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 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 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₌₀ₗ=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 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.