[Paper Review] Advancing the matter bispectrum estimation of large-scale structure: a comparison of dark matter codes

Johnathan Hung, J. Fergusson+1 more|arXiv (Cornell University)|Feb 5, 2019

Galaxies: Formation, Evolution, Phenomena5 references6 citations

TL;DR

This paper introduces MODAL-LSS, an efficient and optimal method to reconstruct the full matter bispectrum from 3D density fields, enabling precise comparison of fast dark matter codes (e.g., 2LPT, PM) against GADGET-3 at z=0.5. It demonstrates that boosting the small-scale power in fast codes significantly improves bispectrum accuracy and reveals that non-Gaussian covariance errors plateau, invalidating simple Gaussian approximations for parameter estimation.

ABSTRACT

Cosmological information from forthcoming galaxy surveys, such as LSST and Euclid, will soon exceed that available from the CMB. Higher order correlation functions, like the bispectrum, will be indispensable for realising this potential. The interpretation of this data faces many challenges because gravitational collapse of matter is a complex non-linear process, typically modelled by computationally expensive N-body simulations. Proposed alternatives using fast dark matter codes (e.g. 2LPT or particle-mesh) are primarily evaluated on their ability to reproduce clustering statistics linked to the matter power spectrum. The accuracy of these codes can be tested in more detail by looking at higher-order statistics, and in this paper we will present an efficient and optimal methodology (MODAL-LSS) to reconstruct the full bispectrum of any 3D density field. We make quantitative comparisons between a number of fast dark matter codes and Gadget at redshift $z=0.5$. This will serve as an important diagnostic tool for dark matter/halo mock catalogues and lays the foundation for realistic high precision analysis with the galaxy bispectrum. In particular, we show that the lack of small-scale power in the bispectrum of fast codes can be ameliorated by a simple `boosting' technique for the power spectrum. We also investigate the covariance of the MODAL-LSS bispectrum estimator, demonstrating the plateauing of non-Gaussian errors in contrast to simple Gaussian extrapolations. This has important consequences for the extraction of information from the bispectrum and hence parameter estimation. Finally we make quantitative comparisons of simulation bispectra with theoretical models, discussing the initial parameters required to create mock catalogues with accurate bispectra.

Motivation & Objective

• To develop a robust, efficient, and optimal method for reconstructing the full 3D matter bispectrum from large-scale structure simulations.
• To quantitatively compare the performance of fast dark matter codes (e.g., 2LPT, PM) against high-precision GADGET-3 simulations at z=0.5 in reproducing the matter bispectrum.
• To assess the impact of small-scale power inaccuracies in fast codes and propose a 'boosting' technique to correct them.
• To investigate the non-Gaussian covariance structure of the bispectrum estimator and challenge the validity of Gaussian extrapolations.
• To provide a diagnostic framework for generating high-precision mock halo and galaxy catalogues with accurate bispectra.

Proposed method

• Proposes MODAL-LSS, a novel bispectrum reconstruction method based on spherical harmonic decomposition of the 3D density field, enabling full-spectrum estimation.
• Employs a tetrapyd-based representation of the bispectrum to efficiently store and reconstruct the 3D bispectrum coefficients across all wavevector triangles.
• Uses Fast Fourier Transforms (FFTs) and tetrapyd-based grids to compute the mode coupling matrix $\gamma_{nm}$, ensuring consistency and minimizing bias in bispectrum estimation.
• Validates the accuracy of $\gamma_{nm}$ by testing the expectation value $\langle \beta^{R}_n \rangle = 0$ and $\langle \beta^{R}_n \beta^{R}_n \rangle = 1$, ensuring unbiased and properly normalized estimators.
• Applies the method to GADGET-3 simulations and fast codes at $z=0.5$, comparing bispectra across different $k_{\text{max}}$ and grid resolutions.
• Analyzes the covariance matrix of the bispectrum estimator, identifying the plateauing of non-Gaussian errors and contrasting them with Gaussian approximations.

Experimental results

Research questions

• RQ1 How accurately can fast dark matter codes (2LPT, PM) reproduce the matter bispectrum compared to GADGET-3 at $z=0.5$?
• RQ2 To what extent does the lack of small-scale power in fast codes degrade bispectrum estimation, and can this be corrected?
• RQ3 What is the true nature of the bispectrum covariance matrix, and does it deviate significantly from Gaussian assumptions?
• RQ4 How does the MODAL-LSS method ensure unbiased and optimal reconstruction of the full bispectrum across all triangle configurations?
• RQ5 What are the minimal simulation box sizes and grid resolutions required to reliably capture the physically relevant $k$-modes in bispectrum analysis?

Key findings

• The MODAL-LSS method enables optimal, efficient, and unbiased reconstruction of the full 3D matter bispectrum, with $\langle \beta^{R}_n \rangle = 0.0001 \pm 0.0048$ and $\langle \beta^{R}_n \beta^{R}_n \rangle = 0.997 \pm 0.045$ when using FFT-based $\gamma_{nm}$, confirming high accuracy.
• The lack of small-scale power in fast codes leads to significant bispectrum inaccuracies, but this is effectively mitigated by a simple 'boosting' technique that enhances the power spectrum at small scales.
• Non-Gaussian covariance errors in the bispectrum estimator plateau at high $k$-modes, invalidating standard Gaussian extrapolations that overestimate errors and compromise parameter estimation.
• The method remains robust across different FFT grid sizes: bispectrum estimates from $2048^3$ grids are consistent with lower-resolution ones to within 2% down to $41k_F$, indicating stability.
• For reliable results, simulations must have a box size large enough to resolve $k > 41k_F$, with $256^3$ grids or larger recommended to avoid aliasing and loss of physical information.
• Using $\gamma_{nm}$ computed on the tetrapyd with $N_{\text{tetra}} = N_{\text{FFT}}$ (e.g., 1024 or more points) ensures consistency and eliminates bias, especially for large grids where FFT computation becomes impractical.