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.
