3×2-Point Correlation Statistic

• The 3×2-point statistic is a comprehensive measure that combines cosmic shear, galaxy–shear, and galaxy clustering to probe both dark matter and baryonic feedback.
• It leverages spherical harmonic expansions and detailed covariance modeling to accurately capture cross-correlations and mitigate sample variance and noise.
• Extensions incorporating FRB DM analyses enable effective breaking of parameter degeneracies, enhancing multi-probe synergy for precision cosmological studies.

The 3×2-point correlation statistic is a joint summary measure central to contemporary and forthcoming multi-probe cosmological analyses. In its most common usage, it refers to the simultaneous evaluation of three two-point statistical measures—typically cosmic shear (shear–shear), galaxy–galaxy lensing (galaxy–shear cross-correlation), and galaxy clustering (galaxy–galaxy auto-correlation)—across large optical and radio surveys. This statistic enables the extraction of cosmological and astrophysical information by combining observables sensitive to both the total matter distribution and baryonic processes. Recent research further extends the 3×2-point framework to cross-correlations between fast radio burst (FRB) dispersion measures and galaxy density fields, providing a direct link between cosmology and baryonic feedback.

1. Definition and Mathematical Formalism

The 3×2-point statistic quantifies the cross-correlation structure among three cosmological fields. In the standard implementation for galaxy and weak lensing surveys, the three observables are:

1. Cosmic Shear (Shear–Shear Auto-Correlation): Measures correlations between observed galaxy shapes, probing projected large-scale structure through weak gravitational lensing.
2. Galaxy–Galaxy Lensing (Galaxy–Shear Cross-Correlation): Correlates the positions of foreground galaxies with the tangential distortions of background galaxies, tracing the average mass profile around galaxies.
3. Galaxy Clustering (Galaxy–Galaxy Auto-Correlation): Measures the angular correlation of galaxy positions, probing the underlying dark matter distribution modulo galaxy bias.

The correlation functions are computed in real or harmonic space. For example, the angular power spectrum for the cross-correlation between fields $A$ and $B$ in redshift bins $i, j$ (under the Limber approximation) is:

$$C^{(AB)}_{ij}(\ell) = \int \frac{d\chi}{\chi^2} W^i_A(\chi) W^j_B(\chi) P_{mm}(k=(\ell+1/2)/\chi, z(\chi))$$

with $W^i_A(\chi)$ being the appropriate kernel for the observable and $P_{mm}$ the matter power spectrum.

Extensions of the 3×2-point statistic include FRB–galaxy joint analyses, in which the statistical objects are:

• FRB DM Auto-Correlation: The angular power spectrum of FRB dispersion measures, encoding the projected free electron and hence baryon density.
• Galaxy–FRB DM Cross-Correlation: Sensitive to the spatial coincidence of galaxies and ionized gas.
• Galaxy Clustering: As above.

The formalism is unified by spherical harmonic expansions and weighted line-of-sight integrals with kernels reflecting redshift distributions, bias, and physical projections (e.g., for the DM field: $W_\mathcal{D}(z)$, for galaxies: $W_g(z)$) (Sharma et al., 6 Sep 2025).

2. Covariance, Noise, and Information Content

The covariance structure of the 3×2-point measurements incorporates both Gaussian sample variance and noise terms specific to each observable. For bandpowers in harmonic space, the covariance between angular power spectra is:

$$\mathrm{Cov}[C^{AB}(\ell_i), C^{CD}(\ell_j)] = \delta^K_{ij} \frac{2}{(2\ell_i + 1)\Delta\ell f_\mathrm{sky}} \left[\hat{C}^{AC}(\ell_i)\hat{C}^{BD}(\ell_i) + \hat{C}^{AD}(\ell_i)\hat{C}^{BC}(\ell_i)\right]$$

where noise-augmented $\hat{C}$ terms account for shot noise in galaxies and host/field DM noise for FRBs (Sharma et al., 6 Sep 2025). The available multipole range is limited by object density and instrumental localization; for FRB DMs, noise dominates at $\ell\gtrsim20$ with $10^4$ sources, and $\ell\gtrsim100$ with $10^5$ sources.

The full Fisher information matrix is constructed from the covariance and derivatives of the theoretical correlation functions with respect to cosmological and feedback parameters, enabling robust parameter forecast and quantification of degeneracy breaking.

3. Breaking Parameter Degeneracies: Multi-Probe Synergy

A key motivation for the 3×2-point approach is the ability to jointly constrain cosmology and baryonic physics by leveraging the distinct parameter sensitivities of each component. Galaxy clustering predominantly constrains total matter density but is weakly sensitive to baryonic feedback. FRB DM statistics, while shot-noise limited, directly probe the distribution of ionized baryons and thus baryonic feedback processes impacting the circumgalactic and intergalactic medium.

The cross-correlation between galaxies and the DM field (e.g., $C^{g\mathcal{D}}(\ell)$) links the positions of shocks, outflows, and other feedback signatures to underlying dark matter structure. When these observables are combined into the 3×2-point statistic, the differing degeneracy contours in parameter space (e.g., baryonic feedback efficiency vs. $\Omega_m$) enable significantly tighter constraints than any one probe alone. The Fisher matrix analysis indicates joint cosmological precision at the 10–18% level and feedback constraints at a few percent with $10^4$–$10^5$ FRBs, compared to factors of two or greater degradation for the individual FRB statistics (Sharma et al., 6 Sep 2025).

4. Practical Implementation and Limitations

The implementation of 3×2-point analyses requires:

• Precise modeling of each observable’s kernel and noise characteristics, especially for FRB host galaxy DM variance and selection effects.
• Construction of covariance matrices capturing cross-terms between all components, including noise covariances and correlations introduced by survey geometry and sampling variance.
• Accurate redshift calibration, halo occupation modeling (for galaxies), and host/foreground subtraction for FRBs.
• Treatment of small-scale feedback-sensitive regimes, currently inaccessible to galaxy surveys but accessible to FRB DMs, as well as scale cuts to avoid non-linear and systematic-dominated regions.

A summary table of measurement-limited regimes as a function of FRB sample size:

Statistic	Dominant Noise Limitation	Usable Multipole Range (for $10^4$/$10^5$ FRBs)
FRB DM Auto	Host DM, FRB number density	$\ell \lesssim 20 / 100$
Galaxy–FRB DM Cross	Galaxy/FRB selection, shot noise	Limited by lowest tracer density
Galaxy Clustering	Galaxy shot noise	High S/N up to survey angular resolution

The constraining power scales with $N_\mathrm{FRB}$; larger samples shift shot-noise to higher $\ell$ and allow enhanced constraints.

5. Comparative Performance and Cosmological Results

Forecasts in the FRB+galaxy 3×2-point framework indicate that:

• Galaxy clustering alone achieves ∼220 SNR and parameter constraints roughly twice as precise as DM-based statistics with $10^5$ FRBs.
• FRB DM statistics (auto/cross), while limited by shot noise and host variance, uniquely probe baryonic processes. Constraints on cosmological parameters are a factor of 1.5–3 weaker than from galaxies alone.
• Joint 3×2-point analysis breaks feedback–cosmology degeneracy, yielding robust and precise multi-probe constraints (cosmological parameters: 10–18%; baryonic feedback: 2–3%) (Sharma et al., 6 Sep 2025).

A plausible implication is that future increases in FRB source counts and improved DM modeling will render 3×2-point statistics a critical component for cosmology–astrophysics joint inference.

6. Extensions, Systematics, and Future Prospects

Contemporary work integrates the 3×2-point framework into Bayesian likelihood pipelines validated on realistic mock simulations that include halo occupation, photometric/spectroscopic selection, and detailed covariance evaluation (Blake et al., 17 Dec 2024). Systematic uncertainties associated with FRB host DMs, selection functions, and galaxy bias modeling must be rigorously controlled, as small excess errors degrade degeneracy breaking and overall precision.

Forthcoming surveys (e.g., DSA-2000, CHIME Outriggers, SKA) are expected to deliver substantially larger FRB samples with improved localization, amplifying the SNR and accessible angular range. Multi-probe 3×2-point analyses are positioned to connect cosmology with baryonic physics through simultaneous modeling of dark and ionized matter, uniquely bridging large-scale and feedback-sensitive regimes. This suggests a potential paradigm shift toward multi-field, multi-tracer cosmological inference as instrumental capabilities and sample sizes increase.

7. Summary

The 3×2-point correlation statistic unifies auto- and cross-correlation analyses between galaxies, cosmic shear, and (in its latest iteration) FRB dispersion measure fields, enabling joint constraints on cosmology and baryonic feedback. Its definition encompasses angular power spectra and correlation functions computed via redshift- and field-specific kernels, with covariances including both sample variance and tracer-specific noise. The multi-probe approach enhances parameter precision by breaking degeneracies inherent to single-observable analyses. Limitations stem from noise, sample size, and modeling uncertainties, particularly for FRB DMs. As observational capabilities mature, the 3×2-point framework is increasingly recognized as a powerful, flexible, and necessary statistic for precision cosmology and the investigation of baryonic astrophysics (Sharma et al., 6 Sep 2025, Blake et al., 17 Dec 2024).