Excursion Set Approach to Primordial Black Holes: Cloud-in-Cloud and Mass Function Revisited

TL;DR

This work reformulates PBH formation during the radiation era within the excursion-set framework, treating the smoothing-scale evolution of the density contrast as a non-Markovian stochastic process with correlated noise. It demonstrates that, unlike halos where the two crossing contributions are equal, PBH formation yields $P_1\neq P_2$, making the total collapse probability $P(>M)=P_1+P_2$ the correct quantity and ensuring a positive-definite mass function. The authors show that neglecting $P_2$ can produce unphysical negative mass-function segments and that the PBH mass function is highly sensitive to the shape of the primordial power spectrum (monochromatic vs extended distributions). The results thus provide a robust theoretical foundation for PBH abundance calculations and clarify when the conventional fudge factor 2 can or cannot be applied in PBH contexts.

Abstract

The abundance and mass function of primordial black holes (PBHs) are often estimated using the Press-Schechter (PS) formalism. In the case of halo formation, the PS formalism suffers from the miscounting of regions collapsing into halos, known as the cloud-in-cloud problem, which is usually corrected by introducing a multiplicative `fudge factor 2'. By analogy, this factor has sometimes been applied to PBH calculations, although its validity has remained unsettled. We reformulate the PS approach for PBHs (forming during radiation-dominated epoch) within the excursion-set framework, where the smoothed density contrast undergoes a stochastic random walk as the smoothing scale varies and collapse is identified with the first threshold crossing. While the halo case is described by a Markovian process, we show that the PBH case is non-Markovian, even when the sharp-k filter Window function is adopted. Decomposing the total collapse probability into two distinct components of the stochastic motion, we numerically confirm that their contributions are exactly equal in the case of halo formation, justifying the fudge factor. For PBHs, however, we demonstrate that this equality no longer holds, and consistent inclusion of both contributions is essential to ensure a positive-definite mass function. Our results clarify the origin of the ambiguity surrounding the fudge factor and establish a robust theoretical foundation for PBH abundance calculations.

Figures (11)

• Figure 1: Schematic picture showing the variation of the smoothed density contrast $\delta (\tau)$ for three cases. At a specific scale $\tau_M$, a region corresponding to the trajectory A is in a halo with mass larger than $M$. A region corresponding to the trajectory B is underdense but is still part of a halo with mass larger than $M$ because $\delta (\tau)$ exceeds $\delta_c$ for some values $\tau< \tau_M$.
• Figure 2: Showing the variation of smoothed density contrast $\delta (\tau )$ (Brownian motion) for the $20$ Langevin trajectories for power law power spectrum \ref{['power-law-spectrum']} for $n=1$ (left-plot) and $n=2$ (right-plot). We set $\mathcal{A}=1$ and the horizontal dashed line shows the typical value of critical density contrast $\delta_c=1.68$.
• Figure 3: Showing the variation of smoothed density contrast $\delta (\tau )$ (Brownian motion) for the $20$ Langevin trajectories for power law power spectrum \ref{['power-law-spectrum']} for $n=0$ (left-plot) and $n=-1$ (right-plot). We set $\mathcal{A}=1$ and the horizontal dashed line shows the typical value of critical density contrast $\delta_c=1.68$. The divergence of the power spectrum at $\tau \rightarrow0$ is regularized by introducing small $\epsilon=10^{-3}$ as $\tau \rightarrow (\tau +\epsilon)$ with $\epsilon \gg$ time-step size used in the simulations.
• Figure 4: Left panel: The black and blue curves represent the number of trajectories computed from $N=10^4$ realizations, corresponding to $P_1$ and $P_2$, respectively, assuming the power-law power spectrum \ref{['power-law-spectrum']} with $n=1$. The green dotted curve shows $N P_1 (\tau_M)$ computed from Eq. (\ref{['P:first-term']}) which is the expected number of trajectories corresponding to $P_1$. The agreement between the black curve and the green dotted curve, within statistical uncertainty, indicates that the simulations were performed correctly. Right panel: Same as the left panel but for $n=2$.
• Figure 5: Left panel: The black and blue curves represent the number of trajectories computed from $N=10^4$ realizations, corresponding to $P_1$ and $P_2$, respectively, assuming the power-law power spectrum \ref{['power-law-spectrum']} with $n=0$. The green dotted curve shows $N P_1 (\tau_M)$ computed from Eq. (\ref{['P:first-term']}) which is the expected number of trajectories corresponding to $P_1$. The agreement between the black curve and the green dotted curve, within statistical uncertainty, indicates that the simulations were performed correctly. Right panel: Same as the left panel but for $n=-1$ and $N=10^3$ realizations. The divergence of the power spectrum at $\tau \rightarrow0$ is regularized by introducing small $\epsilon=10^{-3}$.