Augmented Correlation Functions for Spectroscopic Galaxy Surveys

Abstract: Galaxy redshift surveys encode a wealth of information generated by nonlinear gravitational evolution, galaxy bias, and redshift-space distortions, only part of which is accessible through standard two-point statistics. Motivated by the need for flexible and computationally efficient alternatives, we introduce the augmented correlation function, a general framework in which an arbitrary transformation of the galaxy field defines additional ``latent'' dimensions that extend the standard two-point correlation function and isolate clustering properties averaged out in conventional analyses. As a proof of concept, we study a latent variable constructed from the pairwise gradient of the inverse Laplacian of the galaxy density field, showing that the resulting statistics naturally distinguish clustering regimes associated with infalling and outflowing pairs. Using Fisher forecasts based on $z=1$ halo catalogues from the Quijote simulations within $νΛ\mathrm{CDM}$ cosmology, we find that the augmented correlation systematically yields tighter constraints on all cosmological parameters considered. Although these improvements should be regarded as indicative given the exploratory nature of the analysis and the limitations of Fisher forecasts and simulations, our results demonstrate the potential of augmented correlations as a flexible framework for extracting additional information from spectroscopic galaxy surveys.

• The paper introduces an augmented correlation function that extends traditional two-point statistics via arbitrary field transformations to capture latent clustering details.
• It demonstrates the method using pairwise gradients of the inverse Laplacian, linking latent variables to galaxy infall/outflow velocities and BAO shifts.
• Fisher forecasts on dark matter halos show 2–3× tighter constraints on cosmological parameters, evidencing effective non-Gaussian information recovery.

Augmented Correlation Functions: Enhancing Cosmological Parameter Constraints from Spectroscopic Galaxy Surveys

Motivation and Context

The analysis of large-scale redshift surveys is traditionally grounded in two-point statistics—primarily the galaxy power spectrum and correlation function—which provide interpretable connections to fundamental cosmological parameters via perturbation theory. However, the data richness afforded by modern spectroscopic surveys significantly exceeds the information extracted by standard two-point summaries; critical nonlinear information from gravitational evolution, complex galaxy bias, and redshift-space distortions (RSD) is inevitably compressed or erased in such analyses. While higher-order $N$-point functions theoretically recover lost information, their empirical use is hampered by extreme covariance dimensionality, complex modeling, and sensitivity to survey systematics.

A range of alternative statistics have emerged, including wavelet transforms, Minkowski functionals, $k$-nearest-neighbor measures, and minimum spanning trees, all of which attempt to reparameterize the density field to better preserve non-Gaussian information and environmental dependencies. Of particular interest are "marked" and "density-split" two-point statistics, which assign marks or condition on local environments, and void-based statistics. Despite these advances, there remains no systematic or flexible paradigm for incorporating general field transformations into two-point statistics.

Framework: The Augmented Correlation Function

This work introduces the augmented correlation function as a general formalism for systematically extending two-point statistics via arbitrary transformations of the matter or galaxy field, yielding additional "latent" dimensions beyond pairwise separation $s$ and orientation $\mu$. In this approach, a user-defined auxiliary field $\psi(\mathbf{x})$—potentially nonlocal and of any tensor rank—is computed from the density contrast $\delta(\mathbf{x})$ (or another input). For each galaxy pair, this auxiliary field is mapped by a kernel $K_p(\psi_1, \psi_2)$ onto a latent variable $\lambda$, which serves as a continuous or discrete index. The augmented function $\xi_A(\mathbf{x}_1, \mathbf{x}_2, \lambda)$ is then defined such that the conventional two-point function is recovered upon marginalization over $\lambda$; for discrete $k$0, this is a quantile-based sample split.

This construction subsumes marked correlation functions, density-split statistics, and similar analyses as special cases, and allows diagnostic quantiles, alternative response bases (e.g., the Helmert basis), and higher-dimensional latent variable spaces. The estimator for $k$1 is a straightforward generalization of the Landy–Szalay estimator, accounting for the additional latent-variable bin.

Instantiation: Pairwise Gradient of the Inverse Laplacian

As a concrete demonstration, the authors select as $k$2 the gradient of the inverse Laplacian of the galaxy density contrast, i.e., $k$3, filtered at a scale of $k$4. The kernel $k$5 then projects the difference $k$6 along the pair separation $k$7, producing a latent $k$8 that strongly correlates with pairwise infall (negative $k$9) and outflow (positive $s$0) velocities, thereby linking tightly to information about coherent flows and the BAO shift, RSD, and the Alcock–Paczynski (AP) effect. Importantly, this latent variable is formally distinct from direct velocity measurements or true Zel’dovich displacements and is computed from the observed (redshift-space) galaxy field, not auxiliary simulations.

Numerical Implementation and Fisher Forecasts

The information content of the augmented correlation is quantified via Fisher forecasts using the Quijote simulation suite, which provides a large volume ($s$1), several thousand realizations with $s$2CDM parameter variations, and robust halo catalogs at $s$3. The augmented two-point function and its multipoles are measured for dark matter halos, with the auxiliary variable $s$4 binned into quantiles ($s$5 in the main results), yielding $s$6 for each quantile $s$7.

The Fisher matrix is evaluated numerically, using finite-difference derivatives w.r.t. each cosmological parameter and empirical covariance matrices from the simulation ensemble. Forecasted marginalized uncertainties are compared between standard and augmented two-point analyses, with careful attention paid to binning choices (logarithmic in $s$8, multipole expansion in $s$9), random catalog construction, and configuration-dependence.

Results and Empirical Signatures

Analysis reveals:

• Isolation of Distinct Clustering Regimes: Augmented quantiles distinctly trace clustering regimes from infalling to outflowing pairs, with the corresponding correlation function multipoles displaying systematic trends—e.g., enhanced quadrupole for infalling pairs associated with strong RSD and negative $\mu$0. The BAO peak exhibits systematic quantile-dependent shifts, consistent with the physical interpretation of the latent variable.
• Constraint Improvements for Cosmological Parameters: The augmented correlation yields consistently tighter constraints on all cosmological parameters tested (i.e., $\mu$1), with improvement factors in marginalized $\mu$2 uncertainties exceeding 2–3 for optimal $\mu$3 and binning configurations.
• Partial Non-Gaussian Information Recovery: Significant gains persist even when the minimum scale used in the analysis remains at $\mu$4, indicating that at least part of the improvement arises from reorganization and less aggressive compression of two-point information, rather than purely higher-order effects.
• Structure of Covariance: Augmented correlation covariances remain dominated by their diagonal blocks within quantiles, with relatively modest off-diagonal correlations, indicating efficiency in capturing additional non-redundant information.
• Configuration Dependence and Convergence: Improvements increase with the number of $\mu$5 quantiles used, at the expense of increased noise in covariance and derivatives, thus requiring a balance for practical applications. Extensive convergence tests are provided.

Theoretical and Practical Implications

The augmented correlation function formalism presents a systematic and flexible means of incorporating arbitrary nonlinear or nonlocal information into two-point clustering analyses. It unifies previously ad-hoc marked and density-split statistics and enables straightforward exploration of the vast space of physically motivated field transformations. In practice, its computational tractability—given efficient pairwise estimators and compatibility with existing simulation-based emulators—makes it ripe for immediate application to next-generation spectroscopic datasets.

From a theoretical perspective, the approach highlights the importance of conditional statistics—analyses that explicitly retain environmental or pairwise information in higher-dimensional latent space—rather than aggressive marginalization. The separation of clustering regimes of physical relevance (e.g., strong infall, outflows, high/low density) via auxiliary fields can potentially improve cosmological inference, reduce susceptibility to modeling errors in non-linear or non-Gaussian regimes, and enhance the interpretability of observed clustering signals.

This flexible framework enables future work to systematically survey alternative latent-variable and kernel constructions (including learnable nonlinear transformations, higher-order field derivatives, topological descriptors, void finders, etc.), and to combine or marginalize over them as required.

Future Prospects and Directions

Future development will focus on:

• Application to Realistic Survey Data: Development of tailored emulators for the augmented statistics and application to actual galaxy redshift surveys, incorporating full survey geometries, systematics, bias models, and selection effects.
• Exploration of Latent Variable Space: Systematic study of the efficacy of alternative auxiliary fields and kernels, higher-rank tensor fields, and connections to marked/density-split/void-based statistics.
• Joint Modeling with Other Statistics: Analytic models for the augmented correlation function, possibly beyond perturbation theory, and integration with higher-order or summary statistics from wavelet or topological analyses.
• Machine Learning Approaches: Exploration of learnable (e.g., neural-network-based) latent variables within the augmented correlation formalism to optimize information extraction.

Conclusion

The augmented correlation function formalism establishes a general framework for extracting enhanced cosmological information from spectroscopic galaxy surveys using two-point statistics augmented by arbitrary latent variables constructed from physically motivated field transformations. Proof-of-concept Fisher forecasts with pairwise gradients of the inverse Laplacian as a latent variable demonstrate systematic tightening of parameter constraints and the isolation of nonlinear clustering regimes inaccessible to standard two-point statistics. This approach offers a promising direction for future clustering analyses, with substantial room for methodological expansion and application to forthcoming large-scale survey data.

Reference: "Augmented Correlation Functions for Spectroscopic Galaxy Surveys" (2605.30305)