⚛️ general relativity

Simulating dust collapse: the impact of gauge choices on primordial black hole formation

This paper investigates why numerical relativity simulations of dust collapse often fail before primordial black hole formation, demonstrating that adapting lapse slicing equations and implementing spatially varying shift damping to address shell-focusing instabilities significantly improves simulation stability and proximity to black hole formation.

Original authors: Robyn L. Munoz, Christian T. Byrnes, Marco Bruni, Ian Hawke

Published 2026-10-19

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Robyn L. Munoz, Christian T. Byrnes, Marco Bruni, Ian Hawke

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast, expanding history of our universe, there are moments when gravity wins. When a region of space becomes dense enough, the pull of matter can overcome the outward push of cosmic expansion, causing that region to collapse inward. If the collapse is strong enough, it can crush matter into a point so dense that not even light can escape, forming a black hole. While we know black holes exist today, formed from the death of massive stars, physicists are deeply interested in a different kind: primordial black holes. These would have formed in the very first moments after the Big Bang, created not by dying stars, but by the raw, chaotic density of the early universe itself. Finding out how often these ancient black holes formed helps scientists understand the conditions of the infant cosmos and whether these invisible objects could be the mysterious "dark matter" that holds galaxies together.

To study this, researchers must simulate the collapse of matter under the extreme rules of general relativity. However, simulating a universe filled with "dust"—a term physicists use for matter that has no internal pressure to resist gravity—is notoriously difficult. In a real star, pressure fights against gravity, but in this early universe scenario, there is no pressure. When a cloud of such dust collapses, the particles do not just slow down; they crash into one another, creating a mathematical singularity where the simulation breaks down. This is like trying to calculate the path of two cars that are driving straight at each other with no brakes; the moment they meet, the math explodes. For decades, computer simulations of this process have failed just as the black hole was about to form, crashing before the event horizon could appear.

A team of researchers set out to solve this specific problem: how to keep a computer simulation running long enough to see a primordial black hole form. They used powerful supercomputers to model the collapse of a dust cloud in a matter-dominated era, a time when the universe was filled with slow-moving particles rather than the fast-moving radiation that dominated earlier. Their goal was not just to watch the collapse, but to understand why the simulations were failing and to find a way to fix the computer code so it could survive the crash. They discovered that the failure was not due to a lack of computing power, but to the way the simulation was "sliced" in time and space. In relativity, time and space are flexible, and the choice of how to measure them—called a "gauge"—can make the difference between a stable simulation and a broken one.

The researchers found that the standard methods used to simulate black holes in empty space did not work for a universe that is expanding. In a typical black hole simulation, the code can freeze time as it approaches the center of the collapse, preventing the math from breaking. But in an expanding universe, the background space itself is stretching, which confuses the standard time-freezing techniques. The simulation would either speed up uncontrollably or fail to slow down enough to handle the crushing density at the center. The team tested many different ways to adjust these time and space measurements. They tried new mathematical rules for how the "clock" of the simulation should tick and how the "grid" of space should shift. They found that a specific new rule, which smoothly transitions between two different ways of measuring time, allowed the simulation to survive the initial chaos of the collapse much longer than before.

However, surviving the collapse was only half the battle. The other major hurdle was the "shift," which controls how the grid points in the simulation move. As the dust collapses, the grid points need to spread out to avoid bunching up at the center, where the density becomes infinite. The researchers discovered that the standard way of moving these grid points failed when the dust focused into a single point. They found that the amount of "damping," or resistance, applied to the movement of the grid points had to be carefully tuned. If the damping was too strong or too weak, the simulation would crash. Crucially, they realized that a single setting for this damping did not work for the entire simulation. The best results came when the damping changed depending on how finely the simulation was divided into smaller pieces, a technique known as mesh refinement.

With these improved tools, the team pushed their simulations further than ever before. They watched the density of the dust rise to extreme levels, and the space around it warped significantly. They looked for signs that a black hole was forming, such as a region where light could no longer escape. Their calculations showed that the Schwarzschild radius—the size a black hole would have for that amount of mass—was growing and, in some directions, becoming larger than the actual size of the collapsing cloud. This is a strong hint that a black hole was forming. They also tracked the behavior of light rays and the curvature of space, finding that the patterns were beginning to match what is expected for a black hole.

Despite these successes, the simulations still did not run long enough to definitively confirm the formation of a stable black hole. The code eventually crashed, but it lasted significantly longer than previous attempts, getting much closer to the moment of formation. The researchers concluded that while they had made major progress, the problem of the "shell-focusing" singularity—where the dust collapses into a point—remains a stubborn barrier. They found that the current methods for moving the grid points are not quite right for this specific type of collapse. To truly see a primordial black hole form in a computer, they suggest that future simulations will need even more sophisticated ways to adjust the grid as the collapse happens, perhaps changing the rules of the simulation dynamically as the density increases.

The work highlights a fundamental challenge in understanding the early universe: the tools we have are powerful, but they are not yet perfect for every scenario. By identifying exactly where and why the simulations fail, the researchers have provided a clear roadmap for the next generation of studies. They have shown that with the right adjustments to how time and space are measured in the code, it is possible to push the simulation closer to the edge of a black hole's formation. While the final step of confirming a primordial black hole in a simulation remains just out of reach, the path forward is now visible. The next step is to refine the shifting of the grid points so that the simulation can hold its ground against the crushing weight of the collapsing dust, finally allowing scientists to watch the birth of these ancient, invisible giants.

Technical Summary: Simulating Dust Collapse and the Impact of Gauge Choices on Primordial Black Hole Formation

Problem Statement
Numerical relativity simulations of dust collapse in a matter-dominated era frequently fail before primordial black hole (PBH) formation can be confirmed. This failure stems from two primary issues: numerical limitations inherent to pressureless fluids and inappropriate gauge choices. Specifically, the absence of pressure leads to shell-crossing (where particles with opposing velocities cross, creating shock fronts) and shell-focusing (where matter converges to a point-like or cigar-like singularity). In the context of PBH formation, shell-focusing causes the simulation to crash as it evolves toward a central singularity. Furthermore, standard gauge conditions used in asymptotically flat black hole simulations (e.g., 1+log slicing and Gamma-driver shift) are not directly transferable to cosmological contexts where the background universe is expanding. The challenge is to adapt these gauges to allow simulations to evolve long enough to potentially reach the threshold of black hole formation without succumbing to numerical instabilities or premature crashes.

Methodology
The authors employ 3-dimensional fully non-linear numerical relativity simulations using the Einstein Toolkit. The study focuses on a pressureless dust fluid (perfect fluid with zero pressure) evolving from an Einstein-de Sitter background with a super-horizon perturbation.

Initial Conditions: The simulations begin with a homogeneous FLRW metric perturbed by a sinusoidal mode in the synchronous comoving gauge. The perturbation is defined via the gauge-invariant comoving curvature perturbation $R_c$ , with an initial maximum density contrast of $\delta \approx 0.03$ .
Numerical Schemes: To handle the steepening fields and potential shock formation, the authors utilize High-Resolution Shock Capturing (HRSC) methods (specifically the piecewise parabolic reconstruction and HLLE Riemann solver via the GRHydro code). They also implement Kreiss-Oliger dissipation to dampen numerical oscillations that could lead to overflow errors.
Gauge Evolution: The core of the investigation involves varying the evolution equations for the lapse ( $\alpha$ $α$ ) and shift ( $\beta^i$ $β^{i}$ ):
- Lapse: The authors test various slicing conditions based on the Bona-Masso family. They compare standard 1+log slicing, shock-avoiding slicing, and cosmological adaptations that subtract the background expansion (e.g., $K - K_{FLRW}$ ). They propose a new evolution equation using a softplus function to smoothly transition between synchronous behavior (during expansion) and puncture-avoiding behavior (during collapse).
- Shift: The study investigates the Gamma-driver shift condition, specifically tuning the damping parameter $\eta$ . The authors explore whether a fixed $\eta$ is sufficient or if a spatially varying damping term is required to effectively "spread out" the grid observers and avoid resolving the shell-focusing singularity too early.
- Mesh Refinement: To manage computational cost while resolving the growing non-linearities, the authors use a dynamic mesh refinement strategy (AddRefinements) that triggers higher resolution levels as the density contrast increases, rather than maintaining high resolution throughout the entire simulation.

Key Contributions

Gauge Adaptation for Cosmology: The paper demonstrates that standard black hole gauges fail in cosmological dust collapse without modification. The authors propose a novel lapse evolution equation utilizing a softplus function ( $g_{s+}$ ) to handle the transition from cosmological expansion to gravitational collapse, preventing the lapse from growing unphysically large during the expansion phase while allowing it to decrease during collapse.
Shift Damping and Resolution Dependence: A critical finding is that the optimal shift damping parameter $\eta$ is not universal but depends on the mesh refinement level. Higher resolution levels require significantly lower damping values to remain stable, suggesting that a spatially varying or resolution-dependent damping strategy is necessary for shell-focusing singularities.
Stabilization Techniques: The combination of HRSC, dissipation, and the proposed gauge choices allows simulations to evolve significantly further than previous attempts, reaching density contrasts and proper times close to the theoretical collapse threshold.
Diagnostic Analysis: The authors apply multiple diagnostics to assess black hole formation, including the Hoop conjecture (comparing Schwarzschild radius to circumferential radius), outgoing null ray expansion ( $\Theta_{out}$ ), and the decomposition of the Kretschmann scalar into Ricci and Weyl contributions.

Results

Simulation Stability: Simulations using the synchronous gauge crash immediately upon shell-focusing. The inclusion of HRSC and dissipation extends stability only when paired with a puncture-style gauge. The proposed softplus lapse evolution combined with tuned shift damping allows the simulation to proceed past the turnaround point and deep into the collapse phase.
Lapse Behavior: The new lapse equation successfully prevents the lapse from growing to values ( $\alpha \sim 20$ ) that destabilize the simulation during the expansion phase. However, in the final stages, the lapse begins to increase again in certain regions due to the oscillation of the extrinsic curvature trace around zero, eventually causing the simulation to stop.
Black Hole Indicators:
- Hoop Conjecture: In the best-performing simulations, the ratio of the Schwarzschild radius to the circumferential radius ( $2M/\Re$ ) exceeds unity in certain directions, suggesting the formation of a trapped region.
- Null Ray Expansion: Some grid points in the central region show $\Theta_{out} < 0$ , indicating a trapped region. However, the resolution is insufficient to numerically resolve a continuous apparent horizon surface.
- Curvature Invariants: The Kretschmann scalar evolves toward the Schwarzschild profile ( $48M^2/r^6$ ). The decomposition shows a transition from Ricci-dominated curvature (matter-dominated) to Weyl-dominated curvature (vacuum-like), a signature of black hole formation. However, at the very center ( $r \to 0$ ), the simulations remain Ricci-dominated, indicating that the Weyl-dominated phase characteristic of a formed horizon has not yet been fully reached.
Failure to Resolve: Despite reaching proximity to black hole formation conditions, the simulations ultimately crash. The authors attribute this to the inability of the current shift evolution to indefinitely postpone the shell-focusing singularity without introducing numerical instabilities, and the lapse equation's inability to maintain a stable decrease to zero in the final moments.

Significance and Claims
The paper modestly claims that while it has not definitively simulated the formation of a primordial black hole from dust collapse, it has significantly advanced the understanding of why such simulations fail. The work characterizes the specific interplay between gauge choices, numerical resolution, and the shell-focusing singularity.

The authors assert that their improved gauge choices (specifically the softplus lapse and resolution-dependent shift damping) allow simulations to evolve "significantly further" than previous attempts, reaching a state where black hole diagnostics begin to show positive indicators. However, they conclude that a stable, long-term simulation capable of resolving the apparent horizon requires further refinement of the shift evolution equation, particularly regarding spatially varying damping terms. The paper positions itself as a necessary step toward developing a "shell-focusing singularity-avoiding gauge" suitable for cosmological dust collapse, noting that future work must address the specific nature of the singularity in dust fluids, which differs from the vacuum singularities typically studied in black hole mergers.

Technical Summary: Simulating Dust Collapse and the Impact of Gauge Choices on Primordial Black Hole Formation

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