Clustering of Primordial Black Holes in Excursion Set Theory

The research investigates how Primordial Black Holes (PBHs) cluster within Excursion Set Theory (EST) and demonstrates that enhancing the power spectrum leads to increased PBH formation and clustering in specific mass ranges.

How it works

1. The study extends the EST formalism to compute the joint probability of forming PBH pairs within a specified clustering distance, based on two stochastic trajectories with a shared history.

2. The methodology involves quantifying the probability that two stochastic trajectories share a common history at scales larger than the clustering distance.

3. This is calculated by evaluating the product of the distribution function at the clustering point, P(δcl; Scl), and two conditional first up-crossing distributions, fFU−cond for each PBH.

Key Theoretical Frameworks

(2) Primordial Black Holes Formation in Excursion Set Theory

The PBH abundance is given by the integral:

β = Z ∞ δc P(δ; R) dδ, where P(δ; R) is the probability distribution function of the linear density contrast, smoothed by a k–space window function, Wf2 (k, R). The variance S(R) is related to the curvature power spectrum PR(k), and for significant PBH formation, it is necessary to "enhance the power spectrum at the scale of PBHs formation by a blue-tilted spectral index ns(kPBH) > 1."

(2.2) Single Barrier Approach in Excursion Set Theory

In the context of PBHs, the variance S, which represents mass/radius, is intrinsically linked to the horizon scale at formation. This linkage renders time and mass/radius scales inseparable, allowing for a single barrier that applies uniformly across all scales and times in the RD era. The approach extends to consider the second crossing of trajectories that intersect the constant barrier at larger variances to account for mergers or accretion, though this is noted as being highly sensitive to the step size of the random walk of trajectories.

Results and Findings

1. The study finds a one-to-one correspondence between the blue-tilted spectral index and the mass ranges in which PBHs form and cluster.

2. An enhanced power spectrum not only increases the formation of PBHs in specific mass ranges but also enhances their clustering probability.

3. The clustering probability, P2, is found to decrease asymptotically with increasing clustering distance, approaching a certain value at larger distances.

4. A higher critical density threshold (barrier) leads to a suppression of clustering abundance.

Parameter Dependence

(Figure 3)

The joint probability function P2 shows that the maximum of P2 corresponds to equal-mass PBH pairs. As the spectral index ns increases, the location of the maximum shifts toward higher masses. Furthermore, the maximum value of P2 remains approximately constant for all masses, with a fixed barrier and clustering distance, because a higher ns increases variance such that its peak coincides with the peak of fFU.

(Figure 6)

The analysis demonstrates that P2 increases as δc decreases. Moreover, a lower value of δc results in the peak of P2 moving toward higher-mass PBH pairs. The ratio of variance at clustering distance to the variance for each PBH is given by ω = Scl(2R) / SPBH(R) = 1/2 ns−1. For ns = 2.30, this ratio is ω ≃ 0.4, and substantial portions of their contour plots correspond to physically inadmissible regions, as ω ≥ 0.4.

Conclusion

The key result is the direct correspondence between the spectral index and the mass range of the PBHs formation and clustering. The analysis emphasizes that enhancement of the power spectrum at small scales plays a crucial role in determining the population and clustering of PBHs, providing evidence supporting PBH as a dark matter candidate. These clustered PBHs may leave observable signatures in gravitational lensing data or be detected through gravitational wave signals. Future research could extend this to compute merger rates or explore non-Markovian frameworks.

The gist

An enhanced power spectrum not only increases the formation of PBHs in specific mass ranges but also enhances their clustering probability, establishing a one-to-one correspondence between the blue-tilted spectral index and the mass ranges in which PBHs form and cluster.

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[1] Y. B. Zel’dovich and I. D. Novikov, “The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model,” Sov. Astron., vol. 10, p. 602, 1967.

[2] S. Hawking, “Gravitationally collapsed objects of very low mass,” Mon. Not. Roy. Astron.

Here are specific improvements for AI systems based on the findings of this scientific paper, categorized by capability:

)1. Enhanced Cosmological Parameter Estimation & Dark Matter Modeling:

The paper establishes a direct mapping between the spectral index of primordial fluctuations and the mass range of PBH formation and clustering. This allows AI to move beyond simple parameter fitting for cosmological models (like Planck or CMB analysis).

• AI System Capability: Develop sophisticated Bayesian inference engines that directly map observed features in the matter power spectrum (or inferred power spectrum shape) to physical constraints on PBH abundance, mass distribution, and clustering properties.

• Specific Improvement: Implement a PBH Mass/Clustering Predictor module. Given observational constraints on the primordial spectral index, the AI can predict which mass ranges are most likely to host clustered PBHs and estimate their expected clustering signal (correlation function) before any direct detection is made.

)2. Gravitational Wave (GW) Signal Interpretation & Source Identification:

The paper links specific spectral indices to distinct mass ranges, which are then associated with gravitational wave signals from mergers.

• AI System Capability: Create a GW Signature Classifier trained on the theoretical clustering predictions derived in Section 4 and Figure 3/8.

• Specific Improvement: When analyzing LIGO/Virgo or future GW data, the AI can identify merger events whose inferred masses and spatial distribution are statistically consistent with PBHs formed under specific blue-tilted spectral indices (e.g., distinguishing a signal arising from the intermediate mass range vs. a MACHO range).

)3. Gravitational Lensing Data Extraction & Spatial Clustering Analysis:

The paper explicitly states that the clustering of PBHs produces characteristic imprints detectable via gravitational lensing, and provides analytic forms for the joint probability function, P2 (Eq. 3.5).

• AI System Capability: Build a specialized Lensing Clustering Analyzer.

• Specific Improvement: The AI can process weak gravitational lensing shear maps to search for non-Gaussian clustering signatures characteristic of PBH populations. It will use the derived probability function P2 to calculate the expected signal magnitude for various distance scales (Rcl) and mass pairs (M1, M2), allowing it to statistically distinguish a PBH clustering signal from standard LSS or halo effects.

)4. Constraint-Driven Model Selection and Hypothesis Testing:

The paper shows how changing the critical density threshold (barrier, δc) affects both formation abundance and clustering peak location.

• AI System Capability: Implement an Excursion Set Theory Optimizer.

• Specific Improvement: The AI can automate hypothesis testing by systematically varying the barrier parameter (δc) and spectral index (ns). It can then determine which combination of parameters yields the most physically plausible scenario that matches observational constraints from multiple probes (CMB, LSS, GWs), effectively tuning the underlying inflationary model to fit the data.

)5. Advanced Theoretical Framework Simulation and Prediction:

The core contribution is extending EST to calculate joint probabilities for pairs sharing a common history.

• AI System Capability: Develop a high-fidelity simulator for PBH pair evolution under EST, incorporating the two-trajectory common history logic (Section 3).

• Specific Improvement: The AI can simulate the hierarchical merging and accretion history of clustered PBHs. This allows researchers to predict not just the initial abundance, but how this population evolves over cosmic time—e.g., predicting if a specific clustered configuration will lead to the formation of supermassive black holes (as suggested in Section 5).

We investigate the clustering of Primordial Black Holes (PBHs) within the framework of Excursion Set Theory (EST). The EST formalism is extended to compute the joint probability of forming PBH pairs within a clustering distance, based on two stochastic trajectories with a shared history. Our results show that an enhanced power spectrum not only increases the formation of PBHs in specific mass ranges but also enhances their clustering probability. We find a one-to-one correspondence between the blue-tilted spectral index and the mass ranges in which PBHs form and cluster. Additionally, we demonstrate that the clustering probability decreases asymptotically with increasing clustering distance, while a higher critical density threshold (barrier) leads to a suppression of clustering abundance.