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 the large-scale structure, specifically the amplitude ${\sigma }_8$ of linear matter fluctuations inferred from rest-frame simulated dark matter halos in a comoving volume of $8(h^{-1}\mathrm{G}\mathrm{p}\mathrm{c}{)}^3$. Our constraint on ${\sigma }_8$ is entirely due to nonlinear information, and obtained by explicitly sampling the initial conditions along with bias and noise parameters via a Lagrangian EFT-based forward model, LEFTfield. The comparison with a simulation-based inference analysis employing 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 $k_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.10h{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ ($0.12h{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$), the field-level approach yields a factor of 3.5 (5.2) improvement on the ${\sigma }_8$ constraint, 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.