Effect of stochastic kicks on primordial black hole abundance and mass via the compaction function

TL;DR

This work investigates how stochastic kicks during ultra-slow-roll inflation reshape primordial black hole (PBH) formation across asteroid, solar, and supermassive seeds. By simulating $10^8$ patches with $4\times10^4$ momentum shells and constructing their full radial profiles without a window function, the authors connect stochastic dynamics to the compaction function $\mathcal{C}(r)$ and its average $\bar{\mathcal{C}}(r)$ to predict PBH abundance and mass distributions, including the effects of critical collapse and the radiation-era transfer function. They find that stochasticity induces highly spiky profiles that can boost PBH abundance by up to $\sim 36$ orders of magnitude and shift the mass function to higher masses with broad, multi-order ranges; the transfer function smooths some structure but does not erase the stochastic enhancement. The results imply that observational constraints on PBHs and the required amplitude of the inflationary power spectrum may be significantly altered, and they highlight the need to redo collapse simulations for spiky profiles and to test spherical symmetry with 3D simulations. Overall, the study bridges inflaton potentials to PBH observables, while revealing substantial uncertainty in thresholds and mass-function convergence that must be resolved for robust predictions.

Abstract

We study stochastic effects in viable ultra-slow-roll inflation models that produce primordial black holes. We consider asteroid, solar, and supermassive black hole seed masses. In each case, we simulate $10^8$ patches of the universe that may collapse into PBHs. In every patch, we follow $4\times10^4$ momentum shells to construct its spherically symmetric profile from first principles, without introducing a window function. We include the effects of critical collapse and the radiation era transfer function. The resulting compaction function profiles are very spiky due to stochastic kicks. This can enhance the PBH abundance by up to 36 orders of magnitude, depending on the mass range and collapse criterion. The PBH mass function shifts to higher masses and widens significantly. These changes may have a large effect on observational constraints of PBHs and make it possible to generate PBHs with a smaller amplitude of the power spectrum. However, convergence issues for the mass function remain. The results call for redoing collapse simulations to determine the collapse criterion for spiky profiles.

Figures (12)

• Figure 1: One realisation of the curvature perturbation $\zeta(r)$ and the corresponding compaction function $\mathcal{C}(r)$ and its average $\bar{\mathcal{C}}(r)$ in the asteroid case, together with the smooth fit and the full and Gaussian mean profiles. The right-hand figures show how small-scale variation is smoothed by the transfer function $T(k)$.
• Figure 2: Left: The probability distributions of $\mathcal{C}_{\text{max}}$ and $\bar{\mathcal{C}}_{\text{max}}$ in the asteroid case without the transfer function $T(k)$ (solid), with $T(k)$ (dashed), and from the smooth fit (without $T(k)$) (dotted), all based on the full stochastic calculation. Right: The probability distributions of $\mathcal{C}_{\text{max}}$ and $\bar{\mathcal{C}}_{\text{max}}$ in the asteroid case from the full stochastic calculation (solid), from the mean non-Gaussian profile (\ref{['r_zeta_prime']}) (dashed), and the mean Gaussian profile (dotted), all without $T(k)$.
• Figure 3: Probability distributions of $\mathcal{C}_{\text{max}}$, $\bar{\mathcal{C}}_{\text{max}}$, and $\zeta_{\text{max}}$ in the asteroid case.
• Figure 4: The mass distribution in the asteroid case with the collapse threshold $\mathcal{C}_\text{th}=0.4$ for five cases: without critical scaling, the reverse lognormal distribution fit (without critical scaling), with critical scaling (for $0.4 < \mathcal{C}_{\text{max}} < 0.41$), with the transfer function but no critical scaling, and with the transfer function and critical scaling. The vertical lines mark the monochromatic mass considered in Figueroa:2020jkfFigueroa:2021zah and the mass corresponding to $k_\text{peak}$, at $3.6\times10^{-14}M_\odot$ and $1.4\times10^{-12}M_\odot$, respectively.
• Figure 5: The probability distributions of $\mathcal{C}_{\text{max}}$ and $\bar{\mathcal{C}}_{\text{max}}$ in the solar case, similar to figure \ref{['fig:astP']}.