Multi-scale initial conditions for cosmological simulations

Abstract: We discuss a new algorithm to generate multi-scale initial conditions with multiple levels of refinements for cosmological "zoom-in" simulations. The method uses an adaptive convolution of Gaussian white noise with a real space transfer function kernel together with an adaptive multi-grid Poisson solver to generate displacements and velocities following first (1LPT) or second order Lagrangian perturbation theory (2LPT). The new algorithm achieves RMS relative errors of order 10^-4 for displacements and velocities in the refinement region and thus improves in terms of errors by about two orders of magnitude over previous approaches. In addition, errors are localized at coarse-fine boundaries and do not suffer from Fourier-space induced interference ringing. An optional hybrid multi-grid and Fast Fourier Transform (FFT) based scheme is introduced which has identical Fourier space behaviour as traditional approaches. Using a suite of re-simulations of a galaxy cluster halo our real space based approach is found to reproduce correlation functions, density profiles, key halo properties and subhalo abundances with per cent level accuracy. Finally, we generalize our approach for two-component baryon and dark-matter simulations and demonstrate that the power spectrum evolution is in excellent agreement with linear perturbation theory. For initial baryon density fields, it is suggested to use the local Lagrangian approximation in order to generate a density field for mesh based codes that is consistent with Lagrangian perturbation theory instead of the current practice of using the Eulerian linearly scaled densities.

• The paper introduces a novel algorithm that reduces displacement and velocity errors to the order of 10⁻⁴ in refinement regions.
• It employs adaptive convolution with a real space transfer function kernel and a multi-grid Poisson solver to balance computational cost and precision.
• Galaxy cluster halo resimulations validate the approach, achieving percent-level agreement with key halo characteristics and consistent power spectrum evolution.

An Overview of "Multi-scale initial conditions for cosmological simulations"

The paper "Multi-scale initial conditions for cosmological simulations" by Oliver Hahn and Tom Abel introduces a novel algorithm for generating multi-scale initial conditions equipped to handle cosmological "zoom-in" simulations with multiple levels of refinements. This approach integrates an adaptive convolution of Gaussian white noise with a real space transfer function kernel and employs an adaptive multi-grid Poisson solver. These advancements are designed to achieve precise displacements and velocity fields in line with first or second-order Lagrangian perturbation theory. The motivation is to improve over previous strategies, significantly reducing the errors in displacements and velocities within refinement regions, localized primarily at coarse-fine boundaries, thereby avoiding Fourier-space induced interference issues.

Key Contributions and Results

1. Algorithmic Advancements: The proposed algorithm achieves RMS relative errors in the order of $10^{-4}$ for displacements and velocities in refinement regions, an improvement of approximately two orders of magnitude over previous approaches. The paper emphasizes that errors are confined to boundaries, avoiding Fourier-space interference ringing.
2. Adaptive Convolution and Poisson Solver: The methodology utilizes adaptive convolution and a multi-grid Poisson solver, which balances computational cost and gravitational interaction precision across different scales of refinement. In scenarios using traditional uniform grids, computational demands increase, and velocity fields suffer from discontinuities—both of which this paper addresses effectively.
3. Hybrid Poisson Solver: A hybrid Poisson solver combining multi-grid and FFT-based schemes retains the Fourier space characteristics of conventional approaches. This solver is particularly adept on the finest mesh, maintaining properties typical of uni-grid simulations while preserving the corrected small-scale power distribution.
4. Simulations and Validation: Through a detailed suite of resimulations of a galaxy cluster halo, the authors demonstrate their approach's robustness and accuracy. The results show per cent level agreement with key halo characteristics and correlation functions, substantiating the method's practical viability.
5. Two-component Baryon and Dark-Matter Simulations: The approach is extended to handle two-component simulations, maintaining the expected power spectrum evolution. The paper suggests utilizing the local Lagrangian approximation to initialize baryon density fields, deviating from the commonly adopted Eulerian linear approach. This adjustment offers consistency with Lagrangian perturbation theory, as demonstrated by comparing against linear perturbation predictions.
6. Theoretical Implications: The work provides a more consistent framework for initializing cosmological simulations, handling the transition from large-scale cosmological perturbations to small-scale structure formation processes. This framework is particularly applicable to studies necessitating high resolution in localized areas without compromising the computational resource allocation or introducing non-physical artifacts.

Implications and Future Directions

The proposed method stands to significantly impact cosmological simulations requiring high-fidelity initial conditions, particularly in understanding galaxy formation and large-scale structure. It offers a pathway towards more computationally efficient and accurate simulations, making it feasible to explore scenarios that previously might have been compromised due to resource limits or methodological shortcomings.

The incorporation of this technique with ongoing advancements, such as more sophisticated physical models or hybrid computational architectures, could extend its applicability further. Future developments may explore incorporating machine learning for more optimized error correction and dynamic refinement strategies during simulations, thus advancing both theoretical and applied cosmology fronts.

In summary, Hahn and Abel's contribution presents a crucial step toward more reliable and detailed cosmological simulations, demonstrating that controlled nested grids can achieve unparalleled accuracy in representing both large-scale cosmic environments and fine-scale astrophysical structures.