Revisiting Constraints on Primordial Curvature Power Spectrum from PBH Abundances

This paper derives updated constraints on the amplitude of the primordial curvature power spectrum from primordial black hole abundances by systematically comparing Press-Schechter and peak theory formalisms, revealing that non-spherical collapse effects and the choice of statistical method significantly influence the inferred limits, particularly for broad power spectra.

The Gist

Imagine the early Universe as a giant, bubbling pot of cosmic soup. Most of the time, this soup is smooth and uniform. But sometimes, tiny ripples or "bumps" appear in the density of this soup. If a bump gets big enough and collapses under its own gravity, it can turn into a Primordial Black Hole (PBH)—a black hole born not from a dying star, but from the very first moments of the Universe.

This paper is essentially a detective story trying to figure out how big those "bumps" in the early soup could have been, based on the fact that we haven't found many (or any) of these ancient black holes today.

Here is the breakdown of the story using simple analogies:

1. The Connection: Black Holes as "Fossils"

Think of the Universe's history like a timeline. We can look at the "Cosmic Microwave Background" (the afterglow of the Big Bang) to see the Universe when it was very large and smooth. But that only tells us about the "big picture."

However, if the Universe had huge, jagged bumps in specific, tiny spots, those spots would have collapsed into black holes.

• The Analogy: Imagine you are trying to guess how rough a piece of sandpaper was by looking at the scratches it left on a table. If the table is smooth, the sandpaper was fine. If the table is scratched to pieces, the sandpaper was coarse.
• The Paper's Goal: Since we don't see a universe filled with ancient black holes, we know the "sandpaper" (the early Universe) couldn't have been too rough. This paper calculates exactly how rough it could have been before we would have seen evidence of it.

2. The Problem: We Don't Know the Exact "Collapse Recipe"

The authors point out that while we know black holes form from collapsing bumps, we aren't 100% sure of the exact recipe for how that collapse happens. It's like trying to predict exactly how much water is needed to make a snowball stick, but you don't know if the snow is wet or dry, or if you are packing it with your hand or a machine.

The paper compares two different "recipes" (mathematical methods) used to predict black hole formation:

• Recipe A (Press-Schechter): A classic, simpler method that assumes the collapsing region is a perfect, round ball.
• Recipe B (Peak Theory): A more modern, complex method that looks at the "peaks" of the density waves.

The Finding: When the collapsing region is a perfect ball, both recipes give similar answers. But when the region is a bit lumpy or squashed (like an egg instead of a ball), the recipes start to disagree, especially for the tiniest scales.

3. The Twist: Shape Matters (The "Squashed Ball" Effect)

In the real world, things aren't perfect spheres. A collapsing cloud of gas is often squashed or stretched (ellipsoidal).

• The Analogy: Think of trying to crush a water balloon. If it's a perfect sphere, it's hard to crush. But if you squeeze it into an oval shape, it might burst (collapse) much easier or harder depending on how you squeeze it.
• The Paper's Discovery: The authors found that if you account for these "squashed" shapes, the threshold for forming a black hole gets higher. It's harder to make a black hole out of a squashed blob than a perfect sphere.
• The Result: Because it's harder to make them, the early Universe must have had even bigger bumps (higher energy) to create the black holes we might see. This pushes the "limit" of how rough the early Universe could have been higher up.

4. The Big Reveal: Narrow vs. Wide Peaks

The authors tested two scenarios for the "bumps" in the early soup:

1. Narrow Peaks (Monochromatic): Imagine the bumps are all exactly the same size, like a row of identical marbles.
2. Broad Peaks (Extended): Imagine a mix of sizes, from tiny pebbles to giant boulders.

The Surprise:

• For the identical marbles (Narrow), both math recipes (Press-Schechter and Peak Theory) agreed on the answer.
• For the mix of sizes (Broad), the two recipes gave very different answers, especially for the smallest black holes.

Why does this matter? It means our current understanding of the early Universe is still a bit shaky. Depending on which "recipe" you use, you get a different picture of how violent the early Universe was. The paper shows that for the smallest scales, the difference between the two methods is huge, highlighting that we need better physics to be sure.

Summary: What did they actually do?

The authors took the latest data on how many black holes we don't see and used it to draw a "speed limit" sign for the early Universe.

• They updated the math to include the fact that collapsing clouds aren't perfect spheres.
• They compared two different ways of doing the math.
• They found that while the "speed limit" is similar for simple cases, it gets very confusing for complex, broad scenarios.

The Takeaway: We can use the absence of ancient black holes to map the invisible history of the Universe, but to get the map right, we need to stop assuming everything is a perfect sphere and agree on the best mathematical tools to use. Until then, there is still some wiggle room in our understanding of the Universe's first moments.

Technical Summary

1. Problem Statement

Primordial Black Holes (PBHs) serve as a unique probe for the amplitude of the primordial curvature power spectrum ($P_\zeta(k)$) on small scales, which are inaccessible to standard cosmological observations like the Cosmic Microwave Background (CMB) and Large Scale Structure (LSS). While PBH formation is theoretically linked to the collapse of large-amplitude density perturbations during the radiation-dominated (RD) era, deriving robust constraints on $P_\zeta(k)$ from PBH abundance limits is plagued by significant theoretical uncertainties.

Key sources of uncertainty include:

• Statistical Formalisms: Discrepancies between the Press-Schechter (PS) formalism and Peak Theory in estimating the mass fraction of PBHs ($\beta$).
• Collapse Geometry: The assumption of spherical collapse versus the more realistic ellipsoidal (non-spherical) collapse, which alters the threshold density contrast required for formation.
• Non-linearity: The non-linear relationship between the curvature perturbation ($\zeta$) and the density contrast ($\delta$).
• Mass Functions: The distinction between monochromatic (narrow peak) and extended (broad peak) PBH mass functions.

Previous studies often addressed these factors in isolation or used older observational bounds. This work aims to provide a unified, systematic treatment incorporating the most up-to-date constraints and all major theoretical refinements simultaneously.

2. Methodology

The authors employ a systematic framework to translate PBH abundance limits into upper bounds on $P_\zeta(k)$.

A. Collapse Criteria and Thresholds

• Compaction Function: The study utilizes the compaction function ($C$) rather than simple density contrast to define the collapse threshold. $C$ is conserved on superhorizon scales and allows for a consistent determination of the overdense region's radius ($r_m$).
• Non-linear Relation: The authors derive the threshold density contrast ($\delta_c$) by accounting for the non-linear relation between $\zeta$ and $\delta$. They identify the physically relevant root (Type-I perturbations) for the threshold, denoted as $\delta_{L-,c}$.
• Non-Sphericity: To address the shape of overdense regions, they incorporate the effect of ellipticity ($e$). The threshold for ellipsoidal collapse is modeled as $\delta_{ec} \approx \delta_{sp}(1 + 3e)$, where $\delta_{sp}$ is the spherical threshold. This effectively raises the required density contrast for PBH formation.

B. Mass Fraction Parameter ($\beta$)

The paper compares two statistical approaches to calculate $\beta$ (the fraction of the universe collapsing into PBHs):

1. Press-Schechter (PS) Formalism: Calculates $\beta$ by integrating the probability distribution function (PDF) of the smoothed density contrast above the threshold.
2. Peak Theory: Calculates $\beta$ based on the number density of peaks in the smoothed density field, utilizing the height of peaks in the non-linear field.

C. Mass Functions and Constraints

• Monochromatic Case: Assumes a narrow power spectrum peak ($\Delta \approx 0.1$), producing PBHs of nearly identical mass.
• Extended Case: Assumes a broad power spectrum peak ($\Delta = 1$), producing a wide range of PBH masses.
• Observational Bounds: The authors use the most recent constraints on PBH abundance relative to dark matter ($f_{PBH}$) from Ref. [22], covering a wide mass range. They apply the Carr et al. method to combine constraints from various monochromatic limits to constrain extended mass functions.

3. Key Contributions

1. Unified Framework: This is the first study to simultaneously incorporate non-linear density-curvature relations, non-spherical collapse criteria, and a comparison between PS and Peak theories for both narrow and broad power spectrum peaks.
2. Refined Thresholds: By explicitly calculating the threshold for ellipsoidal collapse using the compaction function, the authors provide a more accurate estimate of the required power spectrum amplitude.
3. Systematic Comparison: The paper provides a direct, side-by-side comparison of constraints derived from PS formalism vs. Peak theory under identical physical assumptions (spherical vs. ellipsoidal).
4. Updated Bounds: The constraints are derived using the latest observational limits on PBH abundance, ensuring the results are current.

4. Key Results

A. Impact of Non-Sphericity

• Including non-spherical (ellipsoidal) collapse significantly increases the inferred amplitude of the primordial power spectrum ($A_p$) required to produce a given PBH abundance.
• This is because ellipsoidal regions require a higher threshold density contrast ($\delta_c$) to collapse compared to spherical regions. Consequently, the constraints on $P_\zeta(k)$ become weaker (allowing for higher amplitudes) when non-sphericity is accounted for.

B. Monochromatic vs. Extended Mass Functions

• Monochromatic Case (Narrow Peak): The constraints on $P_\zeta(k)$ obtained using PS formalism and Peak theory are largely consistent with each other, regardless of whether spherical or ellipsoidal collapse is assumed.
• Extended Case (Broad Peak): A significant discrepancy emerges between the two formalisms.
  • Peak Theory constraints diverge from PS constraints at smaller scales (higher wavenumbers $k$).
  • This difference arises because Peak Theory depends on the parameter $\mu_L$ (related to the second moment of the power spectrum), which is more sensitive to small-scale features than the variance $\sigma_L$ used in PS formalism.
  • The discrepancy highlights that current constraints on broad power spectra are highly sensitive to the choice of statistical formalism.

C. Comparison with Other Probes

• The derived constraints (particularly for the ellipsoidal case) are compared with gravitational wave observatories (LIGO-Virgo-KAGRA, Einstein Telescope, LISA, etc.).
• The results show that PBH constraints remain competitive with, and in some regimes complementary to, future gravitational wave sensitivities for probing the early Universe.

5. Significance

This work underscores that theoretical uncertainties in PBH formation mechanisms are currently the dominant limitation in probing the small-scale primordial power spectrum.

• Robustness: By showing that non-sphericity significantly alters the required power spectrum amplitude, the authors argue that future constraints must move beyond the idealized spherical collapse assumption.
• Formalism Dependence: The finding that PS and Peak theories yield divergent results for broad power spectra indicates that the community lacks a definitive, first-principles derivation of the PBH mass function.
• Future Direction: The paper concludes that reducing these theoretical uncertainties is a crucial step before PBH abundances can be used as a precise tool to probe the final stages of inflation. The provided constraints serve as a more rigorous baseline for future inflationary model building.