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arxiv: 2605.28644 · v1 · pith:VZAYQLNKnew · submitted 2026-05-27 · 🌌 astro-ph.CO

Exploring non-Poisson satellite occupation in HOD models and its impact on 2- and 3-point galaxy clustering

Antoine Rocher This is my paper

Pith reviewed 2026-06-29 10:39 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords halo occupation distributionsatellite occupationgalaxy clusteringConway-Maxwell-Poissoncounts-in-cylindersgalaxy bispectrumnon-Poisson statistics
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The pith

Non-Poisson satellite numbers in halos change small-scale galaxy clustering by up to 10 percent.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard HOD models assume satellite galaxies follow a Poisson distribution at fixed halo mass. This paper replaces that assumption with the Conway-Maxwell-Poisson distribution controlled by a single extra parameter that allows the variance to be sub- or super-Poisson. The resulting changes produce clear shifts in projected two-point clustering and in counts-in-cylinders statistics on small scales, while the tree-level galaxy bispectrum on larger scales remains almost unchanged. A reader should care because small-scale clustering measurements are increasingly used to extract cosmological information, so an incorrect variance assumption could bias those inferences.

Core claim

Replacing the Poisson satellite occupation with the CMP distribution at fixed halo mass alters the variance and thereby shifts the projected correlation function by up to 10 percent, the monopole and quadrupole by up to 5 percent, and counts-in-cylinders by up to 30 percent, while the tree-level bispectrum in the Sugiyama basis changes by less than 2 percent for k_max less than or equal to 0.3.

What carries the argument

The Conway-Maxwell-Poisson distribution, which adds a single dispersion parameter ν to the Poisson model and thereby sets the variance of the satellite count at fixed halo mass.

If this is right

  • Small-scale two-point clustering analyses must allow for non-Poisson satellite variance to avoid systematic errors of several percent.
  • Counts-in-cylinders measurements become a direct probe of the satellite variance parameter.
  • Tree-level bispectrum constraints on scales k less than 0.3 remain robust against this modeling choice.
  • Mock catalogs built with the HODDIES package can incorporate the CMP extension once the numerical scheme for λ is inserted.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Group catalogs from spectroscopic surveys could measure the actual variance of satellite numbers and thereby constrain ν directly.
  • Surveys that combine small-scale clustering with higher-order statistics may need to marginalize over ν to keep cosmological parameter errors unbiased.
  • The limited effect on the bispectrum suggests that any non-Poisson signal is largely absorbed into the two-point function on the scales currently used for cosmology.

Load-bearing premise

The numerical scheme that links the small- and large-λ regimes for the CMP expectation parameter achieves roughly 5 percent accuracy across the relevant range of ν.

What would settle it

A direct count of satellite galaxies per halo at fixed mass that yields a variance outside the range spanned by 0.5 less than ν less than 2, or measured counts-in-cylinders that deviate from the predicted 30 percent shift when ν is varied.

read the original abstract

Understanding the connection between galaxies and dark matter halos is a central challenge in modern cosmology. The Halo Occupation Distribution (HOD) framework provides a widely used statistical description of how galaxies populate dark matter halos, enabling precise modelling of galaxy clustering. A common assumption in standard HOD models is that the number of satellite galaxies follows a Poisson distribution at fixed halo mass. In this work, we revisit this assumption and introduce the Conway-Maxwell-Poisson (CMP) distribution as a minimal extension of of the Poisson model, which add a single parameter, $\nu$, to explore sub- and super-Poisson behaviour. We derive analytical approximations for the CMP expectation parameter $\lambda$ and develop a numerical scheme that smoothly connects small- and large-$\lambda$ regimes, achieving $\sim5\%$ accuracy for $0.5 < \nu < 2$. Using the \texttt{HODDIES} package, we study the impact of non-Poisson satellite occupations on mock galaxy catalogues and clustering statistics. Variations in the variance of the satellite occupation significantly affect small-scale clustering, producing deviations of up to $10\%$ in projected clustering and $5\%$ in the monopole and quadrupole. We further investigate higher-order statistics using counts-in-cylinders (CiC) and the tree-level galaxy bispectrum. CiC statistics are highly sensitive to changes in the variance, with variations up to $\sim30\%$, while the tree-level galaxy bispectrum (in the Sugiyama basis) is only weakly affected ($<2\%$ up to $k_\mathrm{max} = 0.3$). These results suggest that non-Poisson satellite statistics are important for small-scale analyses, but should have a limited impact on cosmological constraints from power spectrum and bispectrum measurements using large scales $k_\mathrm{max} < 0.3$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces the Conway-Maxwell-Poisson (CMP) distribution as a one-parameter extension to the standard Poisson satellite occupation in HOD models, controlled by ν. It derives analytical approximations for the CMP mean parameter λ and presents a numerical scheme connecting small- and large-λ regimes that achieves ~5% accuracy for 0.5<ν<2. The scheme is implemented in the HODDIES package to generate mocks, which are then used to quantify the impact of non-Poisson variance on the projected correlation function (deviations up to 10%), monopole and quadrupole (up to 5%), counts-in-cylinders (up to ~30%), and tree-level galaxy bispectrum (<2% up to k_max=0.3).

Significance. If the numerical scheme is shown to be accurate and the reported percentage shifts are robust to its errors, the results would establish that satellite occupation variance is an important modeling choice for small-scale clustering analyses (particularly CiC) while having only limited impact on large-scale power-spectrum and bispectrum cosmology. The work supplies a concrete, minimal extension and quantitative benchmarks that could be adopted or tested by other HOD implementations.

major comments (1)
  1. [Numerical scheme for CMP λ (methods section describing the approximation and its insertion into HODDIES)] The central results rest on inserting the numerical approximation for CMP λ(ν) into HODDIES. The manuscript states that the scheme achieves ~5% accuracy across 0.5<ν<2, but provides no direct comparison to exact CMP root-finding, no error budget on the resulting <N_sat> or variance at the halo masses that dominate small-scale pairs, and no propagation test showing that a 5% λ error produces <1% bias in the quoted 10% wp or 30% CiC shifts. This validation step is load-bearing for attributing the deviations to the physical non-Poisson effect rather than to the approximation.
minor comments (1)
  1. [Abstract] Abstract contains a typographical repetition: 'minimal extension of of the Poisson model'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for stronger validation of the numerical scheme. We address the single major comment below.

read point-by-point responses

  1. Referee: [Numerical scheme for CMP λ (methods section describing the approximation and its insertion into HODDIES)] The central results rest on inserting the numerical approximation for CMP λ(ν) into HODDIES. The manuscript states that the scheme achieves ~5% accuracy across 0.5<ν<2, but provides no direct comparison to exact CMP root-finding, no error budget on the resulting <N_sat> or variance at the halo masses that dominate small-scale pairs, and no propagation test showing that a 5% λ error produces <1% bias in the quoted 10% wp or 30% CiC shifts. This validation step is load-bearing for attributing the deviations to the physical non-Poisson effect rather than to the approximation.

    Authors: We agree that the current presentation lacks sufficient validation of the numerical scheme. In the revised manuscript we will add (i) a direct comparison of the approximate λ(ν) against exact numerical root-finding over the full range 0.5<ν<2, (ii) an error budget on the resulting mean and variance of N_sat evaluated at the halo masses that dominate the small-scale pair counts, and (iii) a propagation test that injects the maximum 5% λ error into the mock generation and quantifies the induced bias in wp, the multipoles, and CiC. These additions will be placed in the methods section and will demonstrate that the reported percentage shifts remain attributable to the non-Poisson variance rather than to the approximation error. revision: yes

Circularity Check

0 steps flagged

No significant circularity; clustering shifts are direct numerical outputs from HOD mocks

full rationale

The paper's central claims consist of measured percentage deviations in wp, monopole/quadrupole, CiC, and bispectrum obtained by inserting the CMP distribution (with parameter ν) into the HODDIES package and running mock catalogs. These shifts are simulation outputs, not quantities defined in terms of ν or λ by construction. The stated ~5% accuracy of the λ(ν) numerical scheme is an independent approximation claim whose validation is not shown to rely on the clustering results themselves. No self-definitional, fitted-input-as-prediction, or load-bearing self-citation steps are present in the provided text; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard HOD framework plus one new dispersion parameter. No new particles or forces are postulated.

free parameters (1)
  • ν
    Dispersion parameter in the Conway-Maxwell-Poisson distribution that controls deviation from Poisson variance; its value is explored rather than fitted to data in the reported experiments.
axioms (1)
  • domain assumption The Conway-Maxwell-Poisson distribution is a minimal and sufficient extension of the Poisson model for satellite occupation at fixed halo mass.
    Invoked when the authors state that CMP 'add[s] a single parameter, ν, to explore sub- and super-Poisson behaviour'.

pith-pipeline@v0.9.1-grok · 5876 in / 1601 out tokens · 41315 ms · 2026-06-29T10:39:21.432110+00:00 · methodology

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Reference graph

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