Spatial curvature from super-Hubble cosmological fluctuations

Author(s)

Abstract

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 <math display="inline"><msub><mi mathvariant="normal">Ω</mi><mi mathvariant="normal">K</mi></msub></math> today is <math display="inline"><mo>-</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>9</mn></mrow></msup></math> and that its mean is <math display="inline"><mo>+</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>9</mn></mrow></msup></math>, both being overwhelmed by a standard deviation of the order of <math display="inline"><msup><mn>10</mn><mrow><mo>-</mo><mn>5</mn></mrow></msup></math>. 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 <math display="inline"><msub><mi mathvariant="normal">Ω</mi><mi mathvariant="normal">K</mi></msub></math> are obtained if inflation lasts for more than a billion <math display="inline"><mi>e</mi></math>-folds.

Figures

Probability distribution function for $\OmegaKbar =(a H/\ksigma)^2 \OmegaK$ (red curve) for unrealistically large values of $\calPstar = 10^{-3}$ (and $\Ninf=100$), compared to a Gaussian of same mean and variance (black curve). Notice that the most probable value of $\OmegaKbar$ is slightly negative whereas the mean value remains slightly positive.

Probability distribution function for $\OmegaKbar =(a H/\ksigma)^2 \OmegaK$ (red curve) for the currently favored value of $\calPstar = 2.1 \times 10^{-9}$ and for a large number of {\efolds} $\Ninf=10^8$. The variance $\ev{\zetac^2}$ is no longer a small quantity and the distribution acquires heavy tails. Even though the width at half-maximum is $\order{\sqrt{\calPstar}}$, substantial values of $|\OmegaKbar|$ are not rare anymore. For comparison, the black curve shows a Gaussian of same mean and variance.