Spatial curvature endgame: Reaching the limit of curvature determination

Current constraints on spatial curvature show that it is dynamically negligible: $|\Omega_{\rm K}| \lesssim 5 \times 10^{-3}$ (95% CL). Neglecting it as a cosmological parameter would be premature however, as more stringent constraints on $\Omega_{\rm K}$ at around the $10^{-4}$ level would offer valuable tests of eternal inflation models and probe novel large-scale structure phenomena. This precision also represents the "curvature floor", beyond which constraints cannot be meaningfully improved due to the cosmic variance of horizon-scale perturbations. In this paper, we discuss what future experiments will need to do in order to measure spatial curvature to this maximum accuracy. Our conservative forecasts show that the curvature floor is unreachable - by an order of magnitude - even with Stage IV experiments, unless strong assumptions are made about dark energy evolution and the $\Lambda$CDM parameter values. We also discuss some of the novel problems that arise when attempting to constrain a global cosmological parameter like $\Omega_{\rm K}$ with such high precision. Measuring curvature down to this level would be an important validation of systematics characterisation in high-precision cosmological analyses.