Very small mean curvature is a robust prediction of inflation worth rigorous checking. Since current constraints are derived from determinations of the angular-diameter distance to the CMB last-scattering surface, which is also affected by dark energy, they are limited by our understanding of the dark energy. Measurements of luminosity or angular-diameter distances to redshifts in the matter-dominated era can greatly reduce this uncertainty. With a one percent measurement of the distance to z=3, combined with the CMB data expected from Planck, one can achieve sigma(��_k h^2) \sim 10^{-3}. A non-zero detection at this level would be evidence against inflation or for unusually large curvature fluctuations on super-Hubble scales.

7 pages accepted for publication in PRD. Final published version with more on how the curvature constraints depend on the technique used for measuring distance