Simulation of PBH formation in a matter-dominated universe

Technical Summary: Simulation of PBH Formation in a Matter-Dominated Universe

Problem Statement
The formation of Primordial Black Holes (PBHs) is highly sensitive to the equation of state of the dominant cosmic component. While PBH formation during the radiation-dominated era is relatively well-understood, the process during an early matter-dominated epoch remains less certain. In a matter-dominated era, pressure gradients are negligible, allowing overdense regions to collapse more easily. However, the formation criterion is significantly influenced by non-spherical dynamics, inhomogeneity, and ellipticity. Previous analytic estimates often rely on perturbative approximations or the Zel'dovich approximation, which may not hold in the highly nonlinear regime of gravitational collapse. A major challenge in investigating this regime is the implementation of nonspherical symmetry in numerical simulations, particularly when dealing with dust fluids where shell-crossing singularities cause numerical crashes before black hole horizons can be fully resolved.

Methodology
The authors employ fully nonlinear 3+1 numerical relativity using the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism to simulate PBH formation in an expanding matter-dominated background.

• Initial Conditions: The simulations utilize a functional form of the curvature perturbation $\zeta$ that includes ellipticity, creating triaxial configurations. The initial data is constructed using long-wavelength gradient expansion solutions for a dust fluid, characterized by an amplitude parameter $\mu$ and ellipticity parameters ($e, p$).
• Gauge and Coordinates: A novel time slicing condition is adopted, transitioning from an e-folding number-based gauge to a singularity-avoiding "1+log" style slicing as the horizon entry approaches. A non-Cartesian coordinate system is used to enlarge the central region of the simulation box, improving resolution where collapse occurs.
• Matter Descriptions: Two distinct matter models are tested:
  1. Dust Fluid: A pressureless fluid description. The authors note that this description inevitably leads to shell-crossing singularities where fluid density diverges, causing the numerical code to crash.
  2. Collisionless Particles: To circumvent the fluid approximation's breakdown, the authors implement a Particle-in-Cell (PIC) approach within full numerical relativity. Particles are assigned masses to reproduce the desired density inhomogeneity, and their motion is governed by geodesic equations. This allows the simulation to continue past the point where fluid elements would cross.

Key Results

• Spherical Symmetry Validation: The code was validated against the Lemaître-Tolman-Bondi (LTB) solution for spherically symmetric dust collapse. The numerical results successfully reproduced the analytic solution up to the point of horizon formation and correctly identified the threshold between locally naked singularities (PBH formation) and globally naked singularities.
• Dust Fluid Simulations (Non-Spherical):
  • For triaxial collapse, the dust fluid simulations crash due to shell-crossing singularities before an apparent horizon can form unless the initial amplitude $\mu$ is significantly large.
  • The threshold amplitude required to observe horizon formation before the code crashes is found to be approximately twice the value predicted by previous analytic estimations based on the Zel'dovich approximation and hoop conjecture.
  • This suggests that within the valid range of the fluid approximation, PBH formation is harder to achieve than previously thought.
• Collisionless Particle Simulations:
  • When modeling the same system with collisionless particles, the simulations do not crash at shell crossing.
  • Black hole formation (indicated by apparent horizon formation) is observed for initial amplitudes as low as $\mu \approx 0.045$ (for ellipticity $e=0.2$).
  • This threshold is an order of magnitude smaller than the previous analytic estimates and significantly lower than the threshold derived from the dust fluid simulations.
  • However, the particle simulations exhibit significant violations of the Hamiltonian constraint after particle intersections occur, indicating that while the collapse proceeds, the quantitative accuracy of the late-time evolution is limited by the mismatch between the particle description and the geometry.

Significance and Claims
The paper claims to demonstrate a large theoretical uncertainty in the PBH formation criterion during a matter-dominated epoch, contingent on the physical description of the matter field:

1. Fluid Approximation Limit: If the formation of singularities associated with shell crossing in a fluid description is interpreted as a physical suppression mechanism (or if the fluid approximation breaks down before collapse), the threshold for PBH formation is significantly higher than analytic predictions, making PBH formation more difficult.
2. Particle Dynamics: If the collisionless particle description is accepted as a valid extension beyond the single-stream fluid approximation, PBH formation becomes much easier (lower threshold) than previously expected.
3. Theoretical Uncertainty: The discrepancy between the fluid and particle results highlights that the standard analytic criteria (Zel'dovich approximation) may not accurately capture the nonlinear dynamics of triaxial collapse. The authors conclude that the threshold value is highly sensitive to the treatment of the matter field at high densities.

The paper does not claim to definitively resolve the PBH abundance but rather to expose the sensitivity of formation thresholds to the underlying matter model and numerical treatment of singularities. Future work is identified as necessary to refine high-density treatments, improve constraint preservation in particle simulations, and quantify the resulting PBH abundance and gravitational wave signatures.