The k-nearest neighbour cumulative distribution functions (kNN-CDFs) were shown by Coulton, Abel & Banerjee (2024) to respond to equilateral-type primordial non-Gaussianity (PNG) on quasi-nonlinear scales (10-70 Mpc/h) in a manner distinct from changes in cosmological and galaxy-occupation parameters, making them a candidate complement to large-scale bispectrum searches. That demonstration, however, rested on a Zheng-style halo occupation distribution with a single assembly-bias prescription, quasi-Gaussian likelihoods on compressed summaries, and idealized periodic-box samples at fixed redshift. The open problem is to establish the conditions under which the kNN-CDF PNG signal survives broader and more realistic modeling: whether the distinct equilateral-PNG response persists under alternative galaxy-halo connection models and richer assembly-bias freedom, how much primordial bispectrum configuration information is lost by compressing to scale-dependent skewness-like statistics in nonlinear regimes, what likelihood actually describes full kNN-CDF measurements, and how the statistic behaves for observationally representative samples and survey geometries.

Background

Primordial non-Gaussianity (PNG) denotes deviations of the primordial density perturbations from a Gaussian random field; it is commonly parameterized by an amplitude fNLf_{\rm NL} multiplying the leading-order (bispectrum) contribution to the three-point function. PNG shapes such as local, equilateral and orthogonal refer to the configuration dependence of the primordial bispectrum — the scale-and-triangle-shape dependence of the three-point correlations of the initial density field — and different inflationary mechanisms generate different shapes, so measuring fNLf_{\rm NL} shape by shape is a key test of early-Universe physics. Current constraints come mainly from the cosmic microwave background bispectrum and from large-scale-structure two- and three-point functions on large, near-linear scales; extracting PNG from the nonlinear, small-scale regime of the late-time Universe has remained an open problem because galaxy formation physics also generates non-Gaussianity there.

The k-nearest neighbour cumulative distribution function (kNN-CDF) is a clustering statistic defined on a discrete point set (e.g. galaxies): place random query points in the volume and record the CDF of the distance to the k-th nearest data point, as a function of k and of scale. kNN-CDFs are closely related to counts-in-cells (CiC) statistics — the PDF of the number of objects in cells of a given size — since the fraction of query points whose k-th neighbour lies beyond a radius r equals the fraction of cells of radius r containing fewer than k objects. They are fast to compute and capture higher-than-two-point information. The kNN-CDF framework and its two-dimensional generalizations were developed by Yuan, Zamora & Abel (2023), who showed 2D and galaxy-to-galaxy variants are highly informative probes of galaxy clustering.

Coulton, Abel & Banerjee (2024) applied kNN-CDFs to the Quijote-PNG simulations (a suite of N-body simulations with separate local, equilateral and orthogonal PNG extensions, varying also cosmological parameters). They found that for halo samples with Mh<1014 M⊙/hM_h<10^{14}\,M_\odot/h the kNN-CDFs respond to equilateral PNG in a way distinct from all other varied parameters, that this persists in redshift-space galaxy catalogs (catalogs whose positions are inferred from redshifts and hence distorted by galaxies' peculiar velocities), and that it can be distinguished from galaxy modeling within the halo occupation distribution (HOD) framework considered. An HOD is a prescription that populates simulated dark matter halos with galaxies via parameters for central and satellite occupation; assembly bias is the dependence of halo occupation on secondary halo properties beyond mass, such as formation time or concentration. Mapping a subset of the kNN-CDF measurements into the CiC picture, they showed the results are captured by the analytic large-deviation-theory counts-in-cells model of Friedrich et al. (2020) for the bulk of the PDF — meaning the kNN-CDF PNG signal is governed by the scale-dependent skewness of the linear density field, which averages over bispectrum configurations. An analytic Fisher-matrix comparison — a standard forecast method that predicts parameter constraints from a statistic's parameter derivatives and its covariance — of skewness versus the full bispectrum of the linear field suggested the skewness compression retains nearly all information for equilateral PNG but loses substantially for orthogonal PNG. Their forecasts compressed the kNN-CDF measurements to low-dimensional summaries and assumed a Gaussian likelihood; the non-Poissonian shot-noise (a fitted deviation of the discrete count variance from Poisson behavior, following Friedrich et al. 2020/2022) was refit per sample.

The authors state that moving from this demonstration to robust observational PNG constraints with kNN-CDFs requires resolving a set of linked open issues.

Problem Statement

Determine whether, and under what conditions, k-nearest neighbour cumulative distribution functions (kNN-CDFs) can deliver robust observational constraints on primordial non-Gaussianity (PNG) from quasi-nonlinear scales (roughly 10-70 Mpc/h), by resolving the validity conditions that Coulton, Abel & Banerjee (2024) identified as standing between their HOD-limited simulation demonstration and observational deployment. The source states these as several linked future directions sharing the single objective — establishing observational readiness of the kNN-CDF PNG probe — with each addressing a distinct validity condition:

  1. Modeling robustness: determine whether the distinct response of kNN-CDFs to equilateral PNG (and the behavior for local and orthogonal shapes) survives broader galaxy-halo connection modeling than the Zheng-style HOD with the single assembly-bias prescription used in the source — e.g. hybrid or perturbative effective-field-theory-based occupation models, semi-analytic galaxy formation, or hydrodynamical-simulation-informed prescriptions — and whether richer assembly-bias degrees of freedom (velocity bias, halo concentration, tidal environment) degrade or preserve the PNG signal and its separation from late-time galaxy physics.

  2. Information content: quantify how much primordial bispectrum configuration information is lost when kNN-CDF measurements are used in place of the full bispectrum, in observationally realistic nonlinear settings. Through the counts-in-cells mapping the measurements carry the full counts-in-cells PDF, whereas the source's analytic model describes them, for the bulk of that PDF, solely through the scale-dependent skewness of the linear density field; the existing analytic Fisher comparison of skewness versus the full bispectrum on the linear field suggests near-completeness for equilateral PNG but significant loss for orthogonal PNG.

  3. Likelihood: characterize the full sampling distribution of kNN-CDF measurements — tail behavior and correlations across scales, neighbour order k, and parameter dependence — beyond the Gaussian approximation on compressed summaries, as required for optimal inference.

  4. Observational realism: obtain forecast PNG constraints for observationally representative galaxy samples (higher number densities, lower halo masses, requiring higher-resolution simulations than Quijote-PNG), covering such aspects as resolution-dependent effects, non-Poissonian shot-noise modeling across regimes, the impact of survey geometry, masks, light-cone evolution and multi-redshift combinations, and how kNN-CDF small-scale information should be combined with large-scale power-spectrum/bispectrum probes in a joint analysis.

Resolving the objective requires establishing all four validity conditions: any single condition can be resolved on its own, while the objective is resolved only once all four conditions are established. Within a numbered direction, the listed aspects describe the scope of that condition rather than a checklist each of which must be covered separately.

Two further items are acknowledged limitations of the source demonstration rather than enumerated open directions: the analytic counts-in-cells model is expected to hold only for the bulk of the PDF and not in its tails, with possible resolution-dependent deviations, and the non-Poissonian shot-noise parameters were refit per sample rather than derived.

References

  1. Coulton, W. R., Abel, T., Banerjee, A. (2024). Small-scale signatures of primordial non-Gaussianity in k-nearest neighbour cumulative distribution functions. Monthly Notices of the Royal Astronomical Society, 534, 1621-1633. doi:10.1093/mnras/stae2108, arXiv:2309.15151, https://arxiv.org/abs/2309.15151
  2. Yuan, S., Zamora, A., Abel, T. (2023). 2D k-th nearest neighbour statistics: a highly informative probe of galaxy clustering. Monthly Notices of the Royal Astronomical Society, 522, 3935-3947. doi:10.1093/mnras/stad1275, arXiv:2304.10565, https://arxiv.org/abs/2304.10565
  3. Friedrich, O., Uhlemann, C., Villaescusa-Navarro, F., Baldauf, T., Manera, M., Nishimichi, T. (2020). Primordial non-Gaussianity without tails - how to measure fNL with the bulk of the density PDF. Monthly Notices of the Royal Astronomical Society, 498, 464-483. doi:10.1093/mnras/staa2160, arXiv:1912.06621, https://arxiv.org/abs/1912.06621
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