Primordial Black Holes and Cosmological Phase Transitions Report ...

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PBHs and Cosmological Phase Transitions 176 In Sections 11.1 to 11.5 we made use of a few abreviations concerning different contributions to β(tk). We list these abreviations in Table 41. For example, RBE, represents a case for which there are non–negligible contributions from radiation (R), from the QCD Bag Model (B), and from the EW Bag Model (E) but with negligible contributions from the electron–positron annihilation as well as from the QCD Lattice Fit and from the QCD Crossover (if one chooses one of these models instead of the Bag Model). We might have also situations with one or more contributions exceeding the observational limits (these are labeled with an ∗). For exeample, RB ∗ LCE ∗ , represents a case for which we have, besides the contribution from radiation R, contributions from the QCD Lattice Fit (L) or from the QCD Crossover (C). The QCD Bag Model is excluded due to observational constraints (B ∗ ). The same happens for the EW Bag Model (E ∗ ). The contribution from the electron– positron annihilation epoch is negligible in this case. 11.1 Radiation–dominated universe In this section we determine the fraction of the universe going into PBHs at different epochs for a radiation–dominated universe with a running–tilt power spectrum (Section 10). We consider, for t+, all orders of magnitude between 10 −23 s (end of inflation) and 10 8 s. We are interested in a blue spectrum, i.e, a spectrum for which n>1. In fact, if we want to have interesting values for β, then we should have, at least, n 1.22. As an upper limit we consider n ≈ 2.0. This corresponds to a cut–off at tk ∼ 10 5 s (see Figure 93) which excludes PBHs with masses larger than ∼ 10 10 M⊙ (which is equivalent to the mass of the present day largest Supermassive Black Hole (SMBH) candidates; e.g. Natarajan & Treister, 2008). Each pair of the form (t+, nmax) determines a different location and a different value for the maximum value of β(tk). In general one of three things might occur: (1) β(tk) exceeds the observational constraints ⇒ the pair (t+, nmax) must be rejected. (2) β(tk) is negligible (< 10 −100 ) for all values of tk, in which case we take β(tk) = 0. (3) β(tk) is always bellow the observational constraints and, at least during some epoch, above 10 −100 ; these are the cases of interest to us. Let us consider, as an example, the case n =1.30. If t+ = 10 −17 s, then β(tk) exceeds the observational constraints as it is clear from Figure 94. On the other hand, if t+ = 10 −16 s we obtain a valid curve for β(tk) (see Figure 95) with βmax ∼ 10 −17 . As one moves t+ to later epochs the value of βmax becomes smaller (see Figure 95) until, for t+ = 10 −10 s, we reach βmax ∼ 10 −133 (see Figure 96). Thus, we consider that, in the case of n =1.30, there is a window
PBHs and Cosmological Phase Transitions 177 sigla Meaning Table 41: Different contributions to β. B QCD Bag Model BE QCD Bag Model and EW Bag Model BL QCD Bag Model or QCD Lattice Fit BLE (QCD Bag Model or QCD Lattice Fit) and EW Bag Model E EW Bag Model ea e − e + annihilation L QCD Lattice Fit R Radiation RB Radiation and QCD Bag Model RBE Radiation and QCD Bag Model and EW Bag Model RBea Radiation and QCD Bag Model and e − e + annihilation RBL Radiation and (QCD Bag Model or QCD Lattice Fit) RBLC Radiation and (QCD Bag Model or QCD Lattice Fit or QCD Crossover) RBLCE Radiation and (QCD Bag Model or QCD Lattice Fit or QCD Crossover) and EW Bag Model RBLCea Radiation and (QCD Bag Model or QCD Lattice Fit or QCD Crossover) and e − e + annihilation RBLE Radiation and (QCD Bag Model or QCD Lattice Fit) and EW Bag Model RC Radiation and QCD Crossover RCea Radiation and QCD Crossover and e − e + annihilation RE Radiation and EW Bag Model Rea Radiation and e − e + annihilation RL Radiation and QCD Lattice Fit RLea Radiation and QCD Lattice Fit and e − e + annihilation
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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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