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 26 On the other hand we have, from equation (73) Λ R(te) = exp c 3 (tSN 2/3 1/2 teq t+ t− − t0) × tEW+ t− tSN 1/2 new tEW− te tEW+ tEW− teq 1/2 . t+ nqcd × (75) Combining equations (74) and (75) one finds an expression for R(ti). Inserting this expression into equation (38) one obtains, for the scale factor during inflation, the result Λ R(t) = exp c 3 (tSN 2/3 1/2 nqcd teq t+ t− − t0) × × tEW+ t− tSN 1/2 new tEW− te tEW+ × exp (Hi (t − ti)) exp (Hi (te − ti)) ,ti ≤ t ≤ te tEW− teq 1/2 × t+ (76) where Hi corresponds to the value of the Huble parameter during inflation that we assume constant (cf. equation 66). Finally, considering that before inflation the Universe is radiation–dominated, we write Λ R(t) = exp c 3 (tSN 2/3 1/2 nqcd teq t+ t− − t0) × × tEW+ t− tSN 1/2 new tEW− te tEW+ tEW− teq 1/2 × 1/2 t × exp (−Hi (te − ti)) ,tp ≤ t ≤ ti ti t+ (77) where tp represents the Planck time. The scale factor R, the background temperature T and the redshift z at a given epoch are related according to the expression (e.g. Unsöld & Bascheck, 2002) T (t) T0 = R0 = 1 + z (78) R(t) where T0 represents the present day background temperature (T0 ≈ 2.725).
PBHs and Cosmological Phase Transitions 27 The value of t0, i.e., the age of the Universe, can be obtained with the help of expression (e.g. Yao et al., 2006) H0t0 = ∞ 0 dz . (79) (1 + z)H(z) In the case of a flat Universe (Ωm +ΩΛ = 1) expression (79) can be written as (e.g. Yao et al., 2006) H0t0 = 2 3 √ ΩΛ ln 1+√ΩΛ √ , Ωm < 1. (80) 1 − ΩΛ Inserting H0 (see equation 51) into equation (80) with ΩΛ =0.76 we obtain, for the age of the Universe t0 ≈ 4.3 × 10 17 s. (81) Considering that during the cosmological constant domination the Hubble parameter stays constant (cf. equation 65) we have that H(tSN) =H0. Thus, inserting H0 into equation (64) we obtain tSN ≈ 2.8 × 10 17 s. (82) Notice that if Λ = 0 this would correspond to t0. This means that, in the absence of dark energy, the Universe would be younger by a factor of ≈ 1.5 (e.g. Jones & Lambourne, 2004). The value of teq can be obtained with the help of equations (62) and (68) considering that z ≈ 3200 at radiation–matter equality (e.g. Hinshaw et al., 2008) teq ≈ 2.5 × 10 12 s. (83) Proceding the same way, one obtains, for the age of the universe at photon decoupling (z ≈ 1090) tdec ≈ 1.2 × 10 13 s. (84) Taking ti = 10 −35 s we have, from equations (63) and (66), that te ∼ 10 −33 s (85) valid for both N(te) = 50 and N(te) = 70. The calculus of numerical values to t−, t+, tEW− and tEW+ will be discused in Sections 2.4 and 3.2. For the moment, we present on Table 3 all these instants of time as well as the corresponding values for the scale factor. Notice that some of these values may be slightly different, depending on the values one chooses for nqcd, new and t−. In Figure 6 we show R(t) during the QCD transition. Notice that the reduction on the scale factor during a dust–like QCD transition is not very significative. In fact the term t−/t+ is ∼ 1 and, thus, it can be neglected in equations
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Table 49 (continued). PBHs and Cosm
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Table 49 (continued). Radiation EW
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