Stress-Testing Small-Scale Cosmological Inference With Empirical Galaxy--Halo Connection Models

Small-scale galaxy clustering contains substantial cosmological information, but extracting it reliably requires robust modeling of the galaxy--halo connection. In this work, we compare two widely adopted families of empirical models, subhalo abundance matching (SHAM) and the halo occupation distribution (HOD), testing their predictions for galaxy clustering and their recovery of the input cosmology. Using Aemulus-$β$, a new multi-cosmology simulation suite with substructure merger trees, we fit multiple SHAM and HOD models to the small-scale clustering of a stellar-mass-selected mock galaxy catalog generated with the more physical semi-empirical model UniverseMachine. We then use cosmological evidence modeling to infer the structure growth parameter S₈. Our results show that the SHAM model, regardless of the treatment of numerically disrupted subhalos, is able to offer tight, unbiased constraints on S₈ with ~2-3% precision, using simulations of relatively small volumes that span a limited number of cosmologies. In contrast, HOD models, even when accounting for assembly bias, yield weaker and potentially biased cosmological constraints that are sensitive to implementation choices: the inferred S₈ degrades in precision to ~6-11% and can be offset by ~1$σ$. This suggests that the excessive parametric flexibility and model misspecification of most commonly adopted HOD models can degrade and bias the resulting cosmological constraints. We further evaluate each model's performance in predicting higher-order statistics not used in the fit and find that SHAM generally reproduces them more accurately than the tested HOD models. These results highlight the potential of leveraging predictive subhalo-level information from simulations to enable robust cosmological inference on small scales.