Precision measurement of the mean curvature

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 1% measurement of the distance to z=3, combined with the CMB data expected from Planck, one can achieve σ(Ω_kh²)∼10^-3. A nonzero detection at this level would be evidence against inflation or for unusually large curvature fluctuations on super-Hubble scales.