3×2-Point Correlation Statistic

• 3×2-point correlation statistic is a joint analysis of two-point and three-point correlation functions that captures both Gaussian and non-Gaussian signals in cosmological datasets.
• It combines auto- and cross-correlation estimators from observables like galaxy densities and FRB dispersion measures to improve parameter constraints and model accuracy.
• Advanced FFT acceleration and local bias modeling enable efficient computation and robust handling of large-scale structure, breaking traditional degeneracies in cosmological inference.

A 3×2-point correlation statistic refers to the joint statistical analysis of two-point and three-point correlation functions (2PCF and 3PCF) in cosmological datasets, typically incorporating all auto- and cross-correlation data vectors among distinct observables. This approach enables simultaneous measurement of Gaussian (2PCF) and leading non-Gaussian (3PCF) information, thereby improving the constraining power for cosmological and astrophysical parameters. Recent developments have extended 3×2-point frameworks from galaxy clustering contexts to multi-tracer settings, such as joint analyses of fast radio burst (FRB) dispersion measures (DM) and galaxy surveys, integrating both angular and configuration-space formalisms (Slepian et al., 2015, Hoffmann et al., 2018, Sharma et al., 6 Sep 2025).

1. Formal Definitions and Observables

The 3×2-point correlation statistic comprises six fundamental estimators: the auto- and cross-correlation functions of two distinct fields (e.g., galaxy density and FRB DMs), each with their two-point (power spectrum or correlation function) and three-point (bispectrum or three-point correlation function) moments. For projected angular fields (such as those relevant to wide-area surveys), the analyis is conventionally performed in harmonic space, using the angular power spectrum $C_\ell$ and bispectrum $B_{\ell_1\ell_2\ell_3}$, though configuration-space counterparts are standard in 3D datasets.

For fields $A$ and $B$, the two-point function (2PCF) is defined as:

$$\xi_{AB}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_B(\mathbf{x}+\mathbf{r}) \rangle$$

and, for a single field (auto-correlation), as

$$\xi_{A}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_A(\mathbf{x}+\mathbf{r}) \rangle$$

The three-point function (3PCF) for field $A$ is

$$\zeta_{A}(\mathbf{r}_1, \mathbf{r}_2) = \langle \delta_A(\mathbf{x})\, \delta_A(\mathbf{x}+\mathbf{r}_1)\, \delta_A(\mathbf{x}+\mathbf{r}_2) \rangle$$

In multi-probe studies such as FRB DMs and galaxies, one forms auto-spectra $C_\ell^{AA}$, $C_\ell^{BB}$ and cross-spectrum $B_{\ell_1\ell_2\ell_3}$0, with $B_{\ell_1\ell_2\ell_3}$1 denoting angular scale. For the 3×2-point framework, the full data vector at each $B_{\ell_1\ell_2\ell_3}$2 is

$B_{\ell_1\ell_2\ell_3}$3

This triplet forms the core for likelihood analysis, forecasting, and covariance estimation (Sharma et al., 6 Sep 2025). In configuration-space, the analogous data vector is $B_{\ell_1\ell_2\ell_3}$4, supplemented in 3×2-point analyses by the corresponding three-point statistics.

2. Theoretical Models and Bias Frameworks

Accurate interpretation of joint 2PCF and 3PCF data requires physically-motivated models linking observables to the underlying density field. In the context of 21 cm reionization fields, a “local quadratic bias” model is employed:

$B_{\ell_1\ell_2\ell_3}$5

where $B_{\ell_1\ell_2\ell_3}$6 (linear bias) and $B_{\ell_1\ell_2\ell_3}$7 (quadratic bias) are free parameters describing the mapping between 21 cm brightness-temperature fluctuations and the underlying matter density. For the 2PCF and 3PCF, the tree-level predictions in this model are:

$B_{\ell_1\ell_2\ell_3}$8

where $B_{\ell_1\ell_2\ell_3}$9 is the hierarchical configuration, constructed from products of two-point functions (Hoffmann et al., 2018). In joint 3×2-point analyses, both $A$0 and $A$1 are constrained by simultaneously fitting the observed 2PCF and 3PCF, employing a covariance matrix that includes all cross-terms.

In the angular-statistics context relevant to FRB and galaxy surveys, the fields are projected along the line of sight with window functions $A$2, and the Limber approximation is used for efficiency. Feedback and astrophysical effects, such as baryonic feedback from AGN or supernovae, are incorporated via parametric modifications to the matter power spectrum and cross-spectra, with additional nuisance and astrophysical parameters such as $A$3, $A$4, and $A$5 in the “BaryonForge” model (Sharma et al., 6 Sep 2025).

3. Estimation and Fourier-Transform Acceleration

Traditional direct pair and triplet counting for correlation function estimation scales at least quadratically or cubically with sample size, rendering them untenable for surveys with $A$6. Recent advances show that both 2PCF and 3PCF estimators can be recast as discrete convolutions, efficiently computed using fast Fourier transforms (FFTs) on a regular Cartesian grid (Slepian et al., 2015):

• For the 2PCF, the density field is gridded, the forward FFT $A$7 is calculated, and the monopole or multipoles are obtained by filtering in Fourier space, followed by an inverse FFT and binning in separation $A$8. For anisotropic 2PCF, spherical-harmonic kernels or Legendre polynomial projections are used in Fourier space.
• For the 3PCF, the SE15 estimator expresses the measurement in terms of binned spherical-harmonic moments $A$9 of the density field, with the 3PCF multipole $B$0 reconstructed from integrals over products of these moments. FFTs enable rapid evaluation of the required convolutions, reducing the complexity from $B$1 or $B$2 to $B$3.

This approach is critical in the era of large-volume and high particle-count datasets such as DESI, Euclid, and LSST, making full 3×2-point analyses feasible on practical timescales (Slepian et al., 2015).

4. Covariance, Likelihood, and Forecasting Methodologies

The 3×2-point data vector is accompanied by its full Gaussian-block covariance, taking into account auto- and cross-covariances among all spectra. In angular analyses, the block structure at each multipole $B$4 is:

$B$5

where $B$6 includes shot-noise and instrument noise. For FRB DMs, host and field variance terms appear in $B$7 (Sharma et al., 6 Sep 2025).

Parameter constraints are derived via Fisher-matrix forecasting:

$B$8

with derivatives of all three elements of $B$9 computed numerically with respect to each parameter $$\xi_{AB}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_B(\mathbf{x}+\mathbf{r}) \rangle$$0. Resulting marginalized $$\xi_{AB}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_B(\mathbf{x}+\mathbf{r}) \rangle$$1 constraints demonstrate that joint inclusion of all 2PCF and cross-correlation data can break degeneracies between cosmological and feedback parameters otherwise present in individual auto-correlation analyses.

5. Regimes of Validity, Limitations, and Scale Dependence

The local quadratic bias model and tree-level perturbative description underpinning 3×2-point statistics is valid only on sufficiently large scales and at early cosmic times, as demonstrated in 21 cm studies (Hoffmann et al., 2018). In reionization scenarios:

• For scales $$\xi_{AB}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_B(\mathbf{x}+\mathbf{r}) \rangle$$2 Mpc and global neutral fraction $$\xi_{AB}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_B(\mathbf{x}+\mathbf{r}) \rangle$$3, leading-order bias modeling recovers the 2PCF and 3PCF to within 10–20%.
• At smaller scales or late times ($$\xi_{AB}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_B(\mathbf{x}+\mathbf{r}) \rangle$$4), deviations due to patchy ionization, non-local effects, and higher-order terms become pronounced, reducing model accuracy.
• Similar breakdowns are found in FRB-galaxy angular analyses at low FRB densities and high multipoles, where shot and host DM noise dominate.

Non-Gaussian covariance contributions, redshift-space distortions, and observational systematics are not fully encompassed within these frameworks, requiring further model development for future precision cosmology (Sharma et al., 6 Sep 2025, Hoffmann et al., 2018). Extensions to include non-local bias, higher-order moments, and improved treatment of radial kernel and line-of-sight structure have been proposed as necessary for advancing accuracy.

6. Applications and Survey Implications

The 3×2-point paradigm is now integral to analyzing data from current and next-generation surveys:

• Galaxy and LSS Surveys: Fourier-based 2PCF and 3PCF estimators enable rapid computation of full multipole statistics and their covariances across millions of objects, critical for constructing covariance matrices over thousands of mock realizations (Slepian et al., 2015).
• FRB–Galaxy Cross-Correlations: The combination of FRB DM auto-spectra, galaxy clustering, and their cross-correlation provides a multi-probe multi-tracer framework for constraining both astrophysical feedback (e.g., AGN or supernova-driven gas ejection) and cosmological parameters. Joint 3×2-point analyses reduce degeneracies and yield precision on $$\xi_{AB}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_B(\mathbf{x}+\mathbf{r}) \rangle$$5, $$\xi_{AB}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_B(\mathbf{x}+\mathbf{r}) \rangle$$6, and $$\xi_{AB}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_B(\mathbf{x}+\mathbf{r}) \rangle$$7, outperforming pairwise-only analyses especially at modest FRB sample sizes (Sharma et al., 6 Sep 2025).
• 21 cm Cosmology: Local bias 3×2-point estimators allow quantification of the scale- and time-dependence of the bias between matter density and 21 cm brightness, elucidating the morphology and evolution of ionized regions during the EoR (Hoffmann et al., 2018).

In all cases, the 3×2-point approach maintains systematic consistency between data and simulations when identical gridding and FFT pipelines are employed, and support the construction of robust multi-probe covariance matrices required for joint inference of astrophysical and cosmological models.

7. Summary Table: Comparative Overview of 3×2-Point Statistic Elements

Observable Pair	Correlation Function	Typical Role
Galaxy auto	$$\xi_{AB}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_B(\mathbf{x}+\mathbf{r}) \rangle$$8, $$\xi_{AB}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_B(\mathbf{x}+\mathbf{r}) \rangle$$9	LSS clustering/cosmology
FRB DM auto	$$\xi_{A}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_A(\mathbf{x}+\mathbf{r}) \rangle$$0	Baryon/gas physics
Galaxy–FRB DM cross	$$\xi_{A}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_A(\mathbf{x}+\mathbf{r}) \rangle$$1	Feedback/cosmology cross-constraint
21 cm brightness auto (EoR)	$$\xi_{A}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_A(\mathbf{x}+\mathbf{r}) \rangle$$2, $$\xi_{A}(\mathbf{r}) = \langle \delta_A(\mathbf{x})\, \delta_A(\mathbf{x}+\mathbf{r}) \rangle$$3	Patchy reionization, bias modeling

This summary highlights the structure of the 3×2-point approach as employed in current LSS, 21 cm, and FRB-galaxy cosmological research, with all statements traceable to (Slepian et al., 2015, Hoffmann et al., 2018), and (Sharma et al., 6 Sep 2025).