Primordial black hole formation in matter domination

arXiv:2507.18312 · gr-qc, astro-ph.CO · Submitted 2025-07-24 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: Today's paper: "Primordial black hole formation in matter domination".

Jocelyn: This scientific paper investigates the formation threshold and properties of Primordial Black Holes (PBHs) during an early period of Matter Domination,

Vera: First, who's behind it and why it matters.

Title and authors: Vera: So, let's talk about the title and who wrote this paper. "Primordial black hole formation in matter domination" tells us exactly what the focus is: PBHs collapsing specifically during a matter-dominated epoch. The authors are Ebrahimiana, Abolhasanib, and Mirbabayia from the Abdus Salam ICTP and Sharif University of Technology.

Jocelyn: It’s interesting that they're focusing on the matter domination phase because that’s where certain collapse dynamics might behave differently than during inflation or radiation domination, Vera. What do you think about the title itself?

Subrahmanyan: The authors are clearly grounding their theoretical work in a specific cosmological context, which is necessary when you want to connect abstract fluctuation physics to observable outcomes like PBH abundance.

Vera: Exactly. It sets the stage for understanding how the shape of those initial density fluctuations dictates whether we end up with a black hole or something else entirely, which is what this paper is really investigating.

Jocelyn: I wonder if the title implies that they are challenging older assumptions about when these black holes could have formed compared to other epochs.

Subrahmanyan: That’s precisely the implication. The paper suggests that for a cosmologically relevant abundance of PBHs, the effective formation threshold is larger than some previous models suggested in a dust-dominated universe.

Vera: So, essentially, they are refining our understanding of the conditions needed for these black holes to appear during that specific matter-dominated era.

Jocelyn: It sounds like the authors are setting up a comparison between what we see in the CMB and what might be happening at smaller scales during this early epoch.

The paper's summary: Vera: Now, let's get into the actual summary of "Primordial black hole formation in matter domination." The core finding here is that the collapse threshold isn't fixed; it strongly depends on the shape of those initial peaks, and this dependence actually makes them rarer as those peaks become flatter.

Jocelyn: That’s a key point, Vera. So, if you have a slightly flatter fluctuation, it becomes harder for it to collapse into a black hole compared to a very sharp peak. What does that mean for the overall population?

Subrahmanyan: It means that the statistical distribution of peaks plays a huge role; because typical cosmic overdensities aren't perfect top-hats, we have to account for this shape dependence when calculating how many PBHs form.

Vera: And the paper points out that in the extreme case of a top-hat perturbation, Harada et al. argued that velocity dispersion prevents collapse unless the initial peak is larger than a threshold of zeta th ∼ zeta two/five rms.

Jocelyn: But then they immediately narrow that down by including the shape distribution, finding a more realistic effective threshold of zeta th ∼ zeta one/ten rms. That's a significant refinement.

Subrahmanyan: That smaller threshold means that for most peaks to collapse, we need an initial fluctuation level of zeta rms ∼ ten-one which the authors note is much larger than what we observe at the CMB scales.

Vera: So, the implication is that PBH formation during Matter Domination is barely more efficient than what we expect from Radiation Domination. It’s a subtle point about timing and efficiency.

Jocelyn: That suggests that if we are searching for PBHs in the early universe, this paper puts a constraint on how much they could have formed during this specific era compared to other scenarios.

The paper's improvements: Vera: Moving on to the improvements suggested by the authors of "Primordial black hole formation in matter domination," they focus heavily on how typical overdensities, which aren't top-hats, interact with that lower collapse threshold. They find that for a given cosmic abundance, we need zeta rms ∼ zero point one for most PBHs to form from peaks with significantly small second derivatives.

Subrahmanyan: The paper establishes the effective threshold as zeta th ∼ zeta one/ten rms, or log beta ∼ −zeta-one point eight rms, which is a direct consequence of accounting for these typical higher-order derivatives.

Jocelyn: The paper makes a distinction between the dust and w → zero limits, saying that black hole formation in the dust scenario is very sensitive to the profile shape, while in a fluid with constant w, most spherically symmetric peaks collapse into black holes when w → zero.

Vera: That distinction is important because it highlights that velocity dispersion and profile flatness are the deciding factors in the dust scenario, whereas they are more robust in a constant-w fluid model.

Subrahmanyan: The authors also show that numerical N-body simulations confirm this halting mechanism, demonstrating that for a typical profile with A2 = O(one), the threshold for black hole formation is around delta m = O(one).

Jocelyn: It’s interesting how they use these simulations to back up the analytical findings about shell-crossing and velocity dispersion preventing small perturbations from collapsing into black holes.

Conclusion: Vera: So, to wrap up this discussion on "Primordial black hole formation in matter domination," the paper suggests that despite the dynamics of matter domination, PBH formation is constrained by a relatively high effective threshold compared to what some earlier analytical results suggested.

Jocelyn: The main implication for us as observers is that unless we find evidence for these specific fluctuation scales, the abundance of PBHs formed in this era might be much lower than initially estimated.

Subrahmanyan: Theoretically, the finding about spin parameter estimation is also significant; they conclude that the dimensionless spin is typically small when zeta rms is less than one because velocity dispersion plays a more dominant role in preventing collapse than angular momentum.

Vera: That ties it all together: the shape of the peak and velocity dispersion are the primary suppressors for these black holes during this matter-dominated era, which informs our search strategies for gravitational wave signals or other signatures.

Jocelyn: It’s a lot of detail about how initial conditions translate into final black hole properties, and it gives us clearer boundaries for where these objects could have originated.

Subrahmanyan: This paper provides a detailed framework connecting the statistical properties of primordial fluctuations to the physical outcome of collapse during matter domination, which is essential groundwork for future theoretical modeling.

The Abdus Salam ICTP · Sharif University of Technology · IFPU, Institute for Fundamental Physics of the Universe

gr-qc, astro-ph.CO

Submitted: 2025-07-24

Updated: 2026-09-30

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 69/100

The gist: This scientific paper investigates the formation threshold and properties of Primordial Black Holes (PBHs) during an early period of Matter Domination, exploring how peak shape and small-scale

Key concepts

Effective Formation Threshold
This is the minimum initial peak size (zeta) required for primordial fluctuations to collapse into black holes during matter domination. The paper finds this threshold is much larger than earlier estimates, requiring specific fluctuation levels that are more difficult to achieve than those seen in the Cosmic Microwave Background.
Velocity Dispersion Halting Mechanism
This physical process prevents small perturbations from collapsing into black holes by transforming them into virialized structures. When fluctuations grow, they induce random velocities; if these velocities reach a certain level relative to the gravitational potential, collapse stops and re-expansion begins according to the Virial theorem.
Profile Shape Sensitivity
The shape of the initial density fluctuation profile is crucial for PBH formation. While a top-hat profile collapses easily without fluctuations, typical overdensities have different shapes. The competition between the lower collapse threshold and suppression from atypical derivatives dictates which structures form black holes.
Spin Parameter Estimation
The dimensionless spin parameter of formed PBHs is estimated using Tidal Torque Theory. The study concludes that for most PBHs formed in the matter-dominated era, the spin is typically small because velocity dispersion plays a more dominant role than net rotation in determining their final angular momentum.

Terminology

Summary

This scientific paper investigates the formation threshold and properties of Primordial Black Holes (PBHs) during an early period of Matter Domination, exploring how peak shape and small-scale velocity dispersion influence whether rare primordial fluctuations collapse into black holes or form Virialized dark matter halos. The study is significant because it refines the understanding of PBH abundance, suggesting that for cosmologically relevant populations, the effective formation threshold is larger than previously thought in a dust-dominated universe, which has implications for their role as dark matter candidates and sources of gravitational waves.

The Effective Formation Threshold

The paper analyzes how the collapse threshold depends on the shape of primordial fluctuations. It contrasts analytical results for different fluid equations of state:

  1. For a top-hat perturbation in a dust-dominated universe, Harada et al. argued that the growth of velocity dispersion prevents black hole formation unless the initial peak is larger than a threshold: the growth of velocity dispersion prevents the formation of black holes unless the initial peak is larger than ζth ∼ ζ2/5 rms.

  2. Including the shape distribution, a more realistic effective threshold is found to be ζth ∼ ζ1/10 rms.

  3. This requires a specific fluctuation level: this model requires ζrms ∼ 10−1, which is much larger than observed CMB scales. Consequently, PBH formation during Matter Domination is barely more efficient than Radiation Domination.

The Role of Profile Shape and Shell-Crossing

The paper emphasizes that the top-hat profile is unique because it will inevitably collapse into a black hole in the absence of fluctuations. However, typical overdensities are not top-hats. The competition between the lower threshold for collapse and the suppression due to atypical derivatives determines PBH formation history:

  1. For a given cosmologically relevant abundance, we need ζrms ∼ 0.1 for which most PBHs are formed from peaks with significantly small second derivative, but typical higher order derivatives.

  2. The effective threshold is then determined as ζth ∼ ζ1/10 rms, or log β ∼ −ζ−1.8 rms.

  3. A key distinction between the dust and w → 0 limit is that black hole formation is very sensitive to the profile shape in the dust scenario whereas in a fluid with constant w, most spherically symmetric peaks collapse into black holes when w → 0.

The Influence of Velocity Dispersion and Shell-Crossing

Velocity dispersion plays a crucial role in preventing small perturbations from collapsing into black holes by transforming them into Virialized structures. The paper demonstrates this halting mechanism through numerical simulations:

  1. In the context of the top-hat background, fluctuations with wavelengths shorter than its size grow and induce random velocities, leading to a velocity dispersion where σ2vR(t) ∼ GM/R(t)2. If this is reached while R(t) > 2GM, collapse ceases and re-expansion begins per the Virial theorem.

  2. In the dust model (w=0), shell-crossing is a key mechanism: the central density never diverges and shell-crossing is not allowed in a perfect fluid scenario, but once spherical symmetry is relaxed, it can occur.

  3. Numerical N-body simulations confirm this halting mechanism, showing that for a typical profile with A2 = O(1), the threshold for black hole formation is δm = O(1).

Spin Parameter Estimation

The dimensionless spin parameter of PBHs formed in the matter-dominated era is estimated using Tidal Torque Theory (TTT):

  1. TTT suggests that angular momentum reaches its maximum at turnaround time, and subsequent effects are negligible.

  2. The final angular momentum squared is estimated as L2 ∼ GM3rmaxQ2σ2(M), where Q is a dimensionless parameter related to the quadrupole moment of the structure.

  3. For a typical peak with height δm, it can be shown that arms = Lrms / 2GM squared ∼ 1/ν2δm5/2, leading to arms ∼ ζ7/4 rms when most PBHs form around δm ∼ ζ0.1 rms.

  4. The paper concludes that contrary to earlier claims, the spin is typically small when ζrms≪ 1, because the total angular momentum, which requires a net rotation, plays a subdominant role compared to velocity dispersion in PBH formation.

Abundance and Clustering

The production probability of PBHs is estimated using a modified Press-Schechter formalism incorporating the threshold dependence on peak shape:

  1. The production probability is split into three integrals based on the behavior of terms in the Taylor expansion of the curvature profile K(r).

Improvements for AI systems

As a fastidious, diligent researcher, I have analyzed this paper on Primordial Black Hole (PBH) formation during Matter Domination. The core findings revolve around how peak shape and small-scale velocity dispersion modify the standard collapse threshold, leading to a lower effective threshold for typical peaks than previously thought.

Here are the specific improvements to AI systems that could be derived from this research:

)Specific Improvements for AI Systems Derived from the Paper:

  1. Primary Improvement: Enhanced Cosmological Parameter Inference and Constraint Setting in Early Universe Physics.

  2. Specific Capability 1: Precision Threshold Determination for PBH Formation under Varying Fluid Equations of State (w).

  3. Specific Capability 2: Modeling the Effect of Non-Gaussianity and Profile Shape on Structure Formation Thresholds (Connecting Peak Statistics to Physical Outcomes).

  4. Specific Capability 3: Dynamic Simulation and Analysis of Collapse Halting Mechanisms (Integrating Shell-Crossing, Velocity Dispersion, and Virialization in N-body/Hydrodynamic Simulations).

  5. Specific Capability 4: Predictive Modeling of PBH Spin Distributions Based on Formation Mechanism (Linking Peak Statistics to Tidal Torque Theory to predict angular momentum arms).

)What the Improved AI System Can Do:

  1. Precision Threshold Determination for PBH Formation under Varying Fluid Equations of State (w):

The system can accurately calculate the effective black hole formation threshold, accounting for the competition between pressure gradients (for constant w) and velocity dispersion/profile flatness (for w → 0). It can determine whether a given primordial fluctuation amplitude is sufficient to form a PBH across different cosmological epochs.

  1. Modeling the Effect of Non-Gaussianity and Profile Shape on Structure Formation Thresholds:

The AI can analyze the Fourier space decomposition of density fields (using spherical harmonics and power spectrum analysis) to quantify how deviations from Gaussianity (higher-order derivatives, quantified by coefficients like A2, A4) modify the required threshold. It can predict the specific peak shapes that yield lower collapse thresholds for a given fluctuation amplitude.

  1. Dynamic Simulation and Analysis of Collapse Halting Mechanisms:

The system can perform advanced N-body simulations (or utilize learned models based on these simulations) to identify the physical mechanisms—specifically shell-crossing and velocity dispersion—that halt gravitational collapse in non-spherical or realistic density profiles. It can quantify the minimum compression factor required to prevent black hole formation, providing a direct link between initial conditions and final structure size.

  1. Predictive Modeling of PBH Spin Distributions Based on Formation Mechanism:

The AI can estimate the dimensionless spin parameter (arms) by integrating Tidal Torque Theory (TTT) over the distribution of rare peak properties derived from the paper's framework. It can predict if PBHs formed in a specific epoch (Matter Domination vs. Radiation Domination) will have small or large spins, providing a testable prediction for gravitational wave observatories.

Abstract

We study Primordial Black Holes (PBHs) formed by the collapse of rare primordial fluctuations during an early period of Matter Domination. The collapse threshold strongly depends on the shape of the peaks, decreasing as they become flatter and hence rarer. In the extreme limit of a top-hat perturbation, Harada, Kohri, Sasaki, Terada, and Yoo have argued that the growth of velocity dispersion prevents the formation of black holes unless the initial peak is larger than ζ th about ζ rms 2/5. Including the shape distribution of the peaks, we find that for a realistic cosmic abundance of PBHs, the effective threshold is larger, ζ th about ζ rms 1/10. And this model requires ζ rms about 10-1, which is much larger than the observed value at the CMB scales. Hence, PBH formation during Matter Domination is barely more efficient than Radiation Domination. We estimate the dimensionless spin parameter to be a rms about ζ rms 7/4 1, slightly larger than PBHs formed in Radiation Domination.

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