P. Rosselló, F. -S. Kitaura, D. Forero-Sánchez, F. Sinigaglia, G. Favole

A comprehensive analysis of the cosmological large-scale structure derived from galaxy surveys involves field-level inference, which requires a forward modelling framework that simultaneously accounts for structure formation and tracer bias. While structure formation models are well-understood, the development of an effective field-level bias model remains imprecise, particularly in the context of tracer perturbation theory within Bayesian reconstruction methods, which we address in this work. To bridge this gap, we have developed a differentiable model that integrates augmented Lagrangian perturbation theory, nonlinear, nonlocal, and stochastic biasing. At the core of our approach is the Hierarchical Cosmic-Web Biasing Nonlocal (HICOBIAN) model, which provides a positive definite description of tracer bias while incorporating a long- and short-range nonlocal framework via cosmic-web regions and deviations from Poissonity in the likelihood. A key insight of our model is that transitions between cosmic-web regions are inherently smooth, which we implement using sigmoid-based gradient operations. This enables a fuzzy, and, hence, differentiable hierarchical cosmic-web description, making the model well-suited for machine learning frameworks. Our approach accurately reproduces the primordial density field within associated error bars, as validated by two- and three-point statistics in Fourier space. Furthermore, we demonstrate that the methodology approaches the maximum encoded information consistent with Poisson noise. We introduce a Bayesian field-level inference algorithm that leverages the same forward modelling framework used in galaxy, quasar, and Lyman alpha forest mock catalog generation -- including nonlinear, nonlocal and stochastic bias with redshift space distortions -- providing a unified and consistent approach to the analysis of large-scale cosmic structure.