Evolution of a periodic eight-black-hole lattice in numerical relativity

Abstract: The idea of black-hole lattices as models for the large-scale structure of the universe has been under scrutiny for several decades, and some of the properties of these systems have been elucidated recently in the context of the problem of cosmological backreaction. The complete, three-dimensional and fully relativistic evolution of these system has, however, never been tackled. We explicitly construct the first of these solutions by numerically integrating Einstein's equation in the case of an eight-black-hole lattice with the topology of S3.

Overview of "Evolution of a Periodic Eight-Black-Hole Lattice in Numerical Relativity"

This paper presents a numerical study on the concept of black-hole lattices, specifically focusing on the evolution of an eight-black-hole lattice within the framework of numerical relativity. The work is significant because it constructs and evolves a well-defined initial configuration of black holes arranged in a cubic lattice on a topological sphere, $S^3$. This approach offers novel insights into foundational questions related to cosmological backreaction and inhomogeneities in the universe.

Initial Data Construction

The study begins with an explicit construction of initial data for a periodic lattice of eight black holes using the Lichnerowicz-York conformal transverse-traceless decomposition. The authors utilize a3-sphere conformal metric to ensure positive scalar curvature, necessary for the initial configuration in the absence of an extrinsic curvature. With this setup, the initial data are equivalent to slices of $S^3$, implemented through stereographic projection into an asymptotically flat three-space setting.

Numerical Evolution and Formation of Horizons

Using the Einstein Toolkit, the authors evolve this lattice setup to analyze gravitational dynamics over time. This evolution was carried out using a fourth-order accurate finite difference scheme, allowing for investigations into horizon formation and mergers. The study tracked marginally outer trapped surfaces (MOTSs), which provide insight into the behavior and interaction of individual black holes in the lattice. The numerical results illustrated significant phenomena such as the merging of black hole horizons and confirmed their progressive approach, suggesting gravitational interactions in the topology-augmented modulated space.

Implications for Cosmological Modeling

One of the paper's key objectives is to assess how periodic black-hole lattices can serve as models for the large-scale universe. The research explores the concept of "effective" geometry by comparing their configurations to a closed Friedmann-Lemaître-Robertson-Walker (FLRW) universe model. This exploration helps gauge how well a discretely modeled, inhomogeneous universe can mimic the dynamics of a traditional, homogeneous cosmological model. It considers scale factors, edge lengths for discrete lattice "cells", and effective Ricci curvatures, providing a quantitative bridge between idealized cosmological models and realistic, structure-filled universes.

Numerical and Theoretical Considerations

The numerical analysis highlights the challenges of resolving dynamic spacetimes involving multiple black holes, emphasizing adaptive mesh refinement's usefulness. The study also reflects on limitations, including resolution constraints and the loss of well-defined spacetime structures over long simulation times or at high curvature regions.

Future Directions

This paper opens new avenues for numerical and analytical explorations in gravitational physics and cosmology, particularly in refining and understanding the role of inhomogeneities at cosmological scales. Further research would benefit from enhanced resolutions and extended simulations to examine precise backreaction effects, potential universal scaling laws, and implications for dark sector concepts in cosmology.

Conclusion

This comprehensive exploration into eight-black-hole lattices demonstrates numerical relativity's potential for bridging theoretical frameworks with observed cosmic structures. It further contributes valuable computational methods and insights into the broader discourse on the universe's large-scale topology and dynamics.