The trichotomy of primordial black holes initial conditions

This paper demonstrates that the threshold for primordial black hole formation in a radiation-dominated universe depends not only on the compaction function but also on the spatial curvature of the core, leading to a trichotomy of initial conditions (open, closed, or flat) that significantly influence formation probabilities, mass spectra, and the potential connection to Type-II black holes and the NanoGrav signal.

The Gist

Imagine the early universe as a giant, expanding balloon covered in a thin layer of honey. Sometimes, a little too much honey clumps together in one spot. If that clump gets heavy enough, gravity wins, the honey collapses, and a Primordial Black Hole (PBH) is born.

For a long time, scientists thought they knew exactly how heavy that clump needed to be to collapse. They believed it was all about the shape of the clump's edge (the "shell"). If the edge was steep enough, it would collapse. If it was too flat, it would just spread out.

However, this new paper by Germani and Montellà says: "Wait a minute! You're ignoring the center!"

They argue that to know if a black hole will form, you can't just look at the edge of the clump. You also have to look at the core (the very center) and what kind of space it lives in.

The Three Types of "Nests" (Initial Conditions)

The authors propose that the center of these clumps can be one of three things, like three different types of nests for a bird:

1. The Closed Nest (Type-C): Imagine a bowl that curves inward. If the center of your honey clump sits in a bowl that curves inward, gravity gets a little help. It's like rolling a ball down a hill; the shape of the hill helps it roll faster.

   • Result: It's easiest to make a black hole here. You need less honey (less mass) to trigger the collapse.

2. The Flat Nest (Type-F): Imagine the center sits on a perfectly flat table. There's no help, but there's no resistance either.

   • Result: You need a medium amount of honey to make a black hole.

3. The Open Nest (Type-O): Imagine the center sits on a saddle or a hill that curves outward. This shape actually fights against the collapse. It's like trying to roll a ball up a hill; the shape of the hill pushes back.

   • Result: It's the hardest to make a black hole here. You need a huge amount of honey to overcome the resistance and collapse.

The "Shell" vs. The "Core"

The paper introduces a new way of thinking using a "Core and Shell" analogy:

• The Shell: This is the outer ring of the honey clump. This is what scientists used to look at. If the shell is very sharp and thin (like a razor blade), it matters a lot.
• The Core: This is the center.

The Big Discovery:
If the shell is wide and fuzzy (like a soft pillow), the shape of the center doesn't matter much. The whole thing acts like a standard clump, and the old rules work fine.

But, if the shell is very sharp and thin (like a razor blade), the shape of the center becomes the deciding factor.

• If you have a sharp shell on a Closed (bowl) core, the black hole forms easily.
• If you have a sharp shell on an Open (saddle) core, the black hole might not form at all, even if the shell looks the same, because the center is fighting the collapse.

Why Does This Matter?

This changes how we predict how many black holes exist in the universe.

1. The "Sharp" Universe: If the early universe had very specific, sharp spikes in density, the type of core matters a lot. We might have many more black holes than we thought (because the "Closed" cores help them form easily) or fewer (if "Open" cores are common).
2. The "Broad" Universe (The NanoGrav Connection): The paper mentions a recent signal detected by the NanoGrav experiment (which listens for gravitational waves). This signal suggests the universe might have had a very broad, fuzzy distribution of density.
   • If the distribution is broad, the "Core" doesn't matter as much.
   • However, the authors suggest that in this broad scenario, the "Closed" cores (Type-C) might still be the most efficient at making black holes. This would mean the black holes we see today are likely heavier and formed from the "red" (low energy) end of the spectrum, rather than the "blue" (high energy) end.

The "Trichotomy" (The Three-Way Split)

The paper calls this a "Trichotomy" (a split into three). Before, we thought there were two types of black hole formation scenarios. Now, we know there are three distinct paths, depending on whether the center of the clump is Closed, Flat, or Open.

In a nutshell:
To predict if a baby black hole will be born, you can't just look at the baby's skin (the shell). You have to check if the baby is sleeping in a cozy bowl (Closed), a flat mattress (Flat), or a bumpy saddle (Open). The bed they sleep in changes the odds of them waking up as a black hole.

Technical Summary

Here is a detailed technical summary of the paper "The trichotomy of primordial black holes initial conditions" by Cristiano Germani and Laia Montellà.

1. Problem Statement

The formation of Primordial Black Holes (PBHs) depends critically on the threshold amplitude of density perturbations required to trigger gravitational collapse. Historically, this threshold was believed to be determined primarily by the behavior of the compaction function ($C$) or the variable $g$ (derived from the linear smoothed over-density) at its extrema.

• Type-I PBHs: Characterized by a single maximum in the compaction function. The threshold was previously thought to be determined solely by the curvature of the compaction function at this maximum.
• Type-II PBHs: Characterized by a minimum in the compaction function surrounded by two maxima. These were previously thought to require significantly higher thresholds ($g > 4/3$) and were often considered unlikely or discarded in statistical analyses.

Recent numerical simulations suggested that the analytical formulas for Type-I thresholds might fail for certain curvature profiles, particularly those with broad power spectra. The authors identify a gap in understanding: the role of the spatial geometry of the "core" (the super-horizon region surrounding the collapsing shell) has been overlooked. They argue that the threshold is not solely determined by the shell's curvature but also by the Ricci scalar of the spatial geometry at the center.

2. Methodology

The authors employ a combination of analytical gradient expansion and high-precision numerical simulations to investigate PBH formation.

• Theoretical Framework:

  • They utilize the separate universe approach, approximating a local super-horizon patch of the perturbed universe as a Friedmann-Robertson-Walker (FRW) metric with a specific constant 3-dimensional curvature ($K$).
  • They classify initial conditions based on the sign of the Ricci scalar $R$ at the center ($r \to 0$):
    • Type-C (Closed): $R > 0$ (Positive curvature core).
    • Type-F (Flat): $R = 0$ (Flat core).
    • Type-O (Open): $R < 0$ (Negative curvature core).
  • They define a dimensionless curvature parameter for the shell, $w$, and a new parameter for the core-peak ratio, $w_R$, to characterize the sharpness of the perturbation.

• Numerical Simulations:

  • They use the public code SPriBHoS-II to evolve spherically symmetric scalar perturbations in a radiation-dominated, asymptotically flat FRW universe.
  • To efficiently detect black hole formation, they track the Misner-Sharp mass via the auxiliary function $C_0 = 2M/R$. A black hole forms when $C_0$ reaches unity twice (indicating the formation of a trapped surface after horizon crossing).
  • They simulate various profiles for $\zeta(r)$ (the curvature perturbation) corresponding to the three core types (C, F, O) with varying parameters ($\alpha, \beta, \lambda$) to control the core amplitude and shell sharpness.

3. Key Contributions

The paper introduces a "Trichotomy" of initial conditions, fundamentally revising the understanding of PBH formation thresholds:

1. Core Geometry Matters: The threshold for black hole formation is not determined solely by the shell structure (the over-dense region) but significantly by the Ricci scalar of the core.
2. Three Distinct Thresholds:
   • Type-C (Closed Core): The positive curvature of the core assists gravitational collapse. Consequently, Type-C perturbations have the lowest formation threshold among the three types.
   • Type-O (Open Core): The negative curvature of the core resists collapse. Type-O perturbations have the highest formation threshold.
   • Type-F (Flat Core): The threshold lies between Type-C and Type-O.
3. Re-evaluation of Type-II PBHs:
   • Type-II black holes (previously thought to require very high amplitudes) can be generated by Type-O or Type-F cores, or by Type-C cores where the core amplitude is negligible (effectively acting as Type-F).
   • Type-C cores with significant amplitude tend to produce Type-I black holes even at high curvature values ($w$), because the core helps the collapse, preventing the formation of the "trough" characteristic of Type-II.
4. New Threshold Criterion: The authors propose that the transition between Type-I and Type-II behavior in Type-C perturbations is governed by the ratio of the core amplitude to the peak amplitude ($Q = R(0)/R(r_p)$) and the sharpness of the peak ($w_R$).

4. Key Results

• Threshold Saturation: For large values of the shell curvature parameter $w$ (sharp profiles, $w \gtrsim 30$), the thresholds for the three types diverge significantly.
  • Type-C thresholds remain low (favoring Type-I).
  • Type-F and Type-O thresholds rise, favoring Type-II.
• The Separatrix (Eq. 21): The authors derive an empirical formula defining the boundary between Type-I and Type-II behavior for Type-C perturbations:
  $$Q \approx 0.70 - \frac{70.78}{(w_R)^{1.48}}$$
  This implies that for very sharp shells ($w_R \to \infty$), the core amplitude must be at least ~70% of the peak amplitude to maintain a Type-I configuration. If the core is smaller than this, the perturbation behaves as Type-II (effectively flat).
• Statistical Implications:
  • Sharp Power Spectra: For narrow power spectra (typical in many inflationary models), Type-F (flat core) configurations are statistically more probable than Type-C(I) because Type-C(I) requires a central over-density that is statistically suppressed (requiring $\sim 10^{18}$ variances away from the mean). This suggests Type-II PBHs might dominate in these scenarios.
  • Broad Power Spectra: For very broad power spectra (potentially related to the NanoGrav signal), the statistical suppression of Type-C(I) is negligible. In this case, Type-C(I) perturbations dominate the abundance due to their lower threshold. This would lead to a PBH mass spectrum peaked at the Infra-Red (IR) scale rather than the Ultra-Violet (UV) scale.

5. Significance

• Correction of Analytical Models: The paper demonstrates that existing analytical formulas (e.g., Escrivà et al. [3]) for Type-I thresholds are insufficient for Type-O and Type-F profiles with sharp shells, and that Type-C profiles follow these formulas only when the core is negligible.
• Impact on PBH Abundance: By establishing that the core geometry dictates the threshold, the paper suggests that previous estimates of PBH abundance may be inaccurate. The dominance of Type-I vs. Type-II black holes depends heavily on the shape of the primordial power spectrum and the statistical likelihood of specific core geometries.
• Connection to Observations: The findings offer a potential explanation for the NanoGrav signal. If the signal arises from a broad power spectrum, the resulting PBH population would likely be dominated by Type-I black holes formed from Type-C cores, with masses peaking at the IR scale.
• Non-Spherical Considerations: The authors note that while Type-F cores are statistically favored in spherical symmetry, they may correspond to highly non-spherical initial conditions in reality, which could further increase the collapse threshold due to gravitational wave dissipation. This highlights the need for future non-spherical collapse studies.

In summary, Germani and Montellà establish that the spatial curvature of the super-horizon core is a decisive factor in PBH formation, creating a three-way split in formation thresholds that fundamentally alters predictions for PBH abundance and mass spectra.