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 4 However, during inflation (Section 1.3) or in a universe dominated by a cosmological constant, w becomes negative and may not even be constant (e.g. Yao et al., 2006). If w
PBHs and Cosmological Phase Transitions 5 Section 2.3). The conservation of entropy leads to the useful relation (e.g. Schmid et al., 1999) dT 3s = − . (15) d ln R ds/dT In the following we consider the solutions of equation (2) when a single component dominates the energy density. Taking into account that, according to observation, we live in a flat Universe, we consider κ = 0. Note that even if κ = 0 at early times (when the scale factor is smaller) we can neglect the term κ/R 2 in equation (2) as long as w>−1/3 (e.g. Yao et al., 2006). Inserting equation (10) into equation (2), with κ = 0 and Λ = 0, one obtains (e.g. Yao et al., 2006) 2 R(t) ∝ t 3(1+w) . (16) For a radiation–dominated Universe (w =1/3), equation (16) becomes R(t) ∝ t 1/2 . (17) The radiation and matter densities in the Universe decrease as the expansion dilutes the number of atoms and photons. Radiation is also diminished due to the cosmological redshift, so its density falls faster than that of matter. When the age of the Universe was ∼ 10 6 years it became matter–dominated. Now it is appropriate to assume an EoS corresponding to a pressureless gas (w = 0). For this matter–dominated Universe, equation (16) becomes R(t) ∝ t 2/3 . (18) This might also be the case if the Universe experienced a dust–like phase during a phase transition on the radiation–dominated epoch (e.g. Carr et. al., 1994). If the Universe is dominated by a positive cosmological constant Λ then we will have an EoS with w 0 leads to (e.g. Yao et al., 2006) Λ R(t) ∝ exp 3 ct (19) which corresponds to an exponential expansion of the Universe. Assuming that light propagates in Relativistic Cosmology in the same way as it does in General Relativity we will consider now how an observer O receives light from a receding galaxy. Without loss of generality we will take O to be at the origin of coordinates r = 0. Inserting the conditions for a radial null geodesic into the line element (1) we have (e.g. d’Inverno, 1993) dt R(t) dr = ± (1 − kr) 1/2 (20)
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Table 49 (continued). PBHs and Cosm
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Table 49 (continued). Radiation EW
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