How flat is our Universe really?

Distance measurement provides no constraints on curvature independent of assumptions about the dark energy, raising the question, how flat is our Universe if we make no such assumptions? Allowing for general evolution of the dark energy equation of state with 20 free parameters that are allowed to cross the phantom divide, w(z)=−1, we show that while it is indeed possible to match the first peak in the Cosmic Microwave Background with non-flat models and arbitrary Hubble constant, H0, the full WMAP7 and supernova data alone imply −0.12<Ωk<0.01 (2σ). If we add an H0 prior, this tightens significantly to Ωk=0.002±0.009. These constitute the most conservative and model-independent constraints on curvature available today, and illustrate that the curvature-dynamics degeneracy is broken by current data, with a key role played by the Integrated Sachs Wolfe effect rather than the distance to the surface of last scattering. If one imposes a quintessence prior on the dark energy (−1⩽w(z)⩽1) then just the WMAP7 and supernova data alone force the Universe to near flatness: Ωk=0.013±0.012. Finally, allowing for curvature, we find that all datasets are consistent with a Harrison-Zelʼdovich spectral index, ns=1, at 2σ, illustrating the interplay between early and late Universe constraints.