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 112 x 1.75 1.5 1.25 1 0.75 0.5 0.25 t t x1 xy 1 -4.2 -4 -3.8 -3.6 Log 10t1s Figure 44: The function x(t) for the QCD Bag Model (equation 217). The change on the value of w occurs when t = t−, or equivalently, when x =1 (i.e., at the beginning of the transition). Considering that ρ(τ) is a continuous function, we write KkS −3(1+wk) 1 = KsS −3(1+wc) 1 (221) where S1 represents the size of the overdense region at the beginning of the transition. This leads to Ks Kk = S3wc 1 S 3wk 1 = 1 S1 (222) In the case of class E we have wk = 0 and wc =1/3. Now the change of w occurs when t = t+ or, equivalently, when x = y −1 (i.e., at the end of the transition). The continuity of the density, ρ, now leads to Ks Kk = S3wc 2 S 3wk 2 = S2 (223) where S2 represents the size of the overdense region at the end of the transition. Finally, in the case of fluctuations of class C, we have wk = wc =1/3 with an intermediate period during which w = w ′ = 0. Applying the continuity condition for ρ successively at t = t+ and t = t− we obtain, in the case of class C Ks Kk = S3wc 2 S 3wk S 1 3w′ 1 S3w′ 2 = S2 . (224) S1 The expression for S1, which is valid and useful for fluctuations of classes B and C, can be obtained considering that ρ1 is reached from radiation domination
PBHs and Cosmological Phase Transitions 113 (i.e. ρ ∼ S−4 and ρ ∼ R −4 k ). From energy conservation we have the condition ρ1S4 1 = ρkR4 k which can be combined with equation (200) in order to obtain S1 = x 1/4 (1 + δk) 1/4 Rk. (225) The expression for S2, useful for fluctuations of class C, can be obtained considering that ρ2 is reached from the radiation phase (i.e. ρ ∼ S−4 and ρ ∼ R −4 k ). From energy conservation we have the condition ρ2S4 2 = ρkR4 k which can be combined with equation (200) in order to obtain S2C =(xy) 1/4 (1 + δk) 1/4 Rk. (226) On the other hand, the expression for S2 suitable for fluctuations of classes E and F , can be obtained considering that ρ2 is reached from the dust–like phase (i.e. ρ ∼ S−3 and ρ ∼ R −3 k ). From energy conservation we have the condition ρ2S3 2 = ρkR3 k which can be combined with equation (200) in order to obtain S2E =(xy) 1/3 (1 + δk) 1/3 Rk. (227) We are now ready to determine expressions for the turnaround points of classes B, C and E. From equation (212), with the constant Ks/Kk given by equation: (222)–class B, (224)–class C, and (223)–class E; and with S1 given by equation (225) and S2 given by equation (226) in the case of fluctuations of class C and by equation (227) in the case of fluctuations of class E, we obtain 3/4 −1/4 (1 + δk) Sc,B = Rkx , (228) δk δk Sc,C = Rky 1/6 1/2 1+δk , and (229) 2/3 1/6 (1 + δk) Sc,E = Rk(xy) . (230) δ 1/2 k The separation between classes (A,B,C) and classes (D,E) is given by the condition ρ1 = ρk. With the help of equation (200) this becomes δk = x −1 − 1. (231) On the other hand the separation between classes (D,E) and class (F) is given by the condition ρ2 = ρk. With the help of equation (200) this becomes δk =(xy) −1 − 1. (232) The separation between classes A and B can be obtained noting that what distinguishes these classes is the location of the turnaround point. Thus, considering Sc,A (equation 219) equal to Sc,B (equation 228), we obtain x = 1+δk δ2 . (233) k
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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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