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 6 where the + sign corresponds to a receding light ray and the − sign to an aproaching light ray. Consider a light ray emanating from a galaxy P with world line r = r1, at coordinate t1, and received by O at coordinate time t0. Using equation (20) we have (e.g. d’Inverno, 1993) t0 t1 0 dt dr = − . (21) R(t) r1 (1 − kr) 1/2 Next, consider a second light ray emanating from P at time t1 +dt1 and received at O at time t0 + dt0. Thus, we have t0+dt0 t1+dt1 0 dt dr = − . (22) R(t) r1 (1 − kr) 1/2 Comparing equations (21) and (22) it turns out that t0+dt0 t1+dt1 dt R(t) = t0 dt . (23) t1 R(t) Assuming that R(t) does not vary greatly over the intervals dt1 and dt0 we can take it outside the integral, yielding (e.g. d’Inverno, 1993) dt0 dt1 = . (24) R(t0) R(t1) All fundamental particles of the substractum have world lines on which the spatial coordinates are constant and, hence, the metric reduces to ds 2 = dt 2 . Here t measures the proper time along the substractum world lines. The intervals dt1 and dt0 are, respectively, the proper time intervals between the rays as measured at the source and observer. In an expanding Universe we have that t0 >t1 and so R(t0) >R(t1) which means that the observer O will experience a redshift z given by (e.g. d’Inverno, 1993) 1+z = ν0 ν1 = R(t0) dt0 = R(t1) dt1 (25) where ν1 and ν0 are the frequencies measured by the emitter and the receiver, respectively. In a contracting Universe O will detect instead a blue shift. The Hubble parameter H is defined as (e.g. d’Inverno, 1993) H(t) = ˙ R(t) R(t) and the Hubble time is defined as (e.g. Boyanovsky et al., 2006) tH(t) = 1 H(t) (26) R(t) = . (27) ˙R(t)
PBHs and Cosmological Phase Transitions 7 Multiplying tH by the speed of light c one obtains the so called Hubble radius RH (e.g. Boyanovsky et al., 2006) RH(t) = c = cR(t) H(t) ˙R(t) (28) which corresponds to the size of the Observable Universe at a given epoch. The mass contained inside a region with size RH is called the horizon mass and it is given by MH(t) = 4 3 πRH(t) 3 ρ(t). (29) Here we consider for MH(t) the approximation given by (e.g. Carr, 2005) MH(t) ≈ c3 t t ≈ 1015 G 10−23 g (30) s where c is the speed of light in the vacuum and G is the Gravitational constant. This expression is useful in the context of the study of PBHs. It is natural to assume that the mass of a PBH, when it forms, is of the order of MH at that epoch (e.g. Carr, 2005). When t ≈ 10 −23 s we have MH ≈ 10 15 g. These values represent, respectively, the time of formation and the initial mass of the PBHs that are presumed to be explodind by the present time (e.g. Sobrinho, 2003). There are, however, some problems with the stantard Big Bang theory. In order to identify such problems let us start by dividing equation (2) by H 2 1= 8πG κ ρ − 3H2 R2H 2 + c2Λ . (31) 3H2 Consider the case Λ = 0. If κ< 0 the Universe will expand forever and if k>0 the expansion will eventually give way to contraction. Between the two possibilities we have the critical case κ = 0. In this case one obtains from equation (31) the following expression for the density ρc = 3H2 8πG (32) which is called the critical density. The matter density parameter is defined as (e.g. Unsöld & Bascheck, 2002) Ωm = ρ ρc (33) where ρ is the matter density of the Universe. We will introduce here also the quantities (e.g. Covi, 2003) Ωκ = − κ H 2 R 2 ΩΛ = ρΛ ρc (34) = c2Λ . (35) 3H2
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Table 49 (continued). PBHs and Cosm
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Table 49 (continued). Radiation EW
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