Efficient Computation of $N$-point Correlation Functions in $D$ Dimensions

Abstract

We present efficient algorithms for computing the N -point correlation functions (NPCFs) of random fields in arbitrary D-dimensional homogeneous and isotropic spaces. Such statistics appear throughout the physical sciences, and provide a natural tool to describe a range of stochastic processes. Typically, NPCF estimators have O(n N ) complexity (for a data set containing n particles); their application is thus computationally infeasible unless N is small. By projecting onto a suitably-defined angular basis, we show that the estimators can be written in separable form, with complexity O(n 2 ), or O(ng log ng) if evaluated using a Fast Fourier Transform on a grid of size ng. Our decomposition is built upon the D-dimensional hyperspherical harmonics; these form a complete basis on the (D − 1)-sphere and are intrinsically related to angular momentum operators. Concatenation of (N − 1) such harmonics gives states of definite combined angular momentum, forming a natural separable basis for the NPCF. In particular, isotropic correlation functions require only states with zero combined angular momentum. We provide explicit expressions for the NPCF estimators as applied to both discrete and gridded data, and discuss a number of applications within cosmology and fluid dynamics. The efficiency of such estimators will allow higher-order correlators to become a standard tool in the analysis of random fields.

A First Detection of the Connected 4-Point Correlation Function of Galaxies Using the BOSS CMASS Sample

“…In this work, we present a practical method for measuring the nextorder statistic, the 4PCF, and quantify its signal-to-noise in the BOSS dataset. In particular, we extend the NPCF algorithms of [48,49] to measure the connected 4PCF by removing the Gaussian piece (which is degenerate with the 2PCF) at the estimator level. This is unlike previous works, and ensures that our measurement is specifically one of non-Gaussianity, rather than a recapitulation of known physics.…”

“…We begin with a summary of the isotropic 4PCF estimator of [48,49], including discussion of the relevant angular basis [50]. Note that this estimates both the disconnected and connected 4PCF; the removal of the former piece is described in §III.…”

“…Firstly, estimation of the statistic is non-trivial, with naïve approaches having O(N 4 g ) complexity when applied to a dataset containing N g galaxies. Here, we apply the recently-proposed O(N 2 g ) NPCF estimators [48,49] which achieve a significant speed-boost by projecting the statistic onto a separable angular basis obtained from the theory of angular momentum addition [50]. Coupled with a new procedure for removing the disconnected component at the estimator level, this allows the connected isotropic 4PCF to be computed in only a few tens of CPU-hours, including survey geometry corrections.…”

The Power of Locality: Primordial Non-Gaussianity at the Map Level

“…This strategy has been successfully employed on current BOSS data to measure cosmological parameters using the power spectrum [78][79][80] and bispectrum [80][81][82][83][84]. In principle, one can imagine measuring successively higher-order correlators [85,86] to extract additional information from the maps and we anticipate that higher-point information will become increasingly valuable with larger surveys [87][88][89].…”

Modeling Nearest Neighbor distributions of biased tracers using Hybrid Effective Field Theory

“…This was explored, for a similar model to HEFT, in the context of halo samples in Schmittfull et al (2019) and for redshift-space samples of galaxies in Schmittfull et al (2020). Nevertheless, the computational complexity of 𝑁 > 2-point analyses is quite large (however, see Philcox & Slepian (2021) for recent progress in this direction), and so we seek an alternative summary statistic that encodes higher order information, to test the applicability of HEFT. Banerjee & Abel (2021a,b) have proposed a set of new summary statistics to capture the auto and joint clustering of various tracer samples in the context of cosmology.…”