Tomographic 3x2pt Statistics in Cosmology

• Tomographic 3x2pt statistics is a technique that subdivides galaxy surveys into redshift bins to capture the evolution of cosmic structure.
• It integrates galaxy clustering, cosmic shear, and galaxy–galaxy lensing to break parameter degeneracies and boost cosmological inference.
• Advanced binning schemes and covariance estimation, including machine-learning strategies, optimize analyses for Stage-IV surveys like Euclid and LSST.

The tomographic approach to 3×2pt statistics refers to the explicit incorporation of redshift (line-of-sight) information into the joint analysis of three distinct galaxy–survey two-point functions: galaxy clustering, cosmic shear (weak lensing), and galaxy–galaxy lensing. Rather than relying solely on the projection of all sources or galaxies onto the sky, tomography subdivides the sample into redshift bins, enabling the measurement of angular and cross-correlations both within and between bins. This multidimensional approach unlocks additional cosmological information by exploiting the redshift evolution of structure and by breaking degeneracies inherent in lower-dimensional summaries.

1. Formal Definition and Methodological Framework

In the context of photometric and spectroscopic surveys, 3×2pt statistics refers to the set of all possible two-point correlation functions between galaxy density (δ_g) and shear (γ), i.e., galaxy–galaxy clustering (⟨δ_g δ_g⟩), cosmic shear (⟨γγ⟩), and galaxy–galaxy lensing (⟨δ_g γ⟩). The tomographic extension of these statistics entails dividing the galaxy samples into redshift bins labeled by indices i, j, and constructing auto- and cross-correlations between all pairs of bins.

The general tomographic angular power spectrum takes the form: $$C_{\ell}^{XY,ij} = \int dz\, \frac{W_i^{X}(z) W_j^{Y}(z)}{H(z) r^2(z)}\, P_m^{\mathrm{NL}}\left(k = \frac{\ell+1/2}{r(z)}, z\right),$$ where X and Y refer to the observable (galaxy position or shear), $W_i^X(z)$ is the kernel for tracer X in bin i, H(z) is the Hubble parameter, r(z) is comoving distance, and $P_m^{\mathrm{NL}}$ is the non-linear matter power spectrum (Collaboration et al., 21 Jul 2025).

Tomography enhances the analysis by:

• Providing access to the redshift evolution of structure growth
• Efficiently breaking parameter degeneracies, especially for dark energy and primordial initial conditions
• Increasing the number of observable cross-correlation pairs, thus improving the Fisher information matrix for parameter inference (Martinet et al., 2015, Wong et al., 13 Jan 2025, Collaboration et al., 21 Jul 2025)

2. Binning Schemes and Optimization Strategies

Choosing the redshift binning scheme is critical for the efficiency and robustness of a tomographic 3×2pt analysis. Strategies include:

• Equal-number bins: Each bin contains the same number of galaxies; tends to minimize shot noise and is found to provide optimal or near-optimal constraints for the combined 3×2pt observable set (Wong et al., 13 Jan 2025).
• Equal redshift-width or equal comoving distance bins: Binning is linear in z or χ; for cosmic shear alone, bins equally spaced in comoving distance can yield slightly better constraints (by a few percent), but differences are marginal compared to equal-number schemes (Moskowitz et al., 2022, Wong et al., 13 Jan 2025).
• Generalized binning: More sophisticated bin definitions can be constructed by maximizing a figure of merit (FoM), such as the inverse area of the (w₀, wₐ) Fisher forecast ellipse, over free parameters in the bin edge function. One approach defines a generalized metric: $$\mathcal{M} = \int_0^{z_{\text{max}}} [dN/dz]^\alpha [d\chi/dz]^\beta dz,$$ where (α, β) can be tuned to optimize dark energy constraints (Moskowitz et al., 2022).

Machine-learning techniques—including neural networks and self-organizing maps—have also been applied to redefine bin assignments, select or reject galaxies based on photo-z reliability, and iteratively optimize bin edges for signal-to-noise or constraining power (Moskowitz et al., 2022, Alemany-Gotor et al., 24 Jul 2025, Zuntz et al., 2021).

3. Incorporating Observational Realities: Photo-z Errors, Outliers, and Covariance

Realistic tomographic analyses account for a variety of non-idealities:

• Photometric redshift errors: Gaussian-distributed uncertainties and catastrophic outliers affect bin purity, leakage, and covariance. For example, a 5% contamination from catastrophic photo-z outliers can bias the dark energy constraints in a 10-bin 3×2pt analysis to >5σ from the true values if not mitigated (Wong et al., 13 Jan 2025).
• Shape noise and shot noise: Noise sources from intrinsic galaxy ellipticity and finite sampling are added to the theoretical covariance to match survey conditions.
• Covariance estimation: The multidimensionality of tomographic 3×2pt analyses increases the size and complexity of the covariance matrix, which must be accurately estimated either analytically or from large ensembles of simulations (Porth et al., 2023, Wong et al., 13 Jan 2025).

These factors directly inform the optimal number of tomographic bins. In simulated Euclid-like analyses, information gain on dark energy parameters saturates at ≳7–8 bins; marginal returns diminish while requirements on accurate covariance estimation grow (Wong et al., 13 Jan 2025).

4. Cross-correlation Information and Non-Gaussian Statistics

Tomographic approaches can exploit cross-bin (inter-slice) correlations, not only for the standard two-point statistics but also for higher-order quantities (e.g., shear peaks, aperture mass distributions, voids, and three-point functions) (Martinet et al., 2020, Porth et al., 2023). Including cross-bin terms captures the signal from large-scale structures that span multiple redshift intervals, yielding substantial (≈50%) improvements in parameter constraints when compared to auto-bin-only analyses.

Furthermore, non-Gaussian observables such as:

• Full 1D aperture mass distributions (as opposed to only peak or void counts)
• Third-order shear statistics and multipole-decomposed three-point correlation functions

provide independent and highly complementary cosmological information. These break degeneracies (e.g., in S₈, Ωₘ, w₀) not accessible to Gaussian two-point estimators and are computationally tractable in tomographic schemes owing to recent advancements in estimator efficiency (Martinet et al., 2020, Porth et al., 2023).

5. Benefits and Challenges in Tomographic 3×2pt Analyses

Benefits:

• Degeneracy breaking: Redshift tomography provides critical leverage on time-evolving cosmological parameters, including the dark energy equation of state, spatial curvature, and inflationary initial conditions (Collaboration et al., 21 Jul 2025).
• Complementarity: Joint analysis with the CMB (Planck, SO, CMB-S4) further breaks degeneracies and improves constraints on curvature, running, isocurvature, and primordial non-Gaussianity (Collaboration et al., 21 Jul 2025).
• Robustness: By cross-analyzing clustering and lensing (and their cross-correlation), systematics that affect a single probe can be isolated and their impact reduced.

Challenges:

• Covariance management: Increasing bin number improves information, but sharply increases the dimension and estimation requirement of the covariance matrix; errors in covariance propagate to bias in parameter estimation (Wong et al., 13 Jan 2025).
• Photo-z systematics: Outlier contamination, non-representative training sets, and edge effects (e.g., near bin boundaries) must be mitigated, often requiring targeted machine learning strategies (Moskowitz et al., 2022, Alemany-Gotor et al., 24 Jul 2025).
• Binning trade-offs: The optimal binning for lensing, clustering, and their cross-correlation may differ; an iterative or alternating optimization, potentially split by target observable, is often used (Alemany-Gotor et al., 24 Jul 2025).
• Saturation of gains: Beyond about 7–8 bins, additional tomographic slicing yields only marginal improvement at the cost of increased complexity (Wong et al., 13 Jan 2025).

6. Applications and Outlook in Contemporary and Next-Generation Surveys

Stage-IV cosmological surveys—including Euclid, LSST/Rubin, and Roman—have adopted tomographic 3×2pt strategies as the standard for cosmological parameter inference (Wong et al., 13 Jan 2025, Collaboration et al., 21 Jul 2025). Recent studies demonstrate that:

• Optimized tomographic binning via advanced algorithms can yield a factor of ∼2 improvement in the FoM for dark energy (akin to quadrupling the effective survey area), especially if source and lens bins are optimized separately (Alemany-Gotor et al., 24 Jul 2025).
• Tomographic 3×2pt analyses coupled with non-Gaussian summary statistics (peaks, voids, 1D distributions, aperture mass moments) provide substantial leverage to resolve current tensions (e.g., in S₈) and chart the physics of dark energy and inflation (Martinet et al., 2020, Collaboration et al., 21 Jul 2025).
• Full Bayesian field-level inference methods, using tomographic data, now allow direct constraints on primordial non-Gaussianities (f_NL^local) competitive with the CMB (Collaboration et al., 21 Jul 2025).

Looking ahead, further progress depends on advances in redshift inference, sample selection, covariance modeling, and the optimal exploitation of cross-probe and cross-bin information. The tomographic approach to 3×2pt statistics underpins the core analysis architecture for extracting fundamental physics from the next decade of cosmological data.