A Universal Equation to Predict Ω_m from Halo and Galaxy Catalogs

We discover analytic equations that can infer the value of Ω_m from the positions and velocity moduli of halo and galaxy catalogs. The equations are derived by combining a tailored graph neural network (GNN) architecture with symbolic regression. We first train the GNN on dark matter halos from Gadget N-body simulations to perform field-level likelihood-free inference, and show that our model can infer Ω_m with ∼6% accuracy from halo catalogs of thousands of N-body simulations run with six different codes: Abacus, CUBEP³M, Gadget, Enzo, PKDGrav3, and Ramses. By applying symbolic regression to the different parts comprising the GNN, we derive equations that can predict Ω_m from halo catalogs of simulations run with all of the above codes with accuracies similar to those of the GNN. We show that, by tuning a single free parameter, our equations can also infer the value of Ω_m from galaxy catalogs of thousands of state-of-the-art hydrodynamic simulations of the CAMELS project, each with a different astrophysics model, run with five distinct codes that employ different subgrid physics: IllustrisTNG, SIMBA, Astrid, Magneticum, SWIFT-EAGLE. Furthermore, the equations also perform well when tested on galaxy catalogs from simulations covering a vast region in parameter space that samples variations in 5 cosmological and 23 astrophysical parameters. We speculate that the equations may reflect the existence of a fundamental physics relation between the phase-space distribution of generic tracers and Ω_m, one that is not affected by galaxy formation physics down to scales as small as 10 h ⁻¹ kpc.