⚛️ phenomenology

Clustering of Primordial Black Holes in Excursion Set Theory

This paper extends Excursion Set Theory to model Primordial Black Hole clustering, revealing that a blue-tilted power spectrum enhances both PBH formation and clustering probability, while larger clustering distances and higher critical density thresholds suppress clustering abundance.

Original authors: Hamed Kameli, Encieh Erfani

Published 2026-10-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hamed Kameli, Encieh Erfani

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the earliest moments of the universe, fractions of a second after the Big Bang, space was not smooth and uniform. Instead, it was a churning sea of tiny fluctuations, where some regions were slightly denser than others. If a patch of space was dense enough, gravity could overcome the pressure of the expanding universe and cause that region to collapse instantly, forming a black hole. These hypothetical objects, born from the raw density of the infant cosmos rather than the death of a star, are called primordial black holes. While we have never directly observed one, they remain a compelling candidate for dark matter, the invisible substance that holds galaxies together. The key to their existence lies in the "power spectrum," a map that describes how much energy exists at different sizes of these fluctuations. If this map is tilted to favor smaller, denser clumps, the universe could have produced a vast number of these ancient black holes.

A team of researchers has now taken a significant step in understanding how these objects might group together. Using a mathematical framework known as excursion set theory, which treats the formation of cosmic structures as a random walk, they investigated the likelihood of two primordial black holes forming close to one another. Their work reveals a direct link between the shape of the early universe's density map and the specific sizes of black holes that would form in clusters. They found that if the early universe had a specific type of density enhancement, it would not only create more black holes but would also force them to form in pairs or groups within a specific mass range. This clustering is not random; it is a predictable outcome of the physics governing the infant cosmos, suggesting that if these black holes exist, they likely arrived in the universe already holding hands.

The researchers approached this problem by imagining the history of the universe as a series of random paths. In their model, every point in space traces a path as the universe expands and cools. A black hole forms when a path crosses a specific threshold of density. To study clustering, the team looked at two such paths that start together, sharing a common history before they split apart. They calculated the probability that both paths would eventually cross the threshold to form black holes, but only after they had separated by a certain distance. This method allowed them to determine how likely it is to find two black holes of specific masses sitting near each other, rather than just calculating how many black holes exist in total.

Their simulations showed that the shape of the density map, described by something called the spectral index, acts as a dial that controls both the size and the grouping of the black holes. When the researchers adjusted this dial to make the density fluctuations stronger at smaller scales, they found that black holes began to form in distinct mass ranges. For instance, a specific setting produced black holes with masses between one and one hundred times that of our sun, while other settings created objects as light as the moon or as heavy as Jupiter. Crucially, the study demonstrated that these black holes did not just appear in these sizes; they appeared in clusters. The higher the value of this spectral index, the more likely it was to find pairs of black holes forming together within a specific distance.

The team also explored how the distance between these forming black holes affected their chances of appearing as a pair. They found that as the required distance between the two objects increased, the probability of them forming a cluster dropped off rapidly. However, this probability did not vanish completely; it settled into a steady, low value even at large distances. This suggests that while black holes are most likely to be found close to their neighbors, the tendency to cluster is a fundamental property of their formation, persisting even when they are separated by significant space. The researchers also examined the role of the density threshold, the specific point at which a region collapses into a black hole. They discovered that if this threshold is lower, meaning it is easier for a region to collapse, the number of clustered pairs increases significantly.

A major part of this work involved correcting a misunderstanding in previous studies about how to calculate these probabilities. Some earlier approaches tried to apply a method used for ordinary galaxies, which involves tracking how structures grow over time with a changing threshold. The authors argued that this method does not work for primordial black holes because their formation happens at a specific moment when a region re-enters the horizon, making time and mass inseparable in their calculations. By using a single, fixed threshold and focusing on the shared history of the random paths, they avoided a mathematical error that had previously led to an underestimation of how often these black holes might cluster.

The results of this study provide a clear, testable prediction for future observations. If primordial black holes make up a portion of the dark matter, they should not be scattered randomly throughout the cosmos. Instead, they should be found in groups, with the specific sizes of those groups depending on the conditions of the early universe. These clustered black holes could leave distinct signatures in the way light bends around them, a phenomenon known as gravitational lensing, or they could be detected through the gravitational waves produced when they eventually merge. By linking the mass of the black holes directly to their tendency to cluster, this research offers a new way to look for evidence of these ancient objects, turning the search for dark matter into a search for specific patterns in the sky.

Technical Summary: Clustering of Primordial Black Holes in Excursion Set Theory

Problem Statement
Primordial Black Holes (PBHs) are hypothetical candidates for dark matter formed from the gravitational collapse of large density fluctuations in the early Universe. While the formation of PBHs has been extensively studied using the Press–Schechter formalism, peaks theory, and Excursion Set Theory (EST), the spatial clustering of PBHs has received comparatively less attention. Existing literature often relies on standard EST treatments developed for dark matter halos, which may not be directly applicable to PBHs due to fundamental differences in their formation physics. Specifically, the "cloud-in-cloud" problem and the temporal evolution of PBH formation require a distinct theoretical approach. This paper addresses the need to quantify the joint probability of forming PBH pairs within a specific clustering distance, particularly under scenarios involving an enhanced, blue-tilted power spectrum.

Methodology
The authors extend the Excursion Set Theory (EST) framework to compute the joint probability of PBH pair formation. The methodology relies on the following theoretical constructs:

Stochastic Trajectories with Common History: The clustering of two PBHs is modeled by two Markovian random walk trajectories in the $(\delta, S)$ plane (density contrast vs. variance) that share a common history up to a "clustering point" $(S_{cl}, \delta_{cl})$ . After this point, the trajectories evolve independently until they first up-cross a critical density barrier $\delta_c$ , corresponding to the formation of individual PBHs with masses $M_1$ and $M_2$ .
Single-Barrier vs. Two-Barrier Approach: The paper argues against the use of a redshift-dependent (two-barrier) approach for PBHs, which is common in dark matter halo studies. In PBH formation, time and mass are inseparable because formation occurs at horizon crossing ( $k=aH$ ). The variance $S$ inherently contains the temporal evolution, while the barrier $\delta_c$ remains a fixed threshold. The authors contend that applying a time-dependent barrier alongside a time-dependent variance constitutes a "double counting" of temporal effects, leading to an artificial suppression of clustering probabilities. Consequently, they adopt a single, constant barrier approach ( $\delta_c = 0.47$ ).
Forward-Crossing Probability: To calculate the joint probability, the authors employ a forward-crossing approach. They compute the probability of a trajectory reaching the clustering point $(S_{cl}, \delta_{cl})$ without having crossed the barrier previously, multiplied by the conditional first up-crossing probabilities for the two independent trajectories evolving from the clustering point to their respective formation variances $S_1$ and $S_2$ .
Power Spectrum Enhancement: The analysis assumes a blue-tilted spectral index ( $n_s > 1$ ) to enhance the power spectrum at small scales, a necessary condition for significant PBH production.

Key Contributions

Formalism Extension: The paper extends EST to calculate the joint probability of PBH pairs sharing a common history, distinguishing this pairwise quantity from the standard two-point correlation function of the entire PBH population.
Theoretical Clarification: It provides a rigorous argument for using a single, constant barrier in PBH clustering calculations, rejecting the two-barrier approach used in previous studies as theoretically inconsistent for this specific context.
Analytical Solution: The authors derive an analytical expression for the joint probability $P_2(S_1, S_2; S_{cl})$ by integrating over permissible density contrasts, assuming a fixed barrier.

Results
The numerical results, based on the derived formalism, yield the following findings:

Spectral Index and Mass Correspondence: There is a one-to-one correspondence between the blue-tilted spectral index ( $n_s$ $n_{s}$ ) and the mass range in which PBHs form and cluster.
- $n_s \approx 1.68$ : Sub-lunar mass range ( $10^{-13} - 10^{-9} M_\odot$ ).
- $n_s \approx 1.92$ : Earth-Jupiter mass range ( $10^{-6} - 10^{-3} M_\odot$ ).
- $n_s \approx 2.05$ : MACHO mass range ( $10^{-3} - 10^{-1} M_\odot$ ).
- $n_s \approx 2.30$ : Intermediate mass range ( $1 - 10^2 M_\odot$ ).
  Increasing $n_s$ not only increases the abundance of PBHs but also shifts the peak of the clustering probability toward higher masses.
Clustering Distance Dependence: The clustering probability ( $P_2$ ) decreases asymptotically as the clustering distance ( $R_{cl}$ ) increases. For large distances, the probability approaches a constant value because the variance decreases exponentially at large radii, rendering contributions to the integral negligible.
Barrier Dependence: A lower critical density threshold ( $\delta_c$ ) leads to a higher clustering probability and shifts the peak of the distribution toward higher-mass PBH pairs. Conversely, a higher barrier suppresses clustering abundance.
Physical Constraints: The analysis highlights a physical constraint on the variance ratio $\omega = S_{cl}(2R)/S_{PBH}(R) = 1/2^{n_s-1}$ . For $n_s = 2.30$ , $\omega \approx 0.4$ . The authors note that previous studies showing correlation peaks in regions where $\omega \geq 0.4$ may be analyzing physically inadmissible regions.

Significance and Claims
The paper claims that an enhanced blue-tilted power spectrum is a critical factor that simultaneously drives the formation of PBHs in specific mass ranges and enhances their probability of clustering. The study demonstrates that the clustering probability is not merely a statistical artifact but is intrinsically linked to the spectral index and the mass of the forming PBHs.

The authors assert that their results support the viability of PBHs as dark matter candidates, particularly in scenarios where the power spectrum is enhanced at small scales. They suggest that these clustered PBHs could produce characteristic imprints detectable via gravitational lensing and that their eventual mergers could be observed through gravitational wave signals. The paper concludes by proposing that future work could extend this formalism to compute merger rates and explore non-Markovian frameworks, though it acknowledges the increased complexity of reconstructing shared histories in non-Markovian scenarios.

Technical Summary: Clustering of Primordial Black Holes in Excursion Set Theory

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