Probabilistic Lagrangian bias estimators and the cumulant bias expansion

¹ Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain
² Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
³ IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain

^⋆ Corresponding authors: jens.stuecker@univie.ac.at, mpelleje@ed.ac.uk

Received: 19 June 2024
Accepted: 1 May 2025

Abstract

The spatial distribution of galaxies is a highly complex phenomenon that is impossible to predict deterministically at present. However, by using a statistical ‘bias’ relation, it has become feasible to robustly model the average abundance of galaxies as a function of the underlying matter density field. Understanding the properties and parametric description of the bias relation is key to extracting cosmological information from future galaxy surveys. In this work, we contribute to this topic primarily in two ways. First, we have developed a set of ‘probabilistic’ estimators for bias parameters using the moments of the Lagrangian galaxy environment distribution. These estimators include spatial corrections at different orders to measure the bias parameters independently of the damping scale. We report robust measurements of a variety of bias parameters for halos, including the tidal bias and its dependence on spin at a fixed mass. Second, we have proposed an alternative formulation of the bias expansion in terms of ‘cumulant bias parameters’, which describe the response of the logarithmic galaxy density to large-scale perturbations. We find that cumulant biases of halos are consistent with zero at orders of n > 2. This suggests that: (i) previously reported bias relations at the order of n > 2 are an artefact of the entangled basis of the canonical bias expansion; (ii) the convergence of the bias expansion may be improved by expressing it in terms of cumulants; and (iii) the bias function is very well approximated by a Gaussian. We explore these avenues in greater depth in a companion paper.