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 28 (71) to (77). Notice, however, that this will not be the case if one is working locally (i.e., near t− and t+). In Section 2.4, for example, we want to determine numerical values for t− and t+. In that case, it does not make sence to consider t−/t+ = 1. The same idea is valid for the term tEW−/tEW+. According to this, we can replace equations (69) to (73) by the single equation Λ R(t) = exp c 3 (tSN 2/3 1/2 teq t − t0) , te≤t≤ teq (86) tSN teq Equations (67) and (68) remain unchanged. Equation (76) becomes Λ R(t) = exp c 3 (tSN 2/3 1/2 teq te − t0) and equation (77) becomes tSN teq exp (Hi (t − ti)) exp (Hi (te − ti)) ,ti ≤ t ≤ te Λ R(t) = exp c 3 (tSN 2/3 1/2 teq te − t0) tSN teq 1/2 t exp (−Hi (te − ti)) ,tp ≤ t ≤ ti ti (87) (88) In Figure 7 we show R(t) for the entire Universe timeline (i.e., from the Planck time tp up to the present time t0). 1.7 The Cosmic Microwave Background temperature The existence of the Cosmic Microwave Background (CMB) radiation was first predicted by Gamow et al. (1948) but it was only in 1964 that it was observed (serendipitously) by the first time (Penzias & Wilson, 1965). In 1989, NASA launched the Cosmic Background Explorer satellite (COBE), and the initial findings, released in 1990, were consistent with the Big Bang’s predictions regarding the CMB. COBE found a residual temperature of 2.726 K and determined that the CMB was isotropic to about one part in 10 5 (Boggess et al., 1992). During the 1990s, CMB anisotropies were further investigated by a large number of ground–based experiments and the Universe was shown to be almost geometrically flat, by measuring the typical angular size of the anisotropies. The CMB brings us information about the state of the Universe at the photon decoupling epoch (z ≈ 1090) when the photons that reach us now had their last scattering (Section 1.2). The spectrum of the CMB at the present epoch is well described by a blackbody function with (e.g. Boyanovsky et al., 2006) T0 =2.725 ± 0.001K (89)
PBHs and Cosmological Phase Transitions 29 Table 3: The Scale Factor (equations 67–77) for different instants of time during the evolution of the Universe: tp (Planck time), ti (beginning of inflation), te (end of inflation), tEW− (beginning of the EW transition – Section 3.2), tEW+ (end of the EW transition – Section 3.2), t− (beginning of the QCD transition – Section 2.4), t+ (end of the QCD transition – Section 2.4), teq (last scattering surface), tSN (the instant when the Universe starts to accelerate) and t0 (present time). Notice that we have indicated two values for t−. The first one corresponds to the Bag Model results and the second one to the Lattice Fit results (cf. Section 2.4). t(s) R(t) tp ∼ 10 −43 ∼ 10 −60 ti ∼ 10 −35 ∼ 10 −56 te ∼ 10 −33 ∼ 10 −27 tEW− 2.30 × 10 −10 2.9 × 10 −15 tEW+ 3.15 × 10 −10 3.5 × 10 −15 Log 10Rt t− 6.25 × 10 −5 1.4 × 10 −12 (QCD Bag) t− 9.35 × 10 −5 1.9 × 10 −12 (QCD Lattice) t+ 1.08 × 10 −4 2.1 × 10 −12 teq 2.5 × 10 12 3.2 × 10 −4 tSN 2.8 × 10 17 0.73 t0 4.3 × 10 17 1 -11.6 -11.7 -11.8 -11.9 QCD -4.4 -4.3 -4.2 -4.1 -4 -3.9 -3.8 -3.7 Log 10t1s Figure 6: The scale factor during the QCD transition as a function of time. The gray region corresponds to the dust–like epoch, according to the Bag Model (Sections 2.3.1 and 2.4). The curves correspond, from top to bottom, to the: crossover case (nqcd =1/2), Lattice Fit (nqcd =2/3 and t− =9.35 × 10 −5 s) and Bag Model (nqcd =2/3 and t− =6.25 × 10 −5 s).
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Table 49 (continued). PBHs and Cosm
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Table 49 (continued). Radiation EW
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