tk ≈ 1.6 × 10 −5 s) with δc =1/3. The bottom solid curve (in black and red) corresponds to the function (1 − f)δc and the dashed curve corresponds to the identity δ. The intersection point between these two curves at δc1 ≈ 0.18 marks a new threshold for PBH formation. The borders between the different classes of fluctuations are δAB ≈ 0.14 and δBC ≈ 0.10. The top solid curve (in blue) corresponds to the case for which f regards only to the contribution, within the Bag Model, of the period t− < t < t+ (see text for more details). ∆ 1.4 1.2 1 0.8 0.6 0.4 0.2 0 C 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ∆ Figure 72: PBH formation during the QCD transition according to the Lattice Fit for the case x = 2 (tk ≈ 6.1 × 10 −5 s) with δc =1/3. The solid curve corresponds to the function (1 − f)δc and the dashed curve corresponds to the identity δ. The new threshold for PBH formation is now δc1 ≈ 0.29 and the border between fluctuations of classes A and C is δAC ≈ 0.95. ∆1 A
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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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