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

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 $k_\mathrm{max} < 0.3$.