2D-FFTLog: Efficient computation of real space covariance matrices for galaxy clustering and weak lensing

TL;DR

Accurate covariances for two-point functions are essential for cosmological inference but challenging to compute in real space due to double Bessel integrals. The authors present 2D-FFTLog, a two-dimensional generalization of FFTLog, to efficiently compute non-Gaussian real-space covariances for both 3D and projected statistics, with a robust bin-averaging scheme. They validate the method on DES Y3-like and LSST Y1-like surveys, showing agreement with curved-sky covariances and negligible bias in simulated likelihood analyses, while achieving significant speedups ($O(N^2 \log N)$). They release CosmoCov/CosmoLike implementations, enabling rapid covariance calculations for current and future surveys.

Abstract

Accurate covariance matrices for two-point functions are critical for inferring cosmological parameters in likelihood analyses of large-scale structure surveys. Among various approaches to obtaining the covariance, analytic computation is much faster and less noisy than estimation from data or simulations. However, the transform of covariances from Fourier space to real space involves integrals with two Bessel integrals, which are numerically slow and easily affected by numerical uncertainties. Inaccurate covariances may lead to significant errors in the inference of the cosmological parameters. In this paper, we introduce a 2D-FFTLog algorithm for efficient, accurate and numerically stable computation of non-Gaussian real space covariances for both 3D and projected statistics. The 2D-FFTLog algorithm is easily extended to perform real space bin-averaging. We apply the algorithm to the covariances for galaxy clustering and weak lensing for a Dark Energy Survey Year 3-like and a Rubin Observatory's Legacy Survey of Space and Time Year 1-like survey, and demonstrate that for both surveys, our algorithm can produce numerically stable angular bin-averaged covariances with the flat sky approximation, which are sufficiently accurate for inferring cosmological parameters. The code CosmoCov for computing the real space covariances with or without the flat sky approximation is released along with this paper.

Figures (3)

• Figure 1: DES Y3 3$\times$2pt real space flat sky correlation matrices from our FFT method (lower triangle) and the direct quadrature (Quad) integration (upper triangle). The matrices are 900$\times$ 900, and the blocks of covariances between different probes are annotated.
• Figure 2: The SNRs of eigenvalues of the DES Y3 curved sky covariance (upper left) and the SNRs of SVs of the LSST Y1 curved sky covariance (upper right), sorted by magnitude. The eigenvalues and SVs are also compared to the flat sky FFT and Quad covariances, respectively (lower left and right). For each survey, the flat sky eigenvalues and SVs match the curved sky eigenvalues and SVs within a few percent. We only show the 100 (out of 900) eigenvalues of the highest SNRs for DES Y3, and 200 (out of 1560) SVs for LSST Y1.
• Figure 3: The 1$\sigma$ and 2$\sigma$ contours of fitting the simulated 2x2pt data vector with the model, using bin-averaged flat sky covariances and the curved sky covariances for DES Y3 (left) and LSST Y1 (right). For DES Y3, we have two versions of flat sky covariances from the direct quadrature (Quad) integration and our 2D-FFTLog algorithm. For LSST Y1, we only compare the FFT result to the curved sky result.