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The effect of non-Gaussian curvature perturbations on the formation of primordial black holes

J. C. Hidalgo

TL;DR

The paper investigates how non-Gaussian curvature perturbations, encoded by the bispectrum through $f_{\rm NL}$, affect primordial black hole formation. It constructs a non-Gaussian PDF to linear order in the three-point function, incorporating a windowed variance $\Sigma_{\mathcal{R}}^2$ and a scale-dependent quantity $\mathcal{J}$, and then propagates these statistics into the PBH abundance using the Press-Schechter framework. The analysis shows that the high-amplitude tail of perturbations—and hence PBH production—can be enhanced or suppressed depending on the sign of $f_{\rm NL}$, but when constrained by current $f_{\rm NL}$ limits and perturbative validity, the resulting bounds on $\Sigma_{\mathcal{R}}$ remain close to the Gaussian case. Consequently, non-Gaussian perturbations do not significantly alter the standard PBH formation picture, though they provide a way to constrain small-scale perturbations beyond CMB/LSS scales.

Abstract

This paper explores the consequences of non-Gaussian cosmological perturbations for the formation of primordial black holes (PBHs). A non-Gaussian probability distribution function (PDF) of curvature perturbations is presented with an explicit contribution from the three-point correlation function to linear order. The consequences of this non-Gaussian PDF for the large perturbations that form PBHs are then studied. Using the observational limits for the non-Gaussian parameter $f_{NL}$, new bounds to the mean amplitude of curvature perturbations are derived in the range of scales relevant for PBH formation.

The effect of non-Gaussian curvature perturbations on the formation of primordial black holes

TL;DR

The paper investigates how non-Gaussian curvature perturbations, encoded by the bispectrum through , affect primordial black hole formation. It constructs a non-Gaussian PDF to linear order in the three-point function, incorporating a windowed variance and a scale-dependent quantity , and then propagates these statistics into the PBH abundance using the Press-Schechter framework. The analysis shows that the high-amplitude tail of perturbations—and hence PBH production—can be enhanced or suppressed depending on the sign of , but when constrained by current limits and perturbative validity, the resulting bounds on remain close to the Gaussian case. Consequently, non-Gaussian perturbations do not significantly alter the standard PBH formation picture, though they provide a way to constrain small-scale perturbations beyond CMB/LSS scales.

Abstract

This paper explores the consequences of non-Gaussian cosmological perturbations for the formation of primordial black holes (PBHs). A non-Gaussian probability distribution function (PDF) of curvature perturbations is presented with an explicit contribution from the three-point correlation function to linear order. The consequences of this non-Gaussian PDF for the large perturbations that form PBHs are then studied. Using the observational limits for the non-Gaussian parameter , new bounds to the mean amplitude of curvature perturbations are derived in the range of scales relevant for PBH formation.

Paper Structure

This paper contains 6 sections, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The non-Gaussian PDF
  3. Non-Gaussian modifications to the probability of PBH formation
  4. Constraints on non-Gaussian perturbations in the PBH range
  5. Final Remarks
  6. ACKNOWLEDGEMENTS

Figures (3)

  • Figure 1: The fractional departure from Gaussianity is plotted for two types of non-Gaussian distributions $P_{NG}$\ref{['eqn2.2']}. For the potential in Eq. \ref{['eqn3.4']}, $f_{\rm NL} > 0$. The potential of Eq. \ref{['eqn3.5']} gives $f_{\rm NL} < 0$ and its correspondent PDF is shown by a dashed line.
  • Figure 2: The constrains in Table I are plotted together with only the smallest value considered for every mass.
  • Figure 3: A subset of the constraints on $\Sigma_{\mathcal{R}}$ from overproduction of PBHs is plotted for a Gaussian (black line) and non-Gaussian correspondence between $\beta$ and $\Sigma_{\mathcal{R}}$, equations \ref{['eqn4.22']} and \ref{['eqn4.4']} respectively. The green dashed line assumes a constant $f_{\rm NL} = -54$ and the blue dotted line a value $f_{\rm NL} = - 1 / \Sigma_{\mathcal{R}}^2$.