Abstract
The formation and abundance of primordial black holes (PBHs) arising from the curvature perturbation ζ is studied. The non-linear relation between ζ and the density contrast δ means that, even when ζ has an exactly Gaussian distribution, significant non-Gaussianities affecting PBH formation must be considered. Numerical simulations are used to investigate the critical value and the mass of PBHs which form, and peaks theory is used to calculate the mass fraction of the universe collapsing to form PBHs at the time of formation. A formalism to calculate the total present day PBH abundance and mass function is also derived. It is found that the abundance of PBHs is very sensitive to the non-linear effects, and that the power spectrum \U0001d4abζ must be a factor of \U0001d4aa (2) larger to produce the same number of PBHs as if using the linear relation between ζ and δ (where the exact value depends on the critical value for a region to collapse and form a PBH). This also means that the derived constraints on the small-scale power spectrum from constraints on the abundance of PBHs are weaker by the same factor.
Highlights
ArXiv ePrint: 1904.00984 c 2019 The Author(s)
It is found that the abundance of primordial black holes (PBHs) is very sensitive to the non-linear effects, and that the power spectrum Pζ must be a factor of O(2) larger to produce the same number of PBHs as if using the linear relation between ζ and δ
Whether or not a PBH will form depends on the amplitude of the density contrast δm, rather than the amplitude of the curvature perturbation ζ
Summary
We will first describe the general relation between the curvature perturbation ζ and the density contrast δρ/ρb before analysing a specific parametrization of ζ that allows us to vary the profile of δρ/ρb. This allows us, with the help of numerical simulations (see section 4), to span almost all the possible range of values of δc. Throughout this paper, we assume that perturbations large enough to form PBHs are spherically symmetric. This is justified because such peaks must be extremely rare [26], and the perturbation profile is defined using only a radial coordinate r. In the super-horizon regime described the difference between these two gauges is a higher-order correction which can be neglected (see [27] and the references therein)
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