Differentiable fuzzy cosmic web for field-level inference

¹ Instituto de Astrofísica de Canarias, s/n, E-38205 La Laguna, Tenerife, Spain
² Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain
³ Departament de Física Quàntica i Astrofísica, Universitat de Barcelona, Martí i Franquís 1, E08028 Barcelona, Spain
⁴ Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (UB), c. Martí i Franquès, 1, 08028 Barcelona, Spain
⁵ Département d’Astronomie, Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland
⁶ Institut für Astrophysik, Universitüt Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

^★ Corresponding authors: This email address is being protected from spambots. You need JavaScript enabled to view it. , This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 4 June 2025
Accepted: 26 February 2026

Abstract

Context. 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.

Aims. While structure formation models are well understood, the development of an effective field-level bias model remains challenging, particularly in the context of tracer perturbation theory within Bayesian reconstruction methods, which we address in this work.

Methods. To bridge this gap, we developed a differentiable model that integrates augmented Lagrangian perturbation theory and non-linear, non-local, and stochastic biasing. At the core of our approach is the Hierarchical Cosmic-Web Biasing Nonlocal (HICOBIAN) model, which provides a description of a field with a positive number of tracers while incorporating a long- and short-range non-local 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 implemented 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.

Results. We tested the practical implementation of this model through GPU-accelerated computations implemented in JAX, the BRIDGE code, enabling a scalable evaluation of complex biasing. Our approach accurately reproduces the primordial density field within associated error bars derived from Bayesian posterior sampling within a self-specified setting (meaning that inference is performed on data generated by the exact same forward model) 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 also demonstrate that the bias parameters of a higher order non-local-bias model can be accurately reconstructed within the Bayesian framework for bias models with eight parameters.

Conclusions. We introduce a Bayesian field-level inference algorithm that leverages the same forward-modelling framework used in galaxy, quasar, and Lyman-alpha-forest mock-catalogue generation – including non-linear, non-local and stochastic bias with redshift space distortions – providing a unified and consistent approach to the analysis of large-scale cosmic structure.