Effect of stochastic kicks on primordial black hole abundance and mass via the compaction function

This study demonstrates that stochastic kicks in ultra-slow-roll inflation models create spiky compaction profiles that can enhance primordial black hole abundance by up to 36 orders of magnitude and significantly shift their mass function, though convergence issues necessitate further investigation into collapse criteria for such profiles.

The Gist

The Big Picture: Making Black Holes from "Noise"

Imagine the early universe as a giant, calm lake. Usually, we think of this lake as having gentle, predictable ripples (waves). In standard physics, if a ripple gets big enough, it might collapse into a black hole. But this paper asks a different question: What if the lake isn't just rippling, but is actually being constantly poked by invisible, random fingers?

These "pokes" are called stochastic kicks. They are tiny, random jolts caused by quantum mechanics. The authors of this paper discovered that these random jolts don't just make the waves bigger; they make the surface of the lake incredibly spiky and jagged.

The Main Characters

1. The Inflaton Field (The Lake): The energy field that drove the rapid expansion of the universe (inflation).
2. Stochastic Kicks (The Random Pokes): Tiny, random quantum fluctuations that hit the field as it moves.
3. Primordial Black Holes (PBHs): Tiny black holes formed in the very first second of the universe. They are candidates for Dark Matter (the invisible stuff holding galaxies together) or the seeds for the supermassive black holes found in the centers of galaxies today.
4. The Compaction Function (The "Crunchiness" Meter): This is the paper's main tool. Instead of just measuring how high a wave is, this meter measures how "crunchy" or dense the space is. If a spot is "crunchy" enough, it collapses into a black hole.

The Story: From Smooth Hills to Spiky Mountains

1. The Old Way (Smooth Hills)
Previously, scientists thought the universe's early fluctuations were like smooth, rolling hills. To make a black hole, you needed a very tall, smooth hill. If the hill wasn't high enough, nothing happened. This meant the universe had to be "tuned" very precisely to create enough black holes to explain Dark Matter.

2. The New Discovery (Spiky Mountains)
The authors simulated the universe with their "random pokes" (stochastic kicks). They found that these kicks turn the smooth hills into jagged, spiky mountains.

• The Analogy: Imagine a smooth sand dune. Now, imagine a gust of wind blowing sand randomly, creating tiny, sharp spikes all over the dune.
• The Result: Even if the average height of the dune is low, those random spikes can be incredibly high and sharp.

3. The "Crunchiness" Explosion
Because the "Crunchiness Meter" (Compaction Function) is sensitive to how sharp the spikes are, these random jags make the universe look much more prone to collapsing than we thought.

• The Shock: The paper finds that these random spikes can increase the number of black holes formed by up to 36 orders of magnitude.
• Translation: That's like going from finding one grain of sand on a beach to finding enough sand to fill the entire Earth. It's a massive, almost unimaginable increase.

The Three Scenarios Tested

The authors tested this idea on three different sizes of black holes:

1. Asteroid Mass: Tiny black holes that could make up all the Dark Matter.
2. Solar Mass: Black holes about the size of our Sun (like those seen colliding by LIGO).
3. Supermassive Seeds: Giant black holes that act as seeds for the monsters at the center of galaxies.

The Finding: In all three cases, the "spiky" universe created way more black holes than the "smooth" universe.

The Twist: Smoothing Things Out

The universe isn't just a static picture; it evolves. As these spikes form, the radiation in the early universe acts like a smoothing iron.

• The Analogy: Imagine drawing a jagged, scribbled line on a piece of paper. Then, you run a hot iron over it. The sharp points get melted down and smoothed out.
• The Effect: The authors included this "smoothing" (called the transfer function) in their math. It reduced the number of black holes, but not enough to cancel out the effect of the spikes. Even after smoothing, the universe still produced vastly more black holes than previously thought.

Why This Matters (The "So What?")

1. We Don't Need to Tune the Universe as Much
Previously, to get enough black holes for Dark Matter, scientists had to assume the universe's initial energy waves were incredibly strong and perfectly tuned. Now, because the "random pokes" do so much of the heavy lifting, the universe can start with weaker, less perfect waves and still end up with the right amount of black holes. This makes the theory of how the universe began much more natural and less "fine-tuned."

2. The Black Holes Are Bigger and More Varied
In the old smooth model, black holes were all roughly the same size. In this new "spiky" model, the black holes come in a huge variety of sizes, ranging from tiny to massive. This changes how we look for them.

3. New Ways to Look for Them
Because the black holes are different sizes and formed differently, the "echoes" they leave behind (like gravitational waves) will be different. This gives astronomers new clues on where to look with telescopes like LISA (which listens for gravitational waves) or by watching how stars move in dwarf galaxies.

The Catch (The "But...")

The authors are very honest about the limitations:

• The Spikes are Weird: Real black holes might not form from these ultra-sharp, jagged spikes the way our math predicts. The spikes might be so sharp that the pressure of the gas pushes back and prevents the collapse.
• We Need New Simulations: The current "collapse threshold" (the point where a wave becomes a black hole) was calculated for smooth waves. We need new computer simulations to see if these "spiky" waves actually collapse or if they just bounce back.

The Bottom Line

This paper suggests that the early universe was a chaotic, spiky place, not a smooth one. These random quantum "kicks" act like a secret ingredient that turns a few potential black holes into a galaxy of them. If this is true, it solves a major puzzle about Dark Matter and the origins of supermassive black holes, but it also means we need to rewrite the rules of how we calculate black hole formation.

In short: The universe is messier than we thought, and that messiness is exactly what allowed it to create the black holes we see today.

Technical Summary

1. Problem Statement

Primordial Black Holes (PBHs) are a candidate for dark matter and seeds for supermassive black holes (SMBHs). Their formation requires large curvature perturbations ($\zeta$) generated during inflation, typically in Ultra-Slow-Roll (USR) phases.

• The Gap: Previous studies often relied on Gaussian statistics or mean profiles of perturbations. However, in USR inflation, stochastic "kicks" from short-wavelength modes create an exponential tail in the probability distribution of $\zeta$, significantly enhancing PBH abundance.
• The Specific Issue: While the effect of stochasticity on the amplitude of $\zeta$ is known, its effect on the radial profile of the perturbations has been less explored. Specifically, stochastic kicks introduce high-frequency "spikes" into the radial profiles.
• The Question: How do these spiky, non-Gaussian radial profiles affect the compaction function ($C$), which is the standard criterion for PBH collapse? Does the spikiness enhance or suppress collapse compared to smooth profiles, and how does this alter the resulting PBH mass function and abundance?

2. Methodology

The authors employ a first-principles approach using Stochastic Inflation and the $\Delta N$ formalism to simulate the formation of PBHs without relying on ad-hoc window functions.

• Inflationary Model: They utilize single-field inflation models based on a Higgs-like potential with loop corrections, tuned to produce USR phases. Three mass regimes are studied:
  1. Asteroid mass ($\sim 10^{-15} M_\odot$): Potential dark matter.
  2. Solar mass ($\sim M_\odot$): Potential LIGO/Virgo sources.
  3. Supermassive seeds ($\sim 10^3 M_\odot$): Seeds for galactic SMBHs.
• Simulation Procedure:
  • Stochastic Evolution: Instead of solving stochastic differential equations directly at every step, they use an analytical solution for the constant-roll phase where the field evolution is expressed as a sum of Gaussian random variables ($\hat{\xi}_k$).
  • Patch Simulation: They simulate $10^8$ independent patches of the universe.
  • Radial Profiles: For each patch, they construct the spherically symmetric curvature perturbation profile $\zeta(r)$ by summing $4 \times 10^4$ momentum shells (Fourier modes) via inverse Fourier transform. No window function is applied to cut off high-$k$ modes; the power spectrum naturally suppresses them.
  • Compaction Function Calculation: They calculate the compaction function $C(r)$ and its average $\bar{C}(r)$ from the derived $\zeta(r)$.
    • $C(r) = \frac{2}{3}[1 - (1 + r\zeta')^2]$
    • $\bar{C}(r)$ is the volume-averaged compaction function.
  • Collapse Criteria: A PBH is assumed to form if the global maximum of $C(r)$ or $\bar{C}(r)$ exceeds a threshold ($C_{th} = 0.4$ or $\bar{C}_{th} = 0.4$), derived from previous simulations of smooth profiles.
  • Refinements: The study includes the effects of critical collapse (scaling laws near the threshold) and the radiation-era transfer function (linear smoothing of sub-Hubble modes).

3. Key Contributions

1. First-Principles Spiky Profiles: This is the first study to construct full radial profiles of perturbations from stochastic inflation without smoothing or window functions, revealing that stochastic kicks make profiles extremely "choppy" with many narrow peaks.
2. Decoupling of $\zeta$ and $C$: The authors demonstrate that the maximum curvature perturbation ($\zeta_{max}$) is poorly correlated with the maximum compaction function ($C_{max}$). A large $\zeta$ does not guarantee a large $C$, and vice versa, due to the derivative dependence of $C$.
3. Mass Function Evolution: They show that stochastic effects do not just increase abundance but fundamentally reshape the mass distribution, shifting it to higher masses and widening it significantly.
4. Threshold Sensitivity: The study highlights that using collapse thresholds derived from smooth profiles on spiky stochastic profiles may be inaccurate, as pressure gradients from sharp spikes could resist collapse.

4. Key Results

A. Abundance Enhancement

Stochastic kicks enhance PBH abundance by orders of magnitude compared to Gaussian mean profiles or even non-Gaussian mean profiles.

• Mechanism: The derivative dependence of the compaction function amplifies the effect of small-scale stochastic noise.
• Magnitude:
  • Asteroid Mass: Enhancement of $\sim 10^{10}$ to $10^{12}$ over Gaussian mean profiles (depending on threshold).
  • Solar Mass: Enhancement of $\sim 10^9$ over Gaussian mean.
  • Supermassive Seeds: Enhancement of 32 to 36 orders of magnitude over Gaussian mean profiles.
• Transfer Function Effect: Including the linear transfer function (smoothing) reduces the abundance by 2–4 orders of magnitude but does not eliminate the stochastic enhancement.

B. Mass Distribution

• Shift and Broadening: The PBH mass function shifts to higher masses and becomes much wider (spanning 5+ orders of magnitude) compared to the narrow lognormal distributions often assumed.
• Critical Collapse: Including critical scaling shifts the mass distribution slightly toward lower masses but has a smaller effect than the stochastic widening.
• Convergence Issues: The mass distribution does not numerically converge with increasing resolution; higher resolution reveals more high-mass peaks at large radii. This suggests the true mass function may be even broader.

C. Observational Implications

• Power Spectrum Amplitude: Because stochasticity boosts abundance so effectively, the required amplitude of the primordial power spectrum ($P_\zeta$) to match observed dark matter or SMBH seeds can be significantly lower (by 1–2 orders of magnitude) than previously thought.
• Gravitational Waves (GWs): A lower $P_\zeta$ implies weaker induced second-order gravitational waves. This relaxes constraints from LISA and Pulsar Timing Arrays (PTA), which previously ruled out many PBH scenarios due to excessive GW backgrounds.
• Spectral Distortions: The relaxed constraints on $P_\zeta$ also alleviate CMB spectral distortion limits (Silk damping) for SMBH seeds.

5. Significance and Limitations

Significance:

• The paper fundamentally challenges the standard assumption that PBH abundance can be estimated using smooth profiles or simple Gaussian/non-Gaussian statistics.
• It suggests that PBHs can be generated with much smaller inflationary power spectrum amplitudes, potentially resolving fine-tuning issues in inflationary models.
• It provides a mechanism to generate the broad mass functions required to explain both solar-mass mergers and supermassive black hole seeds simultaneously.

Limitations and Future Work:

• Collapse Threshold: The study uses collapse thresholds ($C_{th} \approx 0.4$) derived from smooth profiles. The authors acknowledge that for spiky profiles, high-frequency pressure gradients might prevent collapse, potentially overestimating the abundance. New 3D relativistic collapse simulations with stochastic initial conditions are required to determine the true threshold.
• Spherical Symmetry: The assumption of spherical symmetry may be violated by the large stochastic fluctuations, though non-spherical modes are expected to decay.
• Numerical Convergence: The mass function fails to converge with resolution, indicating that the full extent of the high-mass tail is not yet captured.
• Critical Collapse Range: The applicability of critical scaling laws to these highly non-linear, spiky profiles is uncertain.

Conclusion:
The authors conclude that stochastic effects are a dominant factor in PBH formation, capable of enhancing abundance by up to 36 orders of magnitude and drastically altering the mass function. While the quantitative results depend on the unknown collapse criterion for spiky profiles, the qualitative conclusion—that stochasticity allows for PBH formation with smaller power spectrum amplitudes and broader mass distributions—is robust. This necessitates a re-evaluation of all observational constraints on PBHs.