Spatial curvature from super-Hubble cosmological fluctuations

We revisit how super-Hubble cosmological fluctuations induce, at any time in the cosmic history, a nonvanishing spatial curvature of the local background metric. The random nature of these fluctuations promotes the curvature density parameter to a stochastic quantity for which we derive novel nonperturbative expressions for its mean, variance, higher moments, and full probability distribution. For scale-invariant Gaussian perturbations, such as those favored by cosmological observations, we find that the most probable value for the curvature density parameter ωK today is -10-9 and that its mean is +10-9, both being overwhelmed by a standard deviation of the order of 10-5. We then discuss how these numbers would be affected by the presence of large super-Hubble non-Gaussianities or if inflation lasted for a very long time. In particular, we find that substantial values of ωK are obtained if inflation lasts for more than a billion e-folds.