Pair counting without binning – a new approach to correlation functions in clustering statistics

Abstract

This paper presents a novel perspective on correlation functions in the clustering analysis of the large-scale structure of the universe. We begin with the recognition that pair counting in bins of radial separation is equivalent to evaluating counts-in-cells (CIC), which can be modelled using a filtered density field with a binning-window function. This insight leads to an in situ expression for the two-point correlation function (2PCF). Essentially, the core idea underlying our method is to introduce a window function to define the binning scheme, enabling pair-counting without binning. This approach develops an idea of generalised 2PCF, which extends beyond conventional discrete pair counting by accommodating non-sharp-edged window functions. In the context of multiresolution analysis, we can implement a fast algorithm to estimate the generalised 2PCF. To extend this framework to N-point correlation functions (NPCF) using current optimal edge-corrected estimators, we developed a binning scheme that is independent of the specific parameterisation of polyhedral configurations. In particular, we demonstrate a fast algorithm for the three-point correlation function (3PCF), where triplet counting is accomplished by assigning either a spherical tophat or a Gaussian filter to each vertex of triangles. Additionally, we derive analytical expressions for the 3PCF using a multipole expansion in Legendre polynomials, accounting for filtered field (binning) corrections. Our method provides an exact solution for quantifying binning effects in practical measurements and offers a high-speed algorithm, enabling high-order clustering analysis in extremely large datasets from ongoing and upcoming surveys such as Euclid, LSST, and DESI.