Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Revisiting Constraints on Primordial Curvature Power Spectrum from PBH Abundances

Ashu Kushwaha, Teruaki Suyama

Abstract

Primordial black holes (PBHs) can form in the early Universe, for instance during radiation domination, from the collapse of large-amplitude density perturbations shortly after horizon re-entry. This mechanism establishes an approximate one-to-one correspondence between the PBH mass and the scale of the peak in the primordial curvature perturbations. Consequently, the constraints on PBH abundances can be translated into upper limits on the amplitude of the primordial curvature power spectrum, thereby providing an indirect probe of the last e-folds of inflation corresponding to these smaller scales. We derive constraints on the amplitude of primordial curvature power spectra with both narrow and broad peaks using the most up-to-date bounds on PBH abundances. Given the theoretical uncertainties in PBH formation, we systematically compare the constraints obtained using the Press-Schechter (PS) formalism and peak theory, accounting for the nonlinear relation between curvature perturbations and density contrast. We quantify the impact of spherical versus non-spherical collapse criteria and show that including non-sphericity significantly increases the inferred amplitude of the primordial power spectrum, reflecting the larger threshold density contrast required for PBH formation. We also find that whereas the constraints obtained using the PS formalism and peak theory remain largely similar for the monochromatic case, they differ significantly toward smaller scales in the case of a broad primordial power spectrum. This discrepancy underscores that current constraints remain sensitive to the choice of statistical formalism. Our consistent treatment of monochromatic and extended mass functions provides a systematic mapping based on existing methodologies, while highlighting that reducing these theoretical uncertainties is a crucial step toward probing the early Universe through PBHs.

Revisiting Constraints on Primordial Curvature Power Spectrum from PBH Abundances

Abstract

Primordial black holes (PBHs) can form in the early Universe, for instance during radiation domination, from the collapse of large-amplitude density perturbations shortly after horizon re-entry. This mechanism establishes an approximate one-to-one correspondence between the PBH mass and the scale of the peak in the primordial curvature perturbations. Consequently, the constraints on PBH abundances can be translated into upper limits on the amplitude of the primordial curvature power spectrum, thereby providing an indirect probe of the last e-folds of inflation corresponding to these smaller scales. We derive constraints on the amplitude of primordial curvature power spectra with both narrow and broad peaks using the most up-to-date bounds on PBH abundances. Given the theoretical uncertainties in PBH formation, we systematically compare the constraints obtained using the Press-Schechter (PS) formalism and peak theory, accounting for the nonlinear relation between curvature perturbations and density contrast. We quantify the impact of spherical versus non-spherical collapse criteria and show that including non-sphericity significantly increases the inferred amplitude of the primordial power spectrum, reflecting the larger threshold density contrast required for PBH formation. We also find that whereas the constraints obtained using the PS formalism and peak theory remain largely similar for the monochromatic case, they differ significantly toward smaller scales in the case of a broad primordial power spectrum. This discrepancy underscores that current constraints remain sensitive to the choice of statistical formalism. Our consistent treatment of monochromatic and extended mass functions provides a systematic mapping based on existing methodologies, while highlighting that reducing these theoretical uncertainties is a crucial step toward probing the early Universe through PBHs.
Paper Structure (15 sections, 40 equations, 5 figures)

This paper contains 15 sections, 40 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The compaction function and criterion for the collapse of an overdense region: A brief review
  3. Equivalence between compaction function and averaged (smoothed) density contrast
  4. Effect of non-sphericity of the overdense region
  5. PBH mass function: A brief review
  6. Definition of the PBH mass function
  7. PBH mass function and the mass fraction parameter $\beta$
  8. Mass fraction parameter in PS formalism
  9. Mass fraction parameter in the peak theory
  10. PBH mass in the critical phenomena
  11. Constraints on $\mathcal{P}_\zeta (k)$ from PBHs abundance
  12. Constraints on narrow peaked power spectrum from monochromatic PBH mass function
  13. Constraints on broad peaked power spectrum from extended PBHs mass function
  14. Conclusion
  15. Calculations on relating volume average density contrast with smoothed density contrast

Figures (5)

  • Figure 1: Showing the effect of the mass-ratio $M_{\rm PBH}/M_H$ on the amplitude $\mathcal{A}_p$ and the logmormal power spectrum $\mathcal{P}_{\zeta} (k_p)$ for $\Delta=0.1$ for spherical overdensity case. Solid curves are for constant value $\alpha=0.2$ and dashed curves show the ratio obtained from the near-critical behaviour \ref{['mpbh-mh-scaling']}. The upper panel is for the PS formalism, and the lower panel is for the peak theory. In generating these plots, we choose $\alpha=0.2$ in relation between $\beta$ and $\beta'$ in Eq.\ref{['app-beta-prime-def']} in this work, and near-critical behaviour is assumed only inside the integration in the expression of $\beta$ in Eq.\ref{['beta-ps-final']} and Eq.\ref{['beta-peak-theory-final']}.
  • Figure 2: Showing comparison between Press-Schechter and peak theory on the constraints on $\mathcal{A}_p$ and $\mathcal{P}_{\zeta} (k_p)$ for both spherical and ellipsoidal overdensity case, we use $\Delta=0.1$ and $\alpha=0.2$.
  • Figure 3: Comparison of our constraints (for ellipsoidal case) from Fig. \ref{['fig:Ap-Pzeta-mc-PS-pt']} with other observational limits. The red and black curves show the constraints for peak theory and PS formalism, respectively. The solid Green line shows the region constrained by LVK $O1-O4a$ run LIGOScientific:2025kry. The dotted magenta and blue curves show the projected sensitivities of the LIGO in its final phase (LIGO design) and the Einstein Telescope (ET), curves are taken from Romero-Rodriguez:2021aws.
  • Figure 4: Showing comparison between Press-Schechter and peak theory on the constraints on $\mathcal{A}_p$ and $\mathcal{P}_{\zeta} (k_p)$ for both spherical and ellipsoidal overdensity case, for broad peak case where we use $\Delta=1$ and $\alpha=0.2$. The constraints are obtained from the PBHs abundance against DM, $f_{PBH}$. The oscillatory behaviour for large $k_p$ are the numerical artifact of rapidly oscillating $j_1^2 (k r_m)$ term in $\sigma_L^2$ and $\mu^2_L$ in Eqns.\ref{['sigma-l']} and \ref{['mu-l']}. The oscillating features are insignificant for the peak theory due to the effective cancellation between both terms in Eq.\ref{['beta-peak-theory-final']}, while they become more prominant for PS formalism due to $\sigma_L^2$ dependence in Eq.\ref{['beta-ps-final']}.
  • Figure 5: Comparison of our constraints (for ellipsoidal case) from Fig. \ref{['appfig:Ap-Pzeta-mc-PS-pt']} with other observational limits. The red and black curves show the constraints for peak theory and PS formalism, respectively. The solid Green line shows the region constrained by LVK $O1-O4a$ run LIGOScientific:2025kry. The dotted curves show the projected sensitivities of the LIGO in its final phase (LIGO design) and the ET, SKA, BBO, DECIGO, LISA, which are taken from Romero-Rodriguez:2021awsInomata:2018epa.