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 208 equivalently, the fraction of the universe going into PBHs at that epoch β(tk), we must know the value of the mass variance σ(tk) at that epoch. In order to determine σ(tk) it is also crucial to know the shape of the primordial spectrum of the fluctuations. In Sobrinho & Augusto (2007) we considered different kinds of spectra: i) scale–free power–law spectrum; ii) scale–free power–law spectrum with a pure step; iii) broken scale invariance spectrum; iv) running–tilt power– law spectrum. In the present work we concentrated on the running–tilt power– law spectrum because it is highly supported by recent WMAP observations (Section 10) and, besides that, it possesses a variable index n(k) that might give more power during some epochs relevant to PBH formation. However, the running–tilt power–law spectrum introduces a pair of additional parameters to the equations: a parameter n+ giving the maximum value attained by n(k) and a parameter t+ giving the location of that maximum. At present, the best we can do is to constrain these parameters in accordance with the observational results (Section 10). We have considered, 1.2
PBHs and Cosmological Phase Transitions 209 δc reaching values as low as ≈ 0.091 for a background value δc =1/3 (Figure 66). In Section 7.2 we considered the variation of δc during the QCD Crossover. We introduced a new function f (see equation 262) which takes into account the fact that, during the Crossover, the sound speed decreases but does not vanish. We have done this through an adimensional function α(t) (equation 261) which gives the fraction of the sound speed with respect to the background value (1/ √ 3) at a given moment. We found that, in the case of a Crossover, the reduction on the value of δc is much less pronunced than in the Bag Model case with δc,min ≈ 0.274 for a background value δc =1/3 (Figure 70). In Section 7.3 we considered the variation of δc during the QCD Lattice Fit. In this case we have a period with a vanishing sound speed which is similar to the Bag Model case and also a period during which the sound speed decreases down to zero, resembling the Crossover situation (cf. Figure 30). Thus, we interpret the Lattice Fit as a mixture of both situations and derive an appropriate expression for the function f (see equations 268 to 276). The study was divided, as in the Bag Model case, in before, during and after. As a result, we obtained a reduction of δc from 1/3 to ≈ 0.12 (Figure 80). Tipically, we have curves for β with two peaks: one from the radiation contribution and another from the QCD contribution (e.g. Figures 106f and 109c). Contributions from the QCD Bag Model or from the QCD Lattice Fit are, naturally, more visible than those from the QCD Crossover, since, in the latter, the sound speed never reaches zero. However, in the case of the QCD Crossover we might also reach high values for β (e.g. Figure 104). There are many cases for which the contribution from the QCD (in particular in the case of a Bag Model or a Lattice Fit) exceeds the observational constraints (cf. Tables 44, 45 and 46). Cut from Table 49, in Table 50 we present a list with the ten largest contributions from the QCD Crossover. In each case we have also indicated the contribution from radiation. For the cases shown, the contribution from the EW phase transition is negligible and the contribution from the electron–positron annihilation appears only in two cases (labeled ‘ea’). If one considers, for the QCD phase transition, the Bag Model instead of a Crossover, then the ten cases are excluded due to observational constraints. On the other hand, if one considers the Lattice Fit model, then only the case n+ =1.52 and t+ = 10 −2 s is allowed (labeled ‘L’). Cut from Table 49, in Table 51 we present a list with the ten largest contributions from the QCD Bag Model. In each case we have also indicated the contribution from radiation. Notice that for all ten cases the contribution from the QCD is, by far, much greater than the contribution from radiation. A choice of a Lattice Fit model for the QCD gives similar results for all cases. On the contrary, the Crossover model gives, for these cases, very modest results (see also Table 49). As an interesting situation we mention the case n+ =1.40 and t+ = 10 −7 s, for which we have, besides the contribution from the QCD, an important contribution from the EW phase transition as well as from radiation (Figure 109c). The curve β(tk) spans from ∼ 10 −11 s to ∼ 10 −5 s, showing three noticeable
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CCM-Centro de Ciências Matemática
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Contents List of Figures vii List o
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PBHs and Cosmological Phase Transit
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List of Tables 1 The Universe timel
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