Incorporating curved geometry in cosmological simulations

Spatial curvature is one of the fundamental cosmological parameters that is routinely constrained from observations. The forward modelling of observations, in particular of large-scale structure, often relies on large cosmological simulations. While the so-called separate universe approach allows one to account for the effect of curvature on the expansion rate in small sub-volumes, the non-Euclidean geometry is harder to accommodate. It becomes important when observables are computed over large distances, e.g. when photons travel to us from high redshift. Here we present a fully relativistic framework to run cosmological simulations for curved spatial geometry. The issue of consistent boundary conditions is solved by embedding a spherical cap of the curved spacetime into a hole within a flat exterior, where it can undergo free expansion. The geometric nature of gravity is made explicit in our framework, allowing for a consistent forward modelling of observables inside the curved patch. Our methodology would also work with any Newtonian code to a good approximation, requiring changes only to the initial conditions and post-processing.