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Register a new userPrimordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation
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The formation and abundance of primordial black holes (PBHs) arising from the curvature perturbation $zeta$ is studied. The non-linear relation between $zeta$ and the density contrast $delta$ means that, even when $zeta$ 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 $mathcal{P}_zeta$ must be a factor of $sim2$ larger to produce the same number of PBHs as the linear model (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.
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In the context of transient constant-roll inflation near a local maximum, we derive the non-perturbative field redefinition that relates a Gaussian random field with the true non-Gaussian curvature perturbation. Our analysis shows the emergence of a new critical amplitude $zeta_*$, corresponding to perturbations that prevent the inflaton from overshooting the local maximum, thus becoming trapped in the false minimum of the potential. For potentials with a mild curvature at the local maximum (and thus small non-Gaussianity), we recover the known perturbative field redefinition. We apply these results to the formation of primordial black holes, and discuss the cases for which $zeta_*$ is smaller or of the same order than the critical value for collapse of spherically symmetric overdensities. In the latter case, we present a simple potential for which the power spectrum needs an amplitude 10 times smaller that in the Gaussian case for producing a sizeable amount of primordial black holes.
In the model where Primordial Black Holes (PBHs) form from large primordial curvature (C) perturbations, i.e., CPBHs, constraints on PBH abundance provide in principle constraints on the primordial curvature power spectrum. This connection however depends necessarily on the details of PBH formation mechanism. In this paper we provide, for the first time, constraints on the primordial curvature power spectrum from the latest limits on PBH abundance, taking into account all the steps from gravitational collapse in real space to PBH formation. In particular, we use results from numerical relativity simulations and peak theory to study the conditions for PBH formation for a range of perturbation shapes, including non-linearities, perturbation profile and a careful treatment of smoothing and filtering scales. We then obtain updated PBH formation conditions and translate that into primordial spectrum constraints for a wide range of shapes and abundances. These updated constraints cover a range of scales not probed by other cosmological observables. Our results show that the correct and accurate modelling of non-linearities, filtering and typical perturbation profile, is crucial for deriving meaningful cosmological implications.
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Primordial magnetic field (PMF) is one of the feasible candidates to explain observed large-scale magnetic fields, for example, intergalactic magnetic fields. We present a new mechanism that brings us information about PMFs on small scales based on the abundance of primordial black holes (PBHs). The anisotropic stress of the PMFs can act as a source of the super-horizon curvature perturbation in the early universe. If the amplitude of PMFs is sufficiently large, the resultant density perturbation also has a large amplitude, and thereby, the PBH abundance is enhanced. Since the anisotropic stress of the PMFs is consist of the square of the magnetic fields, the statistics of the density perturbation follows the non-Gaussian distribution. Assuming Gaussian distributions and delta-function type power spectrum for PMFs, based on a Monte-Carlo method, we obtain an approximate probability density function of the density perturbation, and it is an important piece to relate the amplitude of PMFs with the abundance of PBHs. Finally, we place the strongest constraint on the amplitude of PMFs as a few hundred nano-Gauss on $10^{2};{rm Mpc}^{-1} leq kleq 10^{18};{rm Mpc}^{-1}$ where the typical cosmological observations never reach.
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