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

This paper reformulates the Press-Schechter approach for primordial black holes within the excursion-set framework to demonstrate that, unlike halo formation, the non-Markovian nature of the process invalidates the standard "fudge factor" correction, necessitating a consistent inclusion of both stochastic motion components to derive a physically valid mass function.

The Gist

Imagine the early universe as a giant, bubbling pot of soup. In this soup, there are tiny, random swirls of density—some spots are a bit thicker (overdense), and some are a bit thinner (underdense).

Primordial Black Holes (PBHs) are like heavy rocks that form when a swirl in the soup gets so thick that it collapses under its own weight. Scientists want to know: How many of these rocks form, and how heavy are they?

To answer this, they use a mathematical recipe called the Press-Schechter (PS) formalism. Think of this recipe as a way to count how many "thick swirls" exist in the soup.

The Old Problem: The "Cloud-in-Cloud" Confusion

For a long time, when scientists used this recipe to count Dark Matter Halos (the invisible scaffolding that holds galaxies together), they hit a snag called the "Cloud-in-Cloud" problem.

The Analogy:
Imagine you are counting the number of houses in a city.

1. You look at a small patch of land and see a tiny shed. You count it as a house.
2. Then you zoom out and see that the shed is actually inside a large mansion.
3. If you count the shed and the mansion, you are double-counting. You've counted the same house twice!

In the universe, small dense regions often get swallowed up by larger dense regions. The old recipe kept counting the small ones and the big ones, leading to a massive overcount.

The "Fudge Factor 2":
To fix this, the original scientists (Press and Schechter) realized they were only counting half the actual mass. So, they just multiplied their final answer by 2. They called it a "fudge factor" because it felt like a cheat code to make the math work.

Later, a more rigorous method called the Excursion Set Theory (think of it as a "Random Walk" game) proved that this "multiply by 2" rule was actually correct for Dark Matter Halos. It showed that for every path that crosses the "collapse line" once, there is an equally likely path that crosses it, gets pulled back, and crosses it again. The math balanced out perfectly, justifying the factor of 2.

The New Twist: Why PBHs Are Different

The authors of this paper asked: "Does this 'multiply by 2' rule work for Primordial Black Holes?"

Here is where the story changes.

The Difference:

• Dark Matter Halos form slowly over time. Their density changes in a "memoryless" way (like flipping a coin: the next flip doesn't care about the last one). This is called a Markovian process.
• Primordial Black Holes form very quickly during the radiation-dominated era. Their density changes are "sticky" or correlated (like walking in a foggy forest where your next step depends heavily on where you just were). This is called a Non-Markovian process.

The Analogy:

• Halos (Markovian): Imagine a drunk person walking on a straight line. If they stumble left, the next step is random. They might cross a finish line, get pushed back, and cross again. The "multiply by 2" rule works here because the path is unpredictable but fair.
• PBHs (Non-Markovian): Imagine a drunk person walking on a magnetized floor. If they stumble left, the floor pulls them further left. Their path is "correlated." If they cross the finish line, the floor might drag them back hard, or keep them there. The "fair coin flip" logic breaks down.

The Big Discovery

The authors ran massive computer simulations (like running the "drunk walk" game millions of times) to see what happens with PBHs.

1. The Old Way (Using the Fudge Factor 2): When they tried to use the "multiply by 2" rule for PBHs, the math broke. In some mass ranges, it predicted a negative number of black holes.

   • Think of it like this: If you try to bake a cake using a recipe that says "add 2 cups of flour," but you actually need 1 cup, you end up with a mess. If you try to count the "negative cake," it makes no sense.

2. The New Way (No Fudge Factor): They realized that for PBHs, the two parts of the calculation (the "first crossing" and the "re-crossing") are not equal.

   • In the Dark Matter case, Part A = Part B. So, Total = 2 × Part A.
   • In the PBH case, Part A $\neq$ Part B. Sometimes Part B is bigger, sometimes smaller. You cannot just guess and multiply by 2. You have to calculate both parts separately and add them up.

Why This Matters

This paper is a crucial "correction" for the field of cosmology.

• Before: Scientists were guessing whether to multiply by 2 or not when calculating how many black holes formed in the early universe. This led to conflicting results and confusion.
• Now: The authors have shown that the "multiply by 2" rule is wrong for Primordial Black Holes. Using it can lead to impossible results (like negative numbers).

The Takeaway:
To accurately predict how many Primordial Black Holes exist (which could explain Dark Matter or the gravitational waves LIGO detects), we must stop using the old "fudge factor." Instead, we must use a more complex, rigorous method that accounts for the "sticky" nature of the early universe's density fluctuations.

In short: The universe is more complex than a simple coin flip, and for black holes, we need a more sophisticated map to count them.

Technical Summary

Here is a detailed technical summary of the paper "Excursion Set Approach to Primordial Black Holes: Cloud-in-Cloud and Mass Function Revisited" by Ashu Kushwaha and Teruaki Suyama.

1. Problem Statement

The abundance and mass function of Primordial Black Holes (PBHs) are critical for understanding early-universe physics and dark matter candidates. These are frequently estimated using the Press–Schechter (PS) formalism. However, the original PS formalism suffers from the "cloud-in-cloud" problem, where small overdense regions embedded in larger overdense regions are miscounted, leading to an underestimation of the collapsed mass fraction by a factor of two.

In the context of dark matter halo formation, this is resolved by the Excursion Set Theory (Bond et al.), which introduces a multiplicative "fudge factor of 2". This factor arises because the stochastic evolution of the density contrast is a Markovian process (uncorrelated steps) when using a sharp-$k$ filter, making the probability of a trajectory crossing a threshold for the first time equal to the probability of it having crossed previously.

The Core Issue:
There is significant ambiguity in the literature regarding whether this "fudge factor of 2" applies to PBH formation. Some studies apply it, while others do not. Furthermore, naive applications of the PS formalism to PBHs often result in a negative mass function in certain mass ranges, which is physically impossible. The authors aim to rigorously determine if the factor of 2 is valid for PBHs and resolve the issue of negative mass functions.

2. Methodology

The authors reformulate the PS approach for PBHs (formed during the radiation-dominated epoch) within the Excursion Set framework.

• Stochastic Modeling: They model the smoothed density contrast, $\delta(x; R_M)$, at a fixed spatial point as a stochastic random walk (Brownian motion) as the smoothing scale $R_M$ (or effective time $\tau$) varies.
• Collapse Condition: Collapse is identified when the trajectory first crosses a critical threshold $\delta_c$.
• Key Distinction (Markovian vs. Non-Markovian):
  • Halo Formation: With a sharp-$k$ filter, the noise is white (uncorrelated), making the process Markovian. This allows the use of the "reflection principle," proving that the probability of crossing the barrier for the first time ($P_1$) equals the probability of having crossed it earlier ($P_2$), justifying the factor of 2 ($P_{total} = 2P_1$).
  • PBH Formation: The authors derive the two-point correlation function for the noise in the PBH case. They show that even with a sharp-$k$ filter, the noise is colored (correlated) because the variance of the smoothed density contrast depends on the Hubble horizon scale in a specific way during radiation domination. Consequently, the process is Non-Markovian.
• Numerical Simulation: To quantify the effects of non-Markovian noise, the authors:
  1. Derived the exact covariance matrix $C(\tau, \tau')$ for the PBH case.
  2. Generated colored noise using Cholesky decomposition of the covariance matrix.
  3. Solved the Langevin equation numerically for $10^4$ trajectories across various power spectrum shapes (power-law and sharply peaked).
  4. Counted the number of trajectories corresponding to $P_1$ (first crossing at $\tau_M$) and $P_2$ (crossing prior to $\tau_M$).

3. Key Contributions

1. Proof of Non-Markovianity: The paper establishes that the stochastic evolution of density contrasts for PBHs is inherently non-Markovian, even with a sharp-$k$ filter. This is due to the specific dependence of the variance on the smoothing scale in the radiation-dominated era.
2. Invalidation of the "Fudge Factor 2": Through numerical simulations, the authors demonstrate that for PBHs, $P_1 \neq P_2$. Therefore, the total collapse probability is $P(>M) = P_1 + P_2 \neq 2P_1$. The "fudge factor of 2" is mathematically unjustified for PBHs.
3. Resolution of Negative Mass Functions: The authors identify that calculating the mass function using only the first term ($dP_1/dM$) often yields negative values in certain mass ranges. They show that the inclusion of the second term ($dP_2/dM$), which is often neglected, is essential to ensure the total mass function ($dP/dM$) remains positive-definite.
4. Robust Theoretical Framework: They provide a consistent prescription for calculating PBH abundance that avoids the ad-hoc factor of 2 and ensures physical consistency (positive mass function).

4. Results

• Simulation Statistics: Numerical simulations confirm that while $P_1 \approx P_2$ for halo formation (Markovian), $P_1$ and $P_2$ diverge significantly for PBHs (Non-Markovian).
• Mass Function Behavior:
  • The derivative of the total probability, $dP/dM$, remains positive across the mass spectrum.
  • The derivative of the first term alone, $dP_1/dM$, exhibits negative regions (oscillations), particularly for narrow power spectrum peaks.
  • The contribution from $P_2$ (trajectories that crossed the threshold earlier but are still part of the larger structure) is crucial to cancel out these negative regions.
• Dependence on Power Spectrum: The discrepancy between $P_1$ and $P_2$ is most pronounced in regimes where the variance $\sigma^2(\tau)$ decreases rapidly (e.g., $\tau \gg \tau_*$ for a peaked spectrum), a feature unique to the PBH formation dynamics.

5. Significance

This work resolves a long-standing ambiguity in PBH cosmology:

• Theoretical Clarity: It definitively states that the "fudge factor of 2" used in halo formation cannot be applied to PBHs. Using it leads to incorrect abundance estimates.
• Physical Consistency: It provides the necessary correction to prevent unphysical negative mass functions, which have plagued previous literature.
• Methodological Advancement: It establishes that accurate PBH mass function calculations require solving the full non-Markovian stochastic problem (including the $P_2$ term) rather than relying on simplified analytical approximations derived from halo physics.

In conclusion, the paper argues that the calculation of PBH abundance must be based on the sum of both stochastic components ($P_1 + P_2$) derived from the non-Markovian excursion set theory, rather than the simplified $2P_1$ approximation.