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 12 Figure 2: The energy densities of matter and radiation as a function of the scale factor R(t). At a time when R(t)/R(t0) ≈ 10 −4 the energy densities of matter and radiation were equal (R(t0) represents the present day value of the scale factor). It is also represented the energy density due to the cosmological constant, which does not vary with redshift, and is exceeded by the energy densities in matter and radiation at early times (e.g. Jones & Lambourne, 2004). When the universe was ∼ 380000 years old (z ≈ 1090, e.g. Bennett et al., 2003; Hinshaw et al., 2008) and the temperature had dropped to ∼ 3000 K, the number density of free electrons was so low that the universe essentially became transparent and photons could travel unhindered from this time on. This is known as the photon decoupling epoch. The photons released during this epoch become the CMB (Section 1.7). Surrounding every observer in the universe there is a last scattering surface from which the CMB photons have been streaming freely (e.g. Ryden, 2003) becoming redshifted due to the expansion of the universe (Section 1.7). The period between photon decoupling (z ≈ 1090) and the formation of the first luminous objects (z ∼ 11) is referred to as the Cosmic Dark Ages. That is because, during that period, there were no sources of radiation in the Universe, with the exception of the hyperfine 21–cm line of neutral hydrogen (e.g. Hirata & Sigurdson, 2007). Reionization is the second of two major phase changes of hydrogen gas in the Universe (the first was recombination). Reionization occurred once objects started to form in the early Universe, which was energetic enough to ionize neutral hydrogen. As these objects formed and radiated energy, the Universe went from being neutral back to being an ionized plasma, at redshift z ∼ 11 according to WMAP (Wilkinson Microwave Anisotropy Probe) results (Hinshaw et al., 2008). When the universe was ≈ 2.8 × 10 17 s old (≈ 0.7 times the present age) it become dark energy–dominated (see Figure 2). The true nature of this dark
PBHs and Cosmological Phase Transitions 13 energy, which is responsible for the observed non–linear acceleration of the universe, remains unknown (Section 1.8). In Table 1 we present a timeline of the Universe according to the inflationary Big Bang model. 1.3 Inflation In order to explain problems such as ‘flatness’, ‘horizon’ and ‘monopole’ the present paradigm makes use of an inflationary stage of expansion in the very early Universe. During inflation the scale factor R(t) behaves like (e.g. Narlikar & Padmanabhan, 1991) R(t) =R(ti) exp (H(t − ti)) . (38) The scale factor grows exponentially from an initial value Ri = R(ti), corresponding to the instant ti ∼ 10 −35 s when the EW and strong forces separate (e.g. Narlikar & Padmanabhan, 1991, cf. Table 1) to a value Re = R(te) according to (e.g. d’Inverno, 1993) Re Ri = e N(te) where (e.g. Tsujikawa, 2003) N(te) = ln Re Ri = tf ti Hdt ≈ Hte (39) (40) gives the number of e–folds that elapsed during inflation. For example, the value N = 70 means that during inflation the scale factor have grown up by a factor of e 70 (≈ 10 30 ). Although the exact value of N(te) is unknown, in order to solve the mentioned problems, the inflationary stage must last a time interval such that 50 0 which means that the R 2 H 2 term in equation (31) increases during inflation. As a result Ω rapidly approaches unity (see equation 36). After the inflationary period, the evolution of the Universe is followed by the conventional Big Bang phase (Section 1.2) and, despite this, Ω stays of order unity even in the present epoch, solving the flatness problem (e.g. Tsujikawa, 2003). Inflation gives rise to a remarkable phenomenon: physical wavelengths grow faster than the size of the Hubble radius (see equation 28). This means that the region where the causality works is stretched on scales much larger than the Hubble radius, i.e., once a physical wavelength becomes larger than the Hubble radius, it is causally disconnected from physical processes. This solves the horizon problem (e.g. Boyanovsky et al., 2006). Notice that Inflationary cosmology suggests that, even though the observed Universe is incredibly large, it is only an infinitesimal fraction of the entire Universe (e.g. Guth, 2000).
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Table 49 (continued). PBHs and Cosm
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Table 49 (continued). Radiation EW
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