🔭 astrophysics

Press-Schechter Formalism and The PBH Mass Distributions

This paper derives the primordial black hole mass distribution by applying the Press-Schechter formalism with an excursion-set first-crossing construction to Gaussian perturbations, utilizing a sharp- $k$ filter to solve the diffusion equation for the formation fraction and subsequently mapping these results to the present-day dark matter fraction through horizon-entry scaling and radiation-era redshifting.

Original authors: Owais Farooq, Romana Zahoor, Balungi Francis

Published 2026-01-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Owais Farooq, Romana Zahoor, Balungi Francis

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

Imagine the universe right after the Big Bang as a giant, bubbling pot of cosmic soup. In this soup, tiny ripples and bumps (called "perturbations") were constantly forming. Most of these ripples were too small to do anything, but occasionally, a rare, massive bump would grow large enough to collapse under its own gravity, turning into a Primordial Black Hole (PBH).

This paper is essentially a recipe book for predicting how many of these black holes formed, how heavy they are, and how they are distributed today. The authors, Owais Farooq, Romana Zahoor, and Balungi Francis, have created a clear, step-by-step mathematical guide to turn a picture of the early universe's "bumps" into a map of black holes we might see today.

Here is how they do it, broken down into simple concepts:

1. Smoothing the Rough Edges (The "Blur" Filter)

The universe wasn't perfectly smooth; it was a chaotic mess of density. To make sense of it, the authors use a mathematical tool called a "sharp-k filter."

The Analogy: Imagine looking at a high-resolution photo of a mountain range. If you squint your eyes or use a blurry lens, you stop seeing individual rocks and trees and start seeing the general shape of the mountains.
The Science: They "smooth out" the density of the universe at different scales. They ask: "If we look at a chunk of space this big, how much heavier is it than the average?" This gives them a number called the variance (how much the density fluctuates).

2. The Tipping Point (The "Dam" Analogy)

For a black hole to form, a chunk of that soup needs to be heavy enough to collapse.

The Analogy: Think of a dam holding back water. If the water level (density) stays below a certain line, the dam holds. But if a rare, massive wave pushes the water level over the top edge (the threshold, or $\delta_c$ ), the dam breaks, and the water rushes down to form a black hole.
The Science: They calculate the probability of these "rare waves" happening. Because the universe is mostly random (Gaussian statistics), huge waves are very rare, like winning the lottery. The authors use a specific mathematical formula (the "erfc tail") to count exactly how many of these lottery tickets exist.

3. The Random Walk (The "Drunkard's Stroll")

This is the most creative part of their method, called Excursion-Set Theory.

The Analogy: Imagine a drunk person walking on a tightrope. Every step they take is random. Sometimes they step left, sometimes right. We want to know: "What are the odds this person will eventually step off the edge (cross the threshold)?"
The Science: As they look at smaller and smaller chunks of the universe, the density value "walks" up and down like a random path. The authors treat this as a diffusion equation (like heat spreading out). They solve this to find the exact moment the "walker" hits the "edge" (the collapse threshold) for the first time. This gives them a precise map of how many black holes form at different sizes.

4. Connecting the Dots: From Then to Now

The paper doesn't just stop at the moment of creation. It tracks the black holes from the early universe to today.

The Analogy: Imagine a black hole is born as a baby. As the universe expands (like a balloon inflating), the black hole grows heavier relative to the surrounding radiation, but its size in the sky shrinks.
The Science: They use a rule of thumb: Mass is inversely proportional to the square of the scale. If a black hole formed from a very small ripple, it's light. If it formed from a huge ripple, it's heavy. They convert the "formation time" math into "today's" math, accounting for how the universe has cooled and expanded over billions of years.

5. The Final Result: A "Menu" of Black Holes

The end product of their work is a formula that tells us the fraction of dark matter that could be made of black holes of a specific weight.

The Analogy: Imagine a menu at a restaurant. Instead of listing "steak" or "fish," this menu lists "Black Holes weighing 100 tons," "Black Holes weighing 1,000 tons," etc. The paper tells you the probability of ordering each item.
The Science: They provide a way to take a theoretical model of the early universe (the "Primordial Curvature Spectrum") and instantly generate a prediction for the black hole population today. They also show how to check if these predictions match real-world observations (like microlensing or cosmic background radiation) by using a simple "integral test" (adding up the probabilities to see if they stay within safe limits).

Summary

In short, this paper builds a mathematical assembly line:

Input: A picture of the early universe's ripples.
Process: Smooth the ripples, check if they cross the "collapse line" using a random-walk model, and calculate the odds.
Output: A precise prediction of how many black holes of every size exist today and what percentage of the universe's dark matter they might make up.

The authors emphasize that while their current model assumes a "standard" universe (Gaussian randomness, constant collapse rules), their framework is flexible. It can easily be upgraded to include more complex physics (like non-standard randomness or changing collapse rules) without breaking the whole system. It is a foundational tool for anyone trying to understand the hidden population of primordial black holes.

Technical Summary: Press–Schechter Formalism and Primordial Black Hole Mass Distributions

Problem Statement
Primordial black holes (PBHs) serve as a critical link between small-scale primordial fluctuations and late-time probes of compact dark matter. A predictive framework for PBHs requires two essential components: a formation fraction (the probability of collapse) and a mass distribution. While Press–Schechter theory and excursion-set theory have been applied to PBHs in previous literature (e.g., Green et al. 2004; Young et al. 2014), this work aims to provide a self-contained, transparent derivation that maps a primordial curvature spectrum, $P_R(k)$ , directly to a present-day PBH mass distribution, $f_{\text{PBH}}(M)$ . The goal is to establish a constraint-ready representation for testing models against observational data.

Methodology
The paper employs a rigorous derivation based on the Press–Schechter formalism, augmented by the excursion-set first-crossing construction. The methodology proceeds through the following steps:

Smoothing and Variance Definition: The authors define the smoothed density contrast $\delta_R$ and its variance $S(R) = \sigma^2(R)$ . They establish the connection between the variance and the primordial curvature power spectrum $P_R(k)$ via the radiation-era transfer function, accounting for the horizon-entry epoch.
Gaussian Statistics and Thresholds: The derivation assumes Gaussian statistics for the linear density contrast at horizon entry and a constant collapse threshold $\delta_c$ . Under these assumptions, the formation fraction is treated as a rare-event tail of the distribution.
Excursion-Set Construction: Utilizing a sharp- $k$ filter ( $W(kR) = \Theta(1-kR)$ ), the density contrast $\delta(S)$ is modeled as a Markovian random walk in variance time $S$ . The problem of finding the collapse fraction is reduced to solving a diffusion equation with an absorbing barrier at $\delta_c$ .
First-Crossing Density: Using the method of images, the authors derive the first-crossing density $f(S)$ , which represents the probability density of walks crossing the threshold for the first time.
Mass Mapping: The framework maps the formation-era mass function to the present day. This involves:
- Establishing the scaling relation between PBH mass $M$ and the comoving wavenumber $k$ ( $M \propto k^{-2}$ ) during radiation domination.
- Calculating the Jacobian $dS/d \ln M$ for the sharp- $k$ filter.
- Applying redshifting relations to convert the formation fraction into the present-day dark matter fraction per logarithmic mass, $f_{\text{PBH}}(M)$ .
Constraint Integration: The paper outlines a "channel-wise integral criterion" to compare extended mass distributions against observational upper limits ( $f_{\text{max}}(M)$ ) derived from microlensing, CMB anisotropies, and spectral distortions.

Key Contributions and Results

Closed-Form Pipeline: The paper provides a complete analytical pipeline from $P_R(k)$ to $f_{\text{PBH}}(M)$ . Key equations include the first-crossing density $f(S) = \frac{\delta_c}{\sqrt{2\pi}} S^{-3/2} \exp(-\delta_c^2/2S)$ and the present-day fraction scaling $f_{\text{PBH}}(M) \propto M^{-1/2} (d\beta/d \ln M)$ .
Explicit Jacobian: For the sharp- $k$ filter, the authors derive an explicit expression for the differential variance $dS/d \ln M = -\frac{1}{2} P_\delta(k)$ , facilitating the conversion from variance space to mass space.
Constraint-Ready Formalism: The work formalizes the use of the integral criterion $\int (f_{\text{PBH}}(M)/f_{\text{max}}(M)) d \ln M \leq 1$ for extended mass functions, offering a practical method to test models with broad mass distributions against multiple observational channels.
Lognormal Proxy: A lognormal form is presented as a convenient analytic proxy for illustrative extended distributions, particularly those arising from localized features in the power spectrum.

Significance and Claims
The authors position this work as a "flexible baseline" for translating small-scale primordial physics into observationally testable PBH abundance constraints. The significance lies in the transparency and self-containment of the derivation, which explicitly connects the curvature spectrum to the final mass function without relying on opaque numerical steps.

The paper modestly claims that while the current framework assumes Gaussian statistics, a constant threshold, and a sharp- $k$ filter, the pipeline is designed to accommodate theoretically motivated upgrades. These include:

Scale-dependent or profile-dependent collapse criteria (moving barriers).
Non-Markovian statistics arising from non-sharp filters.
Non-Gaussian corrections to tail probabilities.
Critical-collapse scaling to generate extended mass functions from sharply featured power spectra.

Ultimately, the work serves as a foundational tool for interpreting PBH constraints, allowing researchers to input a primordial power spectrum and immediately derive the corresponding present-day dark matter fraction and its consistency with current observational limits.