How Much Information Can Be Extracted from Galaxy Clustering at the Field Level?

We present optimal Bayesian field-level cosmological constraints from nonlinear tracers of cosmic large-scale structure, specifically the amplitude σ_{8} of linear matter fluctuations inferred from rest-frame simulated dark matter halos in a comoving volume of 8  (h^{-1} Gpc)^{3}. Our constraint on σ_{8} is entirely due to nonlinear information, and obtained by explicitly sampling the initial conditions along with tracer bias and noise parameters via a Lagrangian effective field theory-based forward model, leftfield. The comparison with a simulation-based inference of the power spectrum and bispectrum-likewise using the leftfield forward model-shows that, when including precisely the same modes of the same data up to k_{max}=0.10  h Mpc^{-1} (0.12  h Mpc^{-1}), the field-level approach yields a factor of 3.5 (5.2) improvement in the σ_{8} constraint, going from 20.0% to 5.7% (17.0% to 3.3%). This study provides direct insights into cosmological information encoded in galaxy clustering beyond low-order n-point functions.