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 124 7 The threshold δc during the QCD epoch 7.1 Bag Model During a first–order QCD phase transition we replace the lower limit in the condition (251) by (e.g. Cardall & Fuller, 1998) δc → δc(1 − f) (253) where f denotes the fraction of the overdense region spent in the dust–like phase. Therefore, the larger the fraction of time a fluctuation spends in the mixed phase regime, the smaller the required amplitude of the perturbation (at horizon crossing) for collapse into a BH (e.g. Cardall & Fuller, 1998). Next, we present the expressions for f for each class of fluctuations of interest to us (cf. Table 26, e.g. Cardall & Fuller, 1998) fA = 0 (254) fB = S3 c,B − S3 1 S 3 c,B fC = S3 2C − S3 1 S 3 c,C fE = S3 2EF − S3 1 S 3 c,E fF = S3 2EF − S3 1 S 3 c,F =1− x3/2 δ 3 k (1 + δk) 3/2 = (255) x3/4δ 3/2 k y1/2 (y − 1) (256) (1 + δk) 3/4 = (xy)1/2δ 3/2 k (1 − y 1+δk −1 ) (257) = (xy)3/4δ 3/2 k (1 + δk) 3/4 (1 − y−1 ) (258) where the quantities S1, S2C , S2EF , Sc,B, Sc,C, Sc,E, and Sc,F are given by equations (225), (226), (227), (228), (229), (230), and (219), respectively. In the next section we study PBH formation during the QCD phase transition, from fluctuations of classes A, B, C, E and F, according to the Bag Model. We divide the study into Before, During, and After. At the end of the section we compile the results. The study will mostly be done for δc =1/3 but we will also consider the effect of larger values of δc (up to 0.7). 7.1.1 Before the mixed phase When x ≥ 1 we are dealing with fluctuations of classes A, B or C (cf. Figure 45). For a given x we can determine, with the help of equations (233) and (234), the range of amplitudes which correspond to each class. For example, for the case x = 2, we have from equation (233), that 32 δ = 1, and from equation (234) that δ ≈ 0.58. This means that when x = 2 the 32 In order to simplify the writing we represent the density constrast at the horizon crossing time tk by δ (instead of δk).
PBHs and Cosmological Phase Transitions 125 ∆ 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 50: PBH formation during the QCD transition according to the Bag Model for the case x = 2 and δc =1/3. The solid curve corresponds to the function (1 − f)δc and the dashed curve to the identity δ. The pink region corresponds to fluctuations of class B (see text). To the left of this region we have fluctuations of class C (green) and to the right fluctuations of class A (yellow). The borders between the different classes are given by δAB = 1 and δBC ≈ 0.58. Collapse to a BH occurs for values of δ for which the dashed line is above the solid curve (while δ< 1). The intersection point at δ ≈ 0.25 marks a new threshold δc1 for PBH formation (adapted from Cardall & Fuller, 1998). overdensity will be of class C if 0 < δ < 0.58, of class B if 0.58 < δ < 1 and of class A if δ> 1. In order to identify the values of δ for which collapse to a BH occurs (when x = 2 and δc =1/3), we plot in Figure 50 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 254; fB – equation 255; fC – equation 256). We can, then, have PBHs formed from fluctuations of classes B and C only, since fluctuations of class A would lead to the formation of a separate Universe (since for them we always have δ> 1)–Section 6.1. Fluctuations of class C with δ< 0.25 dissipate before forming a PBH. This point δc1 ≈ 0.25 marks a new and lower threshold for PBH formation during the QCD phase transition when x = 2. In Figure 51 we plot the cases: (a) x = 15, (b) x = 30, and (c) x = 90 with δc =1/3 for all the three cases. In the case x = 15 there are two regions for which PBH formation is allowed: i) a region for δ ≥ 1/3, which corresponds to PBH formation from fluctuations of class A during the radiation–dominated Universe; ii) a region between δc1 ≈ 0.15 and δc2 ≈ 0.27 corresponding to the formation of PBHs from fluctuations of classes B and C. The gap between δ =0.27 and δ =1/3 corresponds to: i) fluctuations of class A which dissipate because they have δ< 1/3; ii) fluctuations of class B which dissipate because they do not spend enough time on the dust–like phase, allowing collapse to begin. The case x = 30 is similar to the case x = 15. Notice, however, that now the B ∆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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