Harmony of the Spheres: Extension to All Points of an Algorithm for Producing a Density Field with Given Two-, Three-, and Four-Point Correlation Functions

TL;DR

This paper extends an algorithm to generate 3D density fields with specified higher-order correlation functions at every point, enabling improved simulations for large-scale structure surveys.

Contribution

It demonstrates that the algorithm can produce density fields with correct NPCFs measured at all points, generalizing constrained realizations to higher-order statistics.

Findings

Algorithm can generate density fields with desired NPCFs at all points.

Enables rapid simulation of large-scale structure for survey analysis.

Facilitates testing systematics and covariance estimation.

Abstract

Previous work (Slepian 2024) showed that the Smith-Zaldarriaga (2011) algorithm to realize Cosmic Microwave Background (CMB) maps with any desired harmonic-space bispectrum could be generalized to produce a 3D density field with any desired N-Point Correlation Functions (NPCFs, N = $2, 3, \ldots$) about a particular, specified set of ``primary'' points. This algorithm assured one of having the correct correlations if measured about these specific centers. Here, we show that this algorithm was more general than initially believed, and can in fact be used to produce a density field on a grid that has the correct, desired NPCFs as measured about \textit{every} point on the grid. This paper should be considered the second in the series, and now completes the quest to generalize the idea of ``constrained realization'' (Hoffman and Ribak 1991) to higher-order statistics. This algorithm will be of great use for quickly generating density fields both to produce covariance matrices, and test systematics, for current and future 3D large-scale structure surveys such as Dark Energy Spectroscopic Instrument (DESI), Euclid, Spherex, and Roman.