Excursion-set for Primordial Black Holes I: white noise and moving barrier

This paper defends the excursion-set formalism for deriving primordial black hole mass distributions by demonstrating that noise is strictly white when sampling on a synchronous surface and that cloud-in-cloud effects remain crucial for broad power spectra, thereby establishing the necessity of solving the first-passage-time problem with a moving barrier.

The Gist

Imagine the early universe as a giant, bubbling pot of cosmic soup. Sometimes, this soup gets so thick and clumpy in certain spots that gravity pulls it together so tightly it collapses into a black hole. These are called Primordial Black Holes (PBHs).

Scientists have been trying to figure out exactly how many of these black holes exist and how heavy they are. To do this, they use a mathematical tool called the Excursion-Set Formalism. Think of this tool as a way to track a random walk through a foggy forest to see when (and if) a hiker hits a cliff edge.

This paper by Pierre Auclair, Baptiste Blachier, and Vincent Vennin fixes two major mistakes people were making while using this tool. Here is the story in simple terms:

1. The "Foggy Forest" Analogy (The Random Walk)

Imagine you are walking through a foggy forest, trying to predict when you will fall off a cliff.

• The Cliff: This represents the "formation threshold." If the density of the soup gets high enough (you get close enough to the cliff), a black hole forms.
• The Walk: As you walk, you are looking at the landscape through different-sized binoculars.
  • Wide-angle lens: You see the big picture (large scales).
  • Zoom lens: You see tiny details (small scales).
• The Problem: As you zoom in, the landscape changes. The paper argues that previous scientists were looking at the landscape at the wrong time (specifically, when the "fog" cleared at a specific moment called "Hubble crossing"). This made the path look like it was being pushed by a mysterious wind (colored noise) and made the math break, sometimes predicting "negative black holes" (which is impossible).

The Fix:
The authors say, "Stop looking at the landscape at that specific moment. Instead, take a snapshot of the whole forest at one single, fixed time (a synchronous surface)."

• The Result: Suddenly, the mysterious wind disappears. The path becomes a pure, clean random walk (white noise).
• The Catch: Because you are looking at a fixed time, the "cliff edge" itself starts to move as you zoom in. It's like a cliff that slowly shifts its position as you walk.
• The Solution: They developed a new, efficient computer method to solve the math of "walking toward a moving cliff." This ensures the predictions are accurate and never result in impossible negative numbers.

2. The "Russian Nesting Doll" Problem (Cloud-in-Cloud)

There is a second debate about whether small black holes matter if they are inside big ones.

• The Old View: Some scientists argued that small black holes are irrelevant. They thought, "If a giant black hole forms, it just swallows the small ones inside it, so we only need to count the giants." They believed this "swallowing" (called Cloud-in-Cloud) was so rare it didn't matter.
• The New View: The authors say, "That's only true if the black holes are far apart, like islands in an ocean. But if the universe is full of a continuous 'soup' of fluctuations (broad power spectra), the small and large black holes are mixed together like a dense forest."
• The Analogy: Imagine a forest where a giant tree grows. If the forest is sparse, the giant tree doesn't care about the saplings nearby. But if the forest is incredibly dense, the giant tree's roots and branches will inevitably crush and absorb the saplings growing right underneath it.
• The Consequence: When you ignore this "swallowing," you get the wrong answer. You might predict too many tiny black holes and not enough big ones. In fact, the old math sometimes predicts negative amounts of tiny black holes, which is a clear sign the math is broken. The new method correctly accounts for the big ones eating the small ones.

Why This Matters

This paper is a "repair manual" for the tools cosmologists use to study the universe.

1. It fixes the math: By changing when and how we look at the data, they removed the errors that caused "negative black holes" and confusing noise.
2. It proves the "swallowing" matters: They showed that in realistic scenarios (where the universe has a wide variety of sizes), the big black holes definitely eat the small ones. Ignoring this leads to wrong predictions about how much of the universe is made of dark matter.

In a nutshell: The authors found a better way to map the early universe. They realized we were looking at the map at the wrong angle, which made the terrain look scary and broken. By taking a straight-on snapshot and accounting for the fact that "big things eat small things," they gave us a much more reliable way to count the primordial black holes that might be hiding in our universe today.

Technical Summary

Here is a detailed technical summary of the paper "Excursion-set for Primordial Black Holes I: white noise and moving barrier" by Auclair, Blachier, and Vennin.

1. Problem Statement

The paper addresses two critical issues regarding the application of the excursion-set formalism to the formation of Primordial Black Holes (PBHs):

1. The Nature of Noise and Pathologies: Recent criticisms (e.g., Ref. [29]) argued that the random walks describing PBH formation are subject to colored (correlated) noise when sampling along the Hubble-crossing surface. This was claimed to make the first-passage-time problem intractable and lead to unphysical results, such as negative mass fractions in certain regimes.
2. The Relevance of "Cloud-in-Cloud": Previous studies (e.g., Refs. [31, 32]) suggested that the "cloud-in-cloud" effect (where larger PBHs engulf smaller ones, rendering the smaller ones unobservable) is negligible for PBHs because they form at widely separated scales. Consequently, these studies argued that the standard Press-Schechter (PS) approximation, which neglects barrier crossings, is sufficient.

The authors aim to demonstrate that these criticisms stem from incorrect sampling choices and that the excursion-set formalism is not only robust but necessary for realistic PBH scenarios, particularly those involving broad power spectra.

2. Methodology

A. Sampling Hypersurfaces and Noise Characterization

The authors analyze the coarse-grained density contrast $\delta_R$ as a function of the smoothing scale $R$. They compare two sampling strategies:

• Hubble-Crossing Surface ($R = H^{-1}$): This is the traditional choice where the collapse threshold $\delta_c$ is constant. The authors show that this choice introduces a drift term in the Langevin equation because the density contrast evolves with time on super-Hubble scales. They argue that previous claims of "colored noise" arise from failing to separate this drift from the stochastic noise term. Furthermore, on this surface, the relationship between the scale $R$ and the variance $S$ (the "time" variable in the random walk) is not necessarily monotonic, leading to the reported negative mass fractions.
• Synchronous Hypersurface ($t = t_0$): The authors propose sampling at a fixed cosmic time $t_0$.
  • Result: On this surface, the drift term vanishes entirely. The noise remains strictly white (uncorrelated) provided a sharp top-hat window function is used in Fourier space.
  • Trade-off: The PBH formation threshold $\delta_c$ becomes dependent on the scale $R$ (or the variance $S$), creating a moving barrier problem.

B. Solving the First-Passage-Time Problem

To handle the moving barrier $\delta_c(S)$, the authors employ a numerical framework based on Volterra integral equations of the second kind.

• They derive an implicit relation for the first-passage-time probability distribution $P_{\text{FPT}}(S)$ using a transition probability kernel.
• By choosing a specific kernel function $K(S) = \delta'_c(S)$, they remove singularities near the upper bound of the integration, ensuring numerical stability.
• The problem is discretized into a linear matrix equation: $P_{\text{FPT}} = (I - J\Delta s)^{-1} P_{\text{PS}}$, where $P_{\text{PS}}$ represents the standard Press-Schechter flux and $J$ is a lower triangular matrix accounting for the history of barrier crossings.
• This method is shown to be significantly more efficient and reliable than standard Monte Carlo simulations, especially for rare events.

C. Benchmark Models

The framework is applied to three types of primordial power spectra ($P_\zeta$):

1. Top-hat spectrum: An idealized broad peak.
2. Log-normal spectrum: A smoother peak with a tunable width ($\Delta$).
3. Double log-normal spectrum: Two distinct peaks to test scale separation and mass hierarchies.

3. Key Contributions and Results

1. Resolution of Noise and Pathologies

• White Noise Confirmation: The authors prove that if Fourier modes are uncorrelated, the noise is strictly white when the drift is correctly separated. The "colored noise" issue is an artifact of the Hubble-crossing sampling method.
• Elimination of Negative Mass Fractions: By using a synchronous hypersurface, the mapping between the scale $R$ and the variance $S$ becomes strictly monotonic. This prevents the mathematical breakdown that leads to negative mass fractions in the Press-Schechter approach.

2. Necessity of the Excursion-Set Formalism

• Failure of Press-Schechter: The authors demonstrate that the Press-Schechter approximation (which assumes no barrier crossings) fails significantly when the barrier is moving (scale-dependent).
  • It can predict negative mass functions even for simple power spectra.
  • It drastically overestimates the abundance of low-mass PBHs and underestimates high-mass PBHs compared to the full excursion-set calculation.
• Moving Barrier Importance: Neglecting the time-dependence of the barrier (i.e., assuming $\delta'_c = 0$) leads to quantitatively incorrect results, even for narrow power spectra.

3. Re-evaluation of "Cloud-in-Cloud"

• Broad Spectra: The "cloud-in-cloud" effect is not negligible for broad power spectra. Large PBHs effectively "swallow" smaller ones, suppressing the low-mass tail of the mass function.
• Scale Separation: While cloud-in-cloud is suppressed for widely separated scales (discrete peaks), it becomes dominant in continuous, broad spectra.
• Non-Monotonic Behavior: In double-peak scenarios, increasing the power of a large-scale peak initially increases the abundance of small PBHs (due to the spreading of initial conditions) but eventually decreases it as the large-scale barrier is crossed first (cloud-in-cloud). This nuanced behavior is captured only by the full excursion-set formalism.

4. Significance

• Theoretical Robustness: The paper establishes that the excursion-set formalism is the correct and necessary tool for calculating PBH mass functions in realistic scenarios. It resolves previous mathematical pathologies by shifting the sampling surface to a synchronous frame.
• Observational Implications: The findings suggest that previous estimates of PBH abundance, particularly those relying on Press-Schechter or ignoring barrier motion, may be significantly inaccurate. This impacts constraints on PBHs as dark matter candidates and their role in gravitational wave events (LIGO/Virgo/KAGRA) and pulsar timing arrays (NANOGrav).
• Methodological Advancement: The provided numerical framework for solving first-passage-time problems with moving barriers is efficient and can be applied to other cosmological problems involving stochastic processes and scale-dependent thresholds.

Conclusion

Auclair, Blachier, and Vennin demonstrate that the perceived failures of the excursion-set formalism for PBHs were due to specific sampling choices (Hubble-crossing) rather than fundamental flaws. By adopting a synchronous sampling surface and solving the resulting moving-barrier first-passage-time problem numerically, they show that the formalism is robust, avoids unphysical negative mass fractions, and is essential for accurately modeling PBH formation, especially in scenarios with broad power spectra where cloud-in-cloud effects are significant.