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

The gist Exploring non-Poisson satellite occupation in HOD models and its impact on 2- and 3-point galaxy clustering introduces the Conway–Maxwell–Poisson (CMP) distribution as a minimal extension to standard Halo Occupation Distribution (HOD) models to explore sub- and super-Poisson behavior in satellite galaxy populations, which significantly affects small-scale clustering statistics.

Halo Occupation Statistics

The standard HOD framework describes how galaxies populate dark matter halos through the conditional probability distribution P(Ngal Mh), which is typically decomposed into central and satellite galaxy contributions The number of satellite galaxies is commonly assumed to follow a Poisson distribution with mean λ = ⟨Nsat⟩, meaning P(k) = λ k e −λ k! (2.1) The variance of this Poisson distribution is equal to its mean, Var(Nsat M) = ⟨Nsat(M)⟩ (2.3). This assumption is motivated by the interpretation of satellites as independent tracers within an halo, arising from the superposition of independent merger events However, numerical simulations and observational analyses have suggested that satellite occupation may deviate from strict Poisson behaviour, particularly in regimes of high halo mass.

Describing Satellite Occupation with the Conway–Maxwell–Poisson Distribution

The CMP distribution is introduced as a minimal generalization of the Poisson model for satellite occupation, introducing a single dispersion parameter ν that controls deviations from Poisson behavior while preserving the discrete count nature of the occupation statistics. The probability mass function is given by P(Nsat = k λ, ν) = λ k (k!)νZ(λ, ν) (3.1), where Z(λ, ν) is a normalisation constant defined as Z(λ, ν) = X∞ j=0 λ j (j!)ν (3.2). The Poisson distribution is recovered when ν = 1. The additional parameter ν controls the dispersion of the distribution: ν 1 ⇒ sub-Poissonian.

Mean and Variance Scaling Across λ Values

The mean and variance of the CMP distribution are defined as E[X] = λ ∂∂λ log Z(λ, ν) and Var(X) = λ ∂∂λE[X] (3.3). The primary analytical difficulty lies in the normalizing constant Z(λ, ν), which has no closed form. Different approximations are developed depending on the value of λ as the structure of the CMP distribution changes qualitatively depending on its value. For small λ, E[X] ∼ λ and Var(X) ∼ λ (3.8), meaning it is locally indistinguishable from a Poisson distribution at sufficiently small value of λ. For large λ, the mean of the CMP distribution scales as E[X] ∼ λ 1/ν (3.15), unlike the Poisson distribution where E[X] = λ. A continuous transition between these regimes is constructed using an error function, resulting in a smooth interpolation that allows to recover the expected mean value λ from the CMP distribution on all scales within 5% for 0.6 < ν < 2 and within 10% for ν ∼ 0.5.

Impact of the CMP Distribution on HOD Galaxy Mocks

The CMP distribution is implemented into the HODDIES code to evaluate its impact on galaxy catalogue generation and clustering. The first sanity check validates the scaling with ν described in Section 3.5, showing that the variance increases for ν 1 (sub-Poisson regime). This behavior leads to a broader (narrower) distribution of satellite counts in halos for ν 1). The deviation from Poisson statistics becomes more pronounced with increasing halo mass, reflecting the stronger departure from Poisson statistics at larger λ.

Implications on Clustering Statistics

The impact of non-Poisson satellite occupation is investigated through several clustering statistics. For the 2-point correlation function, a larger variance increases the probability of high-occupancy satellite realizations, thereby enhancing the one-halo contribution at small scales. A larger variance increases the probability of high-occupancy satellite realizations, thereby enhancing the one-halo contribution. For higher-order statistics, Count-inCylinder (CIC) statistics are highly sensitive to changes in the variance, with variations up to ∼ 30%. The CIC distribution shows a higher number of cylinder with higher galaxy counts and oppositely, a smaller number of cylinder with lower galaxy counts when the variance of the satellite occupation is larger (ν < 1) compare to the Poisson case. This highlights the sensitivity of CIC to higher-order moments of the galaxy distribution and to test deviations from Poisson statistics in satellite populations. Furthermore, deviations from Poisson statistics lead to relative differences of up to ∼ 10% in wp and ∼ 5% in ξ0 and ξ2 at small scales. While moderate, these effects can be significant compare to the precision of current observational datasets. The bispectrum measurements are robust against the change of Poisson assumption for kmax < 0.3 and therefore we do not expect strong biases on cosmological constraints. However, a detailed analysis has to be done to investigate more on the impact of non Poisson satellite distribution on cosmological constrain using the bispectrum, and would be the purpose of a future work.

Conclusion

The CMP distribution provides a robust and efficient framework to probe deviations from Poisson satellite statistics in galaxy surveys. It offers a direct and minimally extended way to test the Poisson assumption within the HOD framework using current and upcoming datasets. Future work will focus on constraining the CMP parameter ν using observational data from DESI, exploring its dependence on galaxy properties and halo environment, and assessing the impact of non-Poisson satellite statistics on higher-order clustering observables and cosmological parameter inference.

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The gist Exploring non-Poisson satellite occupation in HOD models and its impact on 2- and 3-point galaxy clustering introduces the Conway–Maxwell–Poisson (CMP) distribution as a minimal extension to standard Halo Occupation Distribution (HOD) models to explore sub- and super-Poisson behavior in satellite galaxy populations, which significantly affects small-scale clustering statistics.

1. Confidence in small-scale clustering analysis can be enhanced by incorporating a single parameter, ν, to control sub- and super-Poissonian satellite occupation behavior within Halo Occupation Distribution (HOD) models. This allows for a minimal and flexible extension of the Poisson model to better capture realistic galaxy populations in mock catalogs.

2. AI systems can generate more physically accurate mock galaxy catalogues by implementing the Conway–Maxwell–Poisson (CMP) distribution into HODDIES, as described in Section 4.1. This directly tests the impact of non-Poisson satellite statistics on mock galaxy catalogues and clustering statistics.

3. The improved AI system can perform precise small-scale clustering measurements by using Count-in-Cylinders (CIC) statistics to probe higher-order moments of the galaxy distribution, as these are particularly sensitive to variations in the variance of the satellite occupation and show variation up to 30% relative to the Poisson case.

4. The system can better constrain cosmological models by performing analyses on two-point correlation function multipoles (monopole and quadrupole) using mock data generated with varying ν, as these observables show a modification of the one-halo term that depends on ν.

5. AI systems can perform more robust cosmological constraint inference by utilizing analytical approximations derived for the CMP mean and variance, allowing them to invert the CMP mean and recover the value of λ corresponding to a target mean occupation, ensuring that the physical interpretation of parameters is preserved.

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, ν, to explore sub- and super-Poisson behaviour. We derive analytical approximations for the CMP expectation parameter λ and develop a numerical scheme that smoothly connects small- and large- λ regimes, achieving about5% accuracy for 0.5 < ν< 2. Using the 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 about30%, while the tree-level galaxy bispectrum (in the Sugiyama basis) is only weakly affected (<2% up to k 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 max < 0.3.