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 142 Table 32: The evolution of δc1 and δc2 (where applicable) as a function of time and as a function of the parameter x for a QCD phase transition according to the Bag Model when δc =1/3 (see Figures 64 and 66). log 10(tk/1 s) x δc1 δc2 -5.3 48.1 0.0963 0.102 -5.2 34.0 0.111 0.135 -5.1 24.1 0.126 0.181 -5.0 17.0 0.143 0.244 -4.9 12.0 0.161 0.333 -4.8 8.5 0.179 – -4.7 6.0 0.197 – -4.6 4.2 0.215 – -4.5 2.9 0.233 – -4.4 2.1 0.249 – -4.3 1.4 0.264 – -4.2 0.98 0.278 – -4.1 0.583 0.290 – -4.0 0.333 0.299 – -3.9 0.197 0.307 – -3.8 0.112 0.315 –
PBHs and Cosmological Phase Transitions 143 where cs0 =1/ √ 3. In the case of the Bag Model (Section 2.3.1) we have α(t) = 1 during the mixed phase and α(t) = 0 otherwise. Now the function f has a more general expression: f = 1 S3 Sc α(t)dS c Si 3 , (262) where Si corresponds to the size of the region when the transition begins. In particular, this expression is valid for the Bag Model. For example, for a fluctuation of class B we recover equation (255) f = 1 S3 Sc,B c,B S1 1 × dS 3 = S3 c,B − S3 1 S3 . (263) c,B In order to apply equation (262) to the QCD Crossover we start by transforming it into an expression where all the quantities are given as a function of time. Thus, taking into account that S = S(t) represents the evolution of the scale factor during the QCD Crossover (equation 86) and that we obtain dS 3 =3S 2 dS =3S dS 3 = 3 2 2 dS dt, (264) dt Λ exp c 3 (tSN 2/3 3 √t teq − t0) tSN t 3/2 dt. (265) eq Considering equations (86) and (265) and that Sc = R(tc) (see equation 240) one obtains, from equation (262), the following result: f = 3 2 1 t 3/2 c tc tk 1 − cs(t) √tdt (266) cs0 where we have taken into account that S1 ≡ tk and Sc ≡ tc. Notice that this result is valid also in the case tk
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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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