What if Planck’s Universe isn’t flat?

Inflationary theory predicts that the observable Universe should be very close to flat, with a spatial-curvature parameter |ΩK|≲10−4. The WMAP satellite currently constrains |ΩK|≲0.01, and the Planck satellite will be sensitive to values near 10−3. Suppose that Planck were to find ΩK≠0 at this level. Would this necessarily be a serious problem for inflation? We argue that an apparent departure from flatness could be due either to a local (wavelength comparable to the observable horizon) inhomogeneity, or a truly superhorizon departure from flatness. If there is a local inhomogeneity, then secondary cosmic microwave background (CMB) anisotropies distort the CMB frequency spectrum at a level potentially detectable by a next-generation experiment. We discuss how these spectral distortions would complement constraints on the Grishchuk-Zel’dovich effect from the low-ℓ CMB power spectrum in discovering the source of the departure from flatness.