On the Significance of Covariance for Constraining Theoretical Models From Galaxy Observables

In this study, we investigate the impact of covariance within uncertainties on the inference of cosmological and astrophysical parameters, specifically focusing on galaxy stellar mass functions derived from the CAMELS simulation suite. Utilizing both Fisher analysis and Implicit Likelihood Inference (ILI), we explore how different covariance structures, including simple toy models and physics-motivated uncertainties, affect posterior distributions and parameter variances. Our methodology utilizes forward modeling via emulators that are trained on CAMELS simulations to produce stellar mass functions based on input parameters, subsequently incorporating Gaussian noise as defined by covariance matrices. We examine both toy model covariance matrices and physically motivated covariance matrices derived from observational factors like the stellar Initial Mass Function (IMF) and photometric aperture size. Our results demonstrate that covariance terms significantly influence parameter inference, often leading to tighter constraints or revealing complex, multimodal posterior distributions. These findings underscore the necessity of accounting for covariance when interpreting astrophysical observations, especially in fields where accurate parameter estimation is critical for model validation and hypothesis testing.