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 work removes a key limitation in constrained realizations of higher-order statistics by showing that a density field with prescribed $N$-point correlation functions (NPCFs) can be realized about every grid point, not just a set of separated primaries. By leveraging independent Gaussian Random Fields on each shell and applying Wick's theorem, the authors prove that cross-correlations between shells around different primaries vanish even when shells overlap, enabling a grid-wide realization of the target NPCFs. The paper provides a detailed recap of the non-Gaussian construction, extends it to cross-sphere configurations, and offers a concrete computational-cost framework showing the approach can be parallelized and potentially faster than traditional N-body or approximate mocks for covariance and systematics studies. Practically, this grid-based constrained realization can produce thousands of fast mocks matching higher-order statistics, with significant implications for covariance estimation and parity-violation searches in current and upcoming large-scale structure surveys (DESI, Euclid, Roman).

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.

Figures (2)

• Figure 1: Here we display two spheres, centered at $\vec{x}$ and $\vec{x}^{\,\prime}$, respectively (cf. Eq. 1. At each center there is a primary galaxy, while the three secondary ones are located in the shells of radius $r_i$ at a given angular coordinate $\hat{r}_i$, here shown by arrows. The correlations are measured out to a radius $R_{\rm max}$, and here we show the configuration treated in Slepian (2024), where the two spheres are taken not to overlap, so that the correlations about one primary would not interfere with those about the second. In the current work, we show this restriction is unnecessary, and that the framework of Slepian (2024) also applies to the more general case, where the two spheres overlap, as shown in Fig. \ref{['fig:two_spheres_overlap']}.
• Figure 2: Here we show the configuration under consideration in this paper, where the spheres around two primaries may overlap each other. However, because the GRFs on each shell and about each primary are independent, the red shells do not "see" the blue shells and vice versa.