Accurate Estimators of Correlation Functions in Fourier Space

TL;DR

This paper introduces a method combining interlaced grids and higher-order interpolation kernels to significantly reduce aliasing in Fourier-space statistics estimation, enabling highly accurate power spectrum and bispectrum measurements for large datasets.

Contribution

It demonstrates that interlaced grids and advanced interpolation kernels together minimize aliasing, improving the accuracy of Fourier-space estimators in cosmological data analysis.

Findings

Aliasing can be effectively reduced using interlaced grids.

Higher-order interpolation kernels further diminish residual aliasing.

Systematic biases below 0.01% achieved up to Nyquist frequency.

Abstract

Efficient estimators of Fourier-space statistics for large number of objects rely on Fast Fourier Transforms (FFTs), which are affected by aliasing from unresolved small scale modes due to the finite FFT grid. Aliasing takes the form of a sum over images, each of them corresponding to the Fourier content displaced by increasing multiples of the sampling frequency of the grid. These spurious contributions limit the accuracy in the estimation of Fourier-space statistics, and are typically ameliorated by simultaneously increasing grid size and discarding high-frequency modes. This results in inefficient estimates for e.g. the power spectrum when desired systematic biases are well under per-cent level. We show that using interlaced grids removes odd images, which include the dominant contribution to aliasing. In addition, we discuss the choice of interpolation kernel used to define density perturbations on the FFT grid and demonstrate that using higher-order interpolation kernels than the standard Cloud in Cell algorithm results in significant reduction of the remaining images. We show that combining fourth-order interpolation with interlacing gives very accurate Fourier amplitudes and phases of density perturbations. This results in power spectrum and bispectrum estimates that have systematic biases below 0.01% all the way to the Nyquist frequency of the grid, thus maximizing the use of unbiased Fourier coefficients for a given grid size and greatly reducing systematics for applications to large cosmological datasets.