Primordial Black Holes and Cosmological Phase Transitions Report ...

More documents

Recommendations

Info

$Kawasaki dynamic in continuum Math. Encounters XXXV ... - CCM$

PBHs and Cosmological Phase Transitions 158 ∆ 1.2 1 0.8 0.6 0.4 0.2 No BH formation BH formation ∆cA No BH formation ∆c2 ∆c1 t t -9 -8 -7 -6 -5 -4 -3 Log 10tk1s Figure 81: The curve in the (log 10(tk/1s), δ) plane indicating which parameter values lead to collapse to a BH in the case of the QCD Lattice Fit when δc =0.7. The vertical dashed lines correspond to tk = t− and tk = t+.
PBHs and Cosmological Phase Transitions 159 8 The threshold δc for PBH formation during the EW phase transition 8.1 Crossover Model (SMPP) During the EW Crossover it is expected a reduction on the value of the PBH formation threshold δc due to the decrease on the sound speed. We adopt for f expression (267) derived for the QCD Crossover (Section 7.2) but now with the sound speed, cs(t), given by equation (180). First, however, we must determine which values of ∆T we will use. We are particularly interested in a value of ∆T for which the threshold δc attains a minimum value (because lower values of δc favour PBH formation). For a given ∆T we determine, with the help of funtion (1 − f)δc, the new threshold δc1 as a function of the horizon crossing time tk. When tk ≪ tEW− or, when tk ≫ tEW+, we get δc1 = δc. Between these two extremes there is a value of tk for which δc1 attains a minimum value δc1,min. For example, when ∆T = 0.001Tc and δc =1/3, we obtain δc1,min ≈ 0.33213 with tk ≈ 8.32 × 10 −11 s. We repeated this procedure for different values of ∆T (0 < ∆T ≤ Tc) and concluded that, in the δc = 1/3 case, our best value is δc1,min ≈ 0.33186, corresponding to having ∆T ≈ 0.013Tc and tk ≈ 8.32 × 10 −11 s. In Figure 82 we show a selection of the obtained results. On Table 35 we show the results for different values of δc. Note that, although, the value of the parameter ∆T remains almost constant the same does not apply to tk. For a larger value of δc, the instant tk, for which we get the lowest δc1,min, is closer to tEW−. 8.1.1 Examples In this section we study the changes on the value of the threshold δc during an EW Crossover, through examples of fluctuations that cross the horizon at different epochs. The study is mainly made for δc = 1/3. We are mostly interested in the case ∆T =0.013Tc (because it is the one that leads to the lowest value of δc) but we also consider the cases ∆T =0.001Tc and ∆T =0.1Tc, with the purpose of comparing the results. Let us start with δc =1/3 and with a fluctuation that crosses the horizon at tk =5.0 × 10 −11 s (i.e., before t1 =2.97 × 10 −10 s, cf. Section 3.2.1). In order to identify the values of δ for which collapse to a BH will occur, in this case, we plot, in Figure 83a, both (1 − f)δc and δ itself as functions of δ. We conclude that the evolution of perturbations entering the horizon at this epoch are not affected by the presence of a Crossover transition. That is because tk occurs sufficiently before the transition. As a second example, we consider a fluctuation that crosses the horizon at tk =7.0×10 −11 s. In this case, we do have a lower threshold for PBH formation δc1 ≈ 0.3330 if ∆T =0.1Tc but not if ∆T =0.001Tc or ∆T =0.013Tc as it is clear from Figure 83b. In Figure 83c we have the case tk ≈ 8.32 × 10 −11 s. This
Page 1:
CCM-Centro de Ciências Matemática
Page 4 and 5:
Contents List of Figures vii List o
Page 6 and 7:
PBHs and Cosmological Phase Transit
Page 8 and 9:
Page 10 and 11:
Page 12 and 13:
List of Tables 1 The Universe timel
Page 14 and 15:
Page 16 and 17:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126: PBHs and Cosmological Phase Transit
Page 175: PBHs and Cosmological Phase Transit
Page 221 and 222: Table 49 (continued). PBHs and Cosm
Page 223 and 224: Table 49 (continued). Radiation EW
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
Page 233 and 234:
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
Page 241 and 242:
show all

Primordial Black Holes and Cosmological Phase Transitions Report ...

Create successful ePaper yourself

Delete template?

Save as template?