Primordial black hole formation in matter domination

This paper investigates primordial black hole formation during an early matter-dominated era, concluding that despite shape-dependent threshold variations, the required large amplitude of primordial fluctuations makes this mechanism barely more efficient than in radiation domination and predicts a slightly larger but still small dimensionless spin parameter.

The Gist

The Big Picture: A Cosmic "What If?"

Imagine the very early universe as a giant, expanding balloon. Usually, we think of this balloon being filled with hot, energetic soup (Radiation Domination). But this paper asks a "What if?" question: What if, for a brief moment, the universe was filled with slow-moving, cold dust instead (Matter Domination)?

In this cold, dusty universe, could tiny clumps of matter collapse to form Primordial Black Holes (PBHs)? These are black holes that didn't form from dying stars, but from the raw material of the Big Bang itself. If they exist, they could explain some of the universe's "missing mass" (Dark Matter) or the heavy black holes we see today.

The Main Problem: The "Velvet Rope" of Gravity

To form a black hole, a clump of matter needs to be heavy enough to crush itself under its own gravity. In a hot universe (Radiation), there is a lot of pressure pushing back, so you need a huge clump to overcome it.

In a cold, dusty universe (Matter), there is no pressure pushing back. You might think, "Great! It should be super easy to make black holes here!"

The authors say: "Not so fast."

They discovered that in a cold, dusty universe, the shape of the clump matters more than you think. If the clump isn't perfectly round and smooth, it falls apart before it can become a black hole.

The Analogy: The "Shell-Game" Collapse

Imagine a group of people (particles of dust) trying to run toward a single point in the center of a room to form a pile.

1. The Ideal Scenario (Perfect Sphere): If everyone starts at the same distance and runs perfectly straight to the center, they all arrive at the exact same time. They pile up, crush together, and boom—Black Hole!
2. The Real Scenario (Rough Edges): In reality, the "clump" isn't a perfect sphere. It's a bit lumpy.
   • The people on the inside run a short distance.
   • The people on the outside run a long distance.
   • Because the inside people run faster (they have less distance to cover), they reach the center first, pass right through it, and start running out the other side.
   • The outside people are still running in.
   • The Crash: The people running out collide with the people running in. This is called Shell-Crossing.

The Result: Instead of a neat pile, you get a chaotic mess. The "outgoing" people push back against the "incoming" people. The clump stops collapsing, bounces back, and settles into a stable, fluffy cloud (a "halo") instead of a black hole.

The "Flatness" Requirement

The authors realized that to avoid this "Shell-Crossing" crash, the clump needs to be incredibly flat on top.

• Think of a Mountain vs. a Table:
  • A Mountain (a typical peak) has steep sides. The top is pointy. The "inner" layers rush down the steep slope much faster than the "outer" layers. They cross paths and crash. No Black Hole.
  • A Table (a "Top-Hat" shape) has a flat top. Everyone on the table is roughly the same distance from the edge. They all slide down at roughly the same speed. They arrive together. Black Hole!

The Catch: A "Table" shape is extremely rare in nature. Most random fluctuations in the universe look like "Mountains." To get a "Table," you need a very specific, unlikely arrangement of matter.

The New Discovery: The "Sweet Spot"

The paper does a deep dive into the math to find the "Goldilocks" zone. They asked: How flat does the clump need to be to form a black hole, and how likely is that to happen?

They found a trade-off:

1. Too Pointy: It crashes (Shell-crossing) and never becomes a black hole.
2. Too Flat: It could become a black hole, but a shape that flat is so incredibly rare that it almost never happens.
3. The Sweet Spot: There is a "just right" level of flatness. It's not a perfect table, but it's flat enough to avoid crashing, and common enough to actually happen.

The Conclusion:
Even with this "sweet spot," the universe still needs a massive amount of initial "jiggle" (fluctuations) to make enough black holes to be interesting.

• The paper calculates that the universe would need to be about 100 times more "jiggly" than what we actually observe in the Cosmic Microwave Background (the afterglow of the Big Bang).
• Translation: If the universe had enough "jiggle" to make these black holes, we would see huge distortions in the early universe that we simply don't see.

Therefore: Primordial black holes forming during this "Matter Domination" era are not a very efficient way to make black holes. They are barely better than the standard "Radiation" era method.

The Spin: Why They Aren't Spinning Tops

Another question was: If these black holes form, are they spinning wildly like tops?

• Old Idea: Maybe the clump was rotating, and that rotation saved it from collapsing, making the final black hole spin fast.
• New Finding: The authors say no. The "Shell-Crossing" (the chaotic collision of particles) creates a "velocity dispersion" (a messy, random jiggling motion) that is much stronger than any organized rotation.
• Analogy: Imagine a crowd of people running into a room. If they are all running in a circle (rotation), they might spin. But if they are just bumping into each other randomly (jiggling), that chaos stops the collapse much more effectively.
• Result: The black holes that do form are likely to be slow-spinning, not fast-spinning.

Summary in One Sentence

The paper argues that while a cold, dusty early universe could theoretically make black holes, the messy nature of collapsing dust (where layers crash into each other) makes it incredibly difficult unless the universe was much "rougher" than we think it was, meaning these black holes are likely not the main source of Dark Matter.

Technical Summary

1. Problem Statement

The paper addresses the formation of Primordial Black Holes (PBHs) during an early Matter-Dominated (MD) era (e.g., an early matter-dominated phase before reheating). A central theoretical puzzle exists regarding the collapse threshold ($\delta_{th}$) for PBH formation in a dust-dominated universe ($w=0$):

• Fluid Limit ($w \to 0$): Standard analytical approaches for perfect fluids suggest that as the equation of state parameter $w$ approaches zero, the threshold for collapse $\delta_{th}$ also approaches zero ($\delta_{th} \propto w$). This implies that even tiny perturbations should collapse into black holes.
• Dust Reality ($w=0$): In a pure dust universe, perfect spherical symmetry is unstable. Without pressure to halt collapse, the inner shells of a perturbation collapse faster than outer shells, leading to shell-crossing. Once shell-crossing occurs, the perfect fluid description breaks down, and the system typically virializes into a halo rather than forming a black hole.
• The Conflict: Previous studies (e.g., Harada et al.) using "top-hat" profiles argued that velocity dispersion prevents collapse unless the initial peak is extremely large ($\zeta_{th} \sim \zeta_{rms}^{2/5}$). However, top-hat profiles are highly atypical. The authors investigate how the shape of the perturbation profile (specifically its flatness) and the presence of small-scale fluctuations ($\zeta_{rms}$) affect the collapse threshold in a realistic MD scenario.

2. Methodology

The authors employ a multi-faceted approach combining analytical derivations, peak theory statistics, and numerical simulations:

• Analytical Framework (Misner-Sharp Equations): They utilize the Misner-Sharp formalism for spherically symmetric collapse in a dust universe. They analyze the evolution of the metric and density, specifically looking at the condition for apparent horizon formation ($R \leq 2GM$) versus the onset of shell-crossing.
• Profile Shape Analysis: Instead of assuming a top-hat profile, they model perturbations using a Taylor expansion of the curvature profile $K(r)$ around the peak:
  $$K(r) \approx K(0) \left( 1 - A_2 \frac{r^2}{r_m^2} - A_4 \frac{r^4}{r_m^4} - \dots \right)$$
  They investigate how the coefficients $A_{2n}$ (representing the "flatness" or higher-order derivatives) influence the collapse time and the competition between gravitational collapse and the growth of velocity dispersion from sub-horizon perturbations.
• Peak Theory Statistics: They apply the theory of random Gaussian fields to calculate the probability distribution of peak shapes (specifically the relationship between the peak height $\nu$ and the shape parameter $x \sim A_2$). This allows them to determine which peak shapes are most likely to form PBHs given a specific abundance constraint.
• Numerical Simulations (N-body): To validate the "halting mechanism" (where velocity dispersion stops collapse), they perform Newtonian N-body simulations.
  • Top-hat backgrounds: Simulating particles with Poisson noise and grid-based initial conditions to measure the minimum compression factor ($b_{min}$) before re-expansion.
  • Gaussian profiles: Simulating particles with Gaussian density profiles to demonstrate that shell-crossing naturally halts collapse for typical profiles ($A_2 \sim O(1)$).

3. Key Contributions and Results

A. The Role of Profile Shape and Shell-Crossing

The authors demonstrate that for a typical peak (where the second derivative $A_2 \sim O(1)$), the collapse is halted by shell-crossing before a black hole can form. The inner shells pass through the center and re-expand, leading to virialization (halo formation) rather than a black hole.

• Result: For typical profiles, the threshold is effectively $\delta_{th} \sim O(1)$.
• Exception: Only exceptionally flat peaks (where $A_2 \ll 1$) can avoid early shell-crossing and collapse to a black hole. The flatter the peak, the lower the threshold, but such peaks are exponentially rare.

B. The Effective Threshold ($\zeta_{th}$)

By balancing the rarity of flat peaks against their lower collapse threshold, the authors derive the effective threshold required to produce a cosmologically relevant abundance of PBHs ($\beta$).

• They find that the dominant contribution to PBH abundance comes from peaks with significantly small second derivatives but typical higher-order derivatives.
• Key Formula: The effective threshold is:
  $$\zeta_{th} \sim \zeta_{rms}^{1/10}$$
  This is significantly larger than the top-hat estimate of $\zeta_{rms}^{2/5}$ but still implies that the required amplitude $\zeta_{rms}$ must be large.

C. Required Amplitude and Comparison to Radiation Domination

To achieve a PBH abundance capable of explaining Dark Matter or gravitational wave events, the required root-mean-square fluctuation amplitude is:
$$\zeta_{rms} \sim 0.1$$
This value is orders of magnitude larger than the observed CMB scales ($\sim 10^{-5}$).

• Significance: The authors conclude that PBH formation during Matter Domination is barely more efficient than during Radiation Domination. The requirement for $\zeta_{rms} \sim 0.1$ makes the MD scenario no more viable for generating PBHs than the standard RD scenario.

D. Spin Parameter

The paper re-evaluates the dimensionless spin parameter ($a_{*}$) of PBHs formed in MD.

• Mechanism: While Tidal Torque Theory suggests angular momentum growth, the authors argue that velocity dispersion (induced by shell-crossing and sub-horizon perturbations) is the dominant factor halting collapse, not angular momentum.
• Result: The spin is typically small:
  $$a_{*} \sim \zeta_{rms}^{7/4} \ll 1$$
  For $\zeta_{rms} \sim 0.1$, $a_{*} \sim 0.01$. This contradicts earlier claims that MD-formed PBHs might have high spins.

E. Clustering and Bias

The authors analyze the clustering of these PBHs. While long-wavelength modes can bias PBH formation (similar to halo formation), they calculate that the bias factor is large ($b_1 \sim 10^2$) but the underlying probability of finding a second PBH is so low ($\sim 10^{-35}$) that the enhancement is negligible. Thus, MD-formed PBHs remain effectively unclustered (Poissonian).

4. Significance

• Resolution of the $w=0$ Paradox: The paper resolves the discrepancy between the $w \to 0$ limit (where $\delta_{th} \to 0$) and the $w=0$ dust case (where $\delta_{th} \sim O(1)$) by showing that shell-crossing is the physical mechanism that prevents collapse for typical profiles.
• Realistic Constraints: It provides a more realistic estimate of PBH formation thresholds by moving beyond idealized top-hat profiles to include the statistical distribution of peak shapes.
• Observational Implications:
  • The requirement of $\zeta_{rms} \sim 0.1$ suggests that MD scenarios are not a "free lunch" for PBH Dark Matter; they require significant enhancement of primordial fluctuations.
  • The prediction of low spin ($a_* \sim 0.01$) offers a distinct observational signature for future gravitational wave detectors (like LISA or Einstein Telescope) to distinguish between PBHs formed in MD vs. other mechanisms.
• Methodological Advance: The combination of analytical peak theory with N-body simulations to model the transition from collapse to virialization provides a robust framework for studying non-linear structure formation in the early universe.

In summary, the paper argues that while an early matter-dominated era allows for PBH formation, the efficiency is heavily suppressed by the shape-dependence of the collapse threshold and the inevitability of shell-crossing for typical perturbations, resulting in a scenario that is not significantly more favorable than the standard radiation-dominated case.