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 $\sigma ({\Omega }_k{h}^2)\sim {10}^{-3}$. A nonzero detection at this level would be evidence against inflation or for unusually large curvature fluctuations on super-Hubble scales.

Figure 1

Uncertainty in ${\Omega }_k{h}^2$ for WMAP (higher curves) and Planck (lower curves) as a function of the fractional error in $D_{0M}$ (left panel), $D_{0M}/r_s^{*}$ (middle panel), and $\sqrt{{\omega }_m}{D}_{0M}$ (right panel) for $z_M=2$ (solid line), $z_M=3$ (dotted line), $z_M=4$ (dashed line), and $z_M=10$ (dotted-dashed line). The straight lines show the amount of error resulting from neglecting the dark energy contribution at $z>z_M$ assuming our fiducial model with the dark energy as a cosmological constant.