Spatial Curvature from Super-Hubble Cosmological Fluctuations

We revisit how super-Hubble cosmological fluctuations induce, at any time in the cosmic history, a non-vanishing 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 non-perturbative 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 $\Omega_\mathrm{K}$ today is $-10^{-9}$, that its mean is $+10^{-9}$, both being overwhelmed by a standard deviation of order $10^{-5}$. 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 $\Omega_\mathrm{K}$ are obtained if inflation lasts for more than a billion e-folds.