Augmented Correlation Function

• Augmented correlation function is a generalization of the two-point correlation function that uses auxiliary latent variables to isolate distinct clustering regimes in galaxy surveys.
• The framework retains standard pair-counting methods while conditioning on an additional variable, allowing quantile splits to differentiate infall and outflow dynamics.
• Application of the method yields improved Fisher forecasts with tighter cosmological parameter constraints, bridging standard two-point and higher-order environmental analyses.

Searching arXiv for the target paper and closely related context. An augmented correlation function is a generalization of the ordinary two-point correlation function 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. In spectroscopic galaxy surveys, the framework is motivated by the observation that nonlinear gravitational evolution, galaxy bias, and redshift-space distortions generate structure that is only partly accessible through standard statistics such as $\xi(s,\mu)$ or its multipoles. The defining move is to retain the pair-counting machinery of a two-point estimator while conditioning pairs on an additional variable derived from a transformation of the field, thereby enlarging pair space beyond separation and line-of-sight angle alone (Bianchi, 28 May 2026).

1. Formal definition in enlarged pair space

The baseline object is the ordinary two-point correlation function of the overdensity field,

$$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$

The augmented construction introduces an auxiliary field $\boldsymbol{\psi}$, defined as an arbitrary deterministic transformation of $\delta$, and a pairwise kernel $K_p$ that combines the values of that field at the two ends of a pair into a latent variable $\lambda$. The augmented correlation function is then

$$\xi_A(\mathbf{x}_1,\mathbf{x}_2,\lambda) = \frac{ \left\langle \delta(\mathbf{x}_1)\delta(\mathbf{x}_2)\, \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }{ \left\langle \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }.$$

The denominator is the volume-weighted distribution of $\lambda$ at fixed pair configuration, and the numerator is the density-contrast-weighted distribution of $\lambda$. The normalization is therefore integral to the definition: the statistic is not merely a weighted two-point function, but a conditional or stratified two-point statistic in an enlarged space. The paper further emphasizes that $\boldsymbol{\psi}$ need not be a vector; it may be scalar or tensor-valued, and the kernel $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$0 may output more than one latent coordinate, $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$1, with $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$2 (Bianchi, 28 May 2026).

This formulation is intended to preserve the computational accessibility of pair-count statistics while making explicit pair properties that the standard $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$3 averages over. A plausible implication is that the augmented object can interpolate between ordinary two-point summaries and statistics that are usually treated as environmental or higher-order descriptors, depending on the choice of $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$4 and $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$5.

2. Estimation, coordinates, and quantile representation

For survey applications, the framework is expressed in the usual redshift-space coordinates $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$6, where $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$7 with $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$8, and $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$9 is the cosine of the angle between $\boldsymbol{\psi}$0 and the line of sight. The practical estimator is a direct extension of Landy–Szalay: $\boldsymbol{\psi}$1 Here $\boldsymbol{\psi}$2, $\boldsymbol{\psi}$3, and $\boldsymbol{\psi}$4 are data–data, data–random, and random–random pair counts in bins of $\boldsymbol{\psi}$5, normalized by the corresponding total number of pairs. Once the latent field and per-object auxiliary values are available, the rest of the pipeline is essentially standard pair counting with one extra binned coordinate (Bianchi, 28 May 2026).

In practice, the latent variable is binned. The paper introduces a quantile coordinate

$\boldsymbol{\psi}$6

where

$\boldsymbol{\psi}$7

is the cumulative distribution of the volume-weighted latent-variable distribution at fixed $\boldsymbol{\psi}$8. Choosing $\boldsymbol{\psi}$9-bin edges so that $\delta$0 is divided into $\delta$1 equal parts yields

$\delta$2

independent of the quantile label $\delta$3. The ordinary two-point function is then exactly the arithmetic mean of the quantile slices,

$\delta$4

where $\delta$5 denotes the augmented correlation in the $\delta$6-th quantile. The paper also presents an alternative Helmert-basis decomposition in which one basis vector is exactly $\delta$7 and the remaining $\delta$8 combinations sum to zero. It further notes that density-split statistics and marked correlations fit inside the same formal structure, depending on how the latent variable is built from the transformed field (Bianchi, 28 May 2026).

3. Proof-of-concept latent variable from inverse-Laplacian gradients

As a proof of concept, the framework is instantiated with an auxiliary field intended to trace pairwise infall. The construction is

$\delta$9

with analogous definitions for $K_p$0 and $K_p$1. In Fourier space this is the gradient of the inverse Laplacian of the density contrast, and, assuming $K_p$2, it satisfies

$K_p$3

The latent variable is then defined by the pairwise kernel

$K_p$4

which produces the scalar $K_p$5 attached to each pair (Bianchi, 28 May 2026).

The sign convention is central: negative $K_p$6 corresponds by construction to infalling pairs, while positive $K_p$7 corresponds to outflowing pairs. The paper explicitly connects $K_p$8 to the Zel’dovich reconstruction equation,

$K_p$9

in the simplified limit $\lambda$0 and $\lambda$1, so that $\lambda$2 resembles a potential-flow field built from the density. A Gaussian smoothing filter with

$\lambda$3

is applied before constructing $\lambda$4. The paper is equally explicit about limitations: because the galaxy field is observed in redshift space and under an assumed cosmology, redshift-space distortions and Alcock–Paczynski distortions propagate into $\lambda$5, so $\lambda$6 is not a clean dynamical observable. It also stresses that the latent variable is intrinsically pairwise: a galaxy can be infalling relative to one neighbor and outflowing relative to another, so the substructures isolated by $\lambda$7-quantiles live in pair space rather than ordinary object space (Bianchi, 28 May 2026).

4. Clustering regimes isolated by latent quantiles

The empirical content of the framework appears in the latent-variable distributions and in the quantile-resolved multipoles. The distribution of $\lambda$8 at fixed $\lambda$9 is described as roughly Gaussian near its core but with exponential tails, with stronger non-Gaussianity and skewness at smaller separations. As separation decreases, the mass-weighted distribution shifts toward more negative $$\xi_A(\mathbf{x}_1,\mathbf{x}_2,\lambda) = \frac{ \left\langle \delta(\mathbf{x}_1)\delta(\mathbf{x}_2)\, \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }{ \left\langle \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }.$$0, meaning that clustered pairs preferentially occupy infall-like latent regions. This suggests that the latent coordinate is not simply a relabeling of separation, but a partition of pair space into clustering regimes that standard $$\xi_A(\mathbf{x}_1,\mathbf{x}_2,\lambda) = \frac{ \left\langle \delta(\mathbf{x}_1)\delta(\mathbf{x}_2)\, \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }{ \left\langle \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }.$$1 averages together (Bianchi, 28 May 2026).

For $$\xi_A(\mathbf{x}_1,\mathbf{x}_2,\lambda) = \frac{ \left\langle \delta(\mathbf{x}_1)\delta(\mathbf{x}_2)\, \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }{ \left\langle \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }.$$2, the leftmost quantile $$\xi_A(\mathbf{x}_1,\mathbf{x}_2,\lambda) = \frac{ \left\langle \delta(\mathbf{x}_1)\delta(\mathbf{x}_2)\, \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }{ \left\langle \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }.$$3 probes the strongest infall and the rightmost quantile $$\xi_A(\mathbf{x}_1,\mathbf{x}_2,\lambda) = \frac{ \left\langle \delta(\mathbf{x}_1)\delta(\mathbf{x}_2)\, \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }{ \left\langle \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }.$$4 the strongest outflow. The resulting multipoles show sharply distinct behavior. The monopoles differ strongly in amplitude and scale dependence. The quadrupoles differ in magnitude and can differ in sign across quantiles. The strongest negative quadrupoles occur for infalling pairs, consistent with stronger redshift-space distortions, whereas strongly outflowing pairs can exhibit quadrupoles near zero or slightly positive on small scales. The paper also reports indications that the BAO peak shifts smoothly across quantiles, from smaller separations in infalling slices to larger separations in outflowing slices. Because the ordinary two-point function is exactly the mean of the quantile slices, differential responses of opposite sign can cancel in the standard average. The paper suggests that, on large scales where pairwise orientation becomes less coherent, the latent coordinate may acquire a more environmental interpretation through the magnitude of $$\xi_A(\mathbf{x}_1,\mathbf{x}_2,\lambda) = \frac{ \left\langle \delta(\mathbf{x}_1)\delta(\mathbf{x}_2)\, \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }{ \left\langle \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }.$$5, though this is presented as interpretation rather than definition (Bianchi, 28 May 2026).

5. Simulation pipeline and forecasting machinery

The implementation uses halo catalogues and random catalogues in a survey-like redshift-space analysis. For the main Quijote setup at $$\xi_A(\mathbf{x}_1,\mathbf{x}_2,\lambda) = \frac{ \left\langle \delta(\mathbf{x}_1)\delta(\mathbf{x}_2)\, \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }{ \left\langle \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }.$$6, each realization contains about $$\xi_A(\mathbf{x}_1,\mathbf{x}_2,\lambda) = \frac{ \left\langle \delta(\mathbf{x}_1)\delta(\mathbf{x}_2)\, \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }{ \left\langle \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }.$$7 halos, corresponding to number density

$$\xi_A(\mathbf{x}_1,\mathbf{x}_2,\lambda) = \frac{ \left\langle \delta(\mathbf{x}_1)\delta(\mathbf{x}_2)\, \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }{ \left\langle \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }.$$8

and the random catalogue has three times the data density. Halo positions are painted onto a $$\xi_A(\mathbf{x}_1,\mathbf{x}_2,\lambda) = \frac{ \left\langle \delta(\mathbf{x}_1)\delta(\mathbf{x}_2)\, \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }{ \left\langle \delta_D\!\left[\lambda-K_p\!\big(\boldsymbol{\psi}(\mathbf{x}_1),\boldsymbol{\psi}(\mathbf{x}_2)\big)\right] \right\rangle }.$$9 grid using Cloud-In-Cell assignment with unit weights; one deinterlacing step is applied and the assignment window is deconvolved. The redshift-space coordinate is

$\lambda$0

and Alcock–Paczynski distortions are parameterized by

$\lambda$1

Pair counts are accumulated in $\lambda$2 logarithmic $\lambda$3-bins over $\lambda$4, $\lambda$5 linear $\lambda$6-bins over $\lambda$7, and $\lambda$8 linear $\lambda$9-bins over $\lambda$0. Angular dependence is compressed to Legendre multipoles,

$\lambda$1

and analogously for each quantile. The initial analysis considers $\lambda$2, but the Fisher forecasts retain only monopole and quadrupole because the hexadecapole is too noisy (Bianchi, 28 May 2026).

The forecasts use the Quijote N-body suite. Each simulation evolves $\lambda$3 dark-matter particles in a periodic box of side length $\lambda$4; massive-neutrino runs also include $\lambda$5 neutrino particles. Halos are identified with FoF using linking length $\lambda$6, with minimum halo mass $\lambda$7. The fiducial $\lambda$8CDM cosmology is

$\lambda$9

The Fisher matrix is

$\boldsymbol{\psi}$0

Centered finite differences are used for $\boldsymbol{\psi}$1, and a one-sided higher-order formula is used for $\boldsymbol{\psi}$2. The covariance is estimated from $\boldsymbol{\psi}$3 fiducial realizations; derivatives use $\boldsymbol{\psi}$4 realizations per cosmology variation, observed along three Cartesian axes, for $\boldsymbol{\psi}$5 effective realizations. The final forecast data vector uses $\boldsymbol{\psi}$6 and $\boldsymbol{\psi}$7, and the Hartlap correction is applied when inverting the covariance matrix (Bianchi, 28 May 2026).

6. Forecast performance, related methodologies, and limitations

For the chosen latent variable, the augmented correlation function yields tighter Fisher constraints for all six cosmological parameters than the standard correlation function. In the most constraining configurations explored, the improvement reaches factors of about $\boldsymbol{\psi}$8 relative to the standard two-point function. Increasing the number of quantiles from $\boldsymbol{\psi}$9 to $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$00 and $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$01 yields further gains in the tested range, although $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$02 is treated as a conservative default because larger $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$03 increases data-vector size and therefore derivative and covariance noise. The relative improvement persists when the minimum separation scale is varied: all constraints weaken as small scales are removed, but the improvement of the augmented statistic remains and in some cases increases. The paper explicitly notes that some quantiles respond in nearly opposite ways to parameters such as $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$04, so averaging them into $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$05 suppresses sensitivity that the augmented representation preserves (Bianchi, 28 May 2026).

The framework is presented as a formal umbrella broad enough to include statistics built from smoothed density fields, density splits, and marked correlations. The proof-of-concept choice of $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$06 also overlaps conceptually with reconstruction, void-based analyses, and transformed-field methods. The paper nevertheless treats the current implementation as exploratory and explicitly qualifies the forecast gains as indicative rather than definitive. The caveats are extensive: the analysis uses Fisher forecasts with a Gaussian-likelihood approximation and neglects the parameter-dependence of the covariance; covariance matrices and derivatives are estimated from finite simulation samples; the neutrino-mass derivative is one-sided and less well converged; the simulations are periodic boxes with halos used as galaxy tracers rather than full survey mocks; the constraining power depends on the choice of auxiliary field, pooling kernel, smoothing scale, latent binning, and compression scheme; and real-data issues such as survey windows, angular masks, incompleteness, fiber collisions, redshift failures, covariance estimation for larger data vectors, and optimal AP/RSD treatment in the transformed field remain open. The paper reports convergence tests indicating covariance-induced uncertainties stable at the $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$07 level for $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$08, and derivative-induced constraints stable at the $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$09 level for $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$10, except $$\xi = \langle \delta(\mathbf{x}_1)\,\delta(\mathbf{x}_2)\rangle, \qquad \delta(\mathbf{x})=\frac{\rho(\mathbf{x})}{\langle \rho \rangle}-1.$$11, which is less well converged (Bianchi, 28 May 2026).

The proposed future directions include smoothed density fields, higher derivatives of the inverse Laplacian, topological and geometric descriptors including void finders, and potentially learned transformations in which the map from the observed field to latent variables is optimized rather than hand-designed. In that sense, the augmented correlation function is less a single statistic than a framework for constructing pair-conditioned clustering observables that keep the practical virtues of two-point estimators while exposing hidden structure in pair space.