3-Point Function & BAO

Some history of past and current 3PCF work by my group and others.

Here, we will focus on the 3PCF. There is an enormous amount of theoretical work on the bispectrum, which may be added in the future. A 3PCF review talk ZS gave at the Canary Islands Summer School on Fuerteventura in 2017 is here

3PCF Algorithms

Counts-in-Cells

The 3PCF in 3D has been considered very expensive to measure. Early algorithms measured the projected (2D) 3PCF and used counts-in-cells to avoid direct counting of triplets; for instance, Peebles and Groth 1975, 1977 and Fry and Peebles 1980.

kd-tree-Based Algorithms

More recently, kd-trees were used to try to accelerate the problem (Zhang & Pen 2004, Moore et al. 2001,  Gray et al. 2004, March 2013). Tree codes struggle on the cosmological scales of interest for BAO and where perturbation theory is most applicable, as the galaxy clustering is nearly random on these scales, while kd-trees offer acceleration mainly for highly clustered datasets. 

Spherical Harmonic-Based Algorithms

Our group has developed a series of algorithms based on Legendre expansion of the 3PCF's dependence on its internal opening angle (the basis first proposed by Szapudi 2004). This algorithm exploited the spherical harmonic addition theorem as well as changing the order of operations between triangle construction and binning to enable an order-N^2 approach. The first work showed how to do this in configuration space, and the second that it could be further sped up using the Convolution Theorem and Fourier Transforms.  

Ultimately, this work provoked generalization of the Legendre polynomials to larger numbers of arguments, and led to RN Cahn and ZS developing the isotropic basis functions (which for two arguments are Legendre polynomials, up to normalization and phase). These in turn led to the ENCORE NPCF algorithm and its GPU-based successor, CADENZA.

Other Approaches

Sabiu et al. (2019) gave an approach based on a graph spatial database that enabled computation of the BOSS 3PCF and small-scale 4PCF.

Anisotropic 3PCF Algorithms

The first algorithm for measuring the anisotropic (line-of-sight dependent) 3PCF was Slepian and Eisenstein 2017; this algorithm was then implemented in Friesen et al. (incl. Slepian) 2017, the GALACTOS code. Since then, my group has improved the line of sight definition in this framework in Garcia and Slepian 2022, which shows how to use the center of a galaxy triplet to define the line of sight yet remain factorizable and thus fast. 

In parallel, Sugiyama and collaborators have developed a formalism using anisotropic 3-argument basis functions to measure the anisotropic 3PCF and bispectrum. The series of papers includes the algorithm and measurements (first detection of non-zero anisotropic bispectrum, using BOSS data, joint anisotropic 2- and 3PCF analysis on mocks, constraints on Modified Gravity, and test of large-scale structure consistency relations); their code is here. 

Umeh (2021) has also developed an efficient scheme to estimate the anisotropic 3PCF multipoles, using FFTLog. 

3PCF Measurements

In 1975, Peebles and Groth did so on the Zwicky galaxy catalog in 2D using counts-in-cells to avoid direct counting of triplets, and in 1977 on Shane-Wirtaanen catalog data. In 1980 Fry and Peebles measured the cluster-galaxy-galaxy 3PCF. 

Measurements of the quasar 3PCF were made in 1991, and for galaxies, by Jing and Borner (Las Campanas redshift survey, 1998), Kayo et al. (SDSS, 2004), Pan and Szapudi (2dFGS, 2005), Nichol et al. (SDSS, 2006; used kd-tree algorithm) Guo et al. (SDSS DR7, 2007), Marin et al. (SDSS DR7 LRGs, 2011), McBride et al. (SDSS, 2011; used kd-tree algorithm, second paper in the series here) and Guo et al. (2015). VIPERS' 3PCF was measured in 2017 (Moresco et al.).

BAO in the 3PCF

Some of Szapudi's (2004) figures (e.g. Fig. 2) show clear evidence of BAO wiggles in the theory model for the 3PCF in the Legendre basis.  Some evidence was found for BAO in the 3PCF in a measurement on SDSS DR9 LRGs by Gaztanaga et al. (2009). 

My work made the first detection of BAO in the 3PCF, measuring the large-scale SDSS BOSS CMASS 3PCF in a series of two papers in 2017 (first in a compressed basis where one triangle side was integrated out and then the full basis). We also constrained the baryon-dark matter relative velocity term in galaxy biasing using these 3PCF measurements in 2017. 

A first detection of BAO in the bispectrum, using SDSS BOSS, was made around the same time (Pearson and Samushia 2018). A first detection of BAO in the 3PCF of galaxy clusters was made by Moresco et al. in 2021.

3PCF Analysis Robustness

Hoffmann et al. (2018) used simulations to analyze consistency of the parameter constraints one would infer from 3PCF vs. from bispectrum. Veropalumbo et al. (2022) used simulations to further explore the impact of covariance and modeling on 3PCF parameter constraints.

3PCF Basis

On the mathematical basis side, Szapudi (2004) suggested decomposing the 3PCF Legendre polynomials in triangle opening angle, with a projected treatment in the analogous basis given by Zheng (2004). 

My work from 2015 on pursued the idea of using the Legendre basis for the 3PCF in both algorithms and modeling, eventually leading to the isotropic basis for N-Point Correlation Functions Cahn and I proposed in 2021.

3PCF Modeling

Modeling work using both the hierarchical ansatz and perturbation theory was done by Peebles, Fry, and collaborators throughout the 1980s: Monte Carlo modeling (Fry and Peebles 1980), hierarchical amplitudes (Fry 1984 papers I and II), relationship of NPCFs to the PDF (1985), 3PCF for biasing (Melott and Fry 1986), from N-body simulations (Fry et al. 1992 and 1993), and separating gravitational clustering and galaxy biasing (Fry 1994). Work on the hierarchical ansatz and related ideas is reviewed in Suto (1993), and the scale dependence of the 3PCF explored with N-body simulations in Matsubara and Suto (1994). 

Buchalter at al. modeled the angular 3PCF (2000). Further modeling was done by Takada and Jain (2003), Wang et al. (2004), and Marin et al. (2008). 

We presented a model of the tree-level 3PC including RSD and local quadratic and tidal tensor biasing in Slepian and Eisenstein (2017). 

An efficient implementation of the sub-leading order 3PCF model using FFTLog was suggested in Guidi et al. (2023).

3PCF Beyond Galaxies

Sensitivity of the Lyman-alpha forest 3PCF to a fluctuating UV background was found in simulations in 2019 by Tie et al., and the use of 3PCF to probe reionization was suggested in 2020 by Jennings et al.

3PCF for the Future

More discussion of 3PCF can be found in a Letter of Interest (LOI) on higher-order statistics submitted by myself and others to Snowmass 2021.