How flat can you get? A model comparison perspective on the curvature of the Universe

Abstract

The question of determining the spatial geometry of the Universe is of greater relevance than ever, as precision cosmology promises to verify inflationary predictions about the curvature of the Universe. We revisit the question of what can be learnt about the spatial geometry of the Universe from the perspective of a three-way Bayesian model comparison. We show that, given current data, the probability that the Universe is spatially infinite lies between 67% and 98%, depending on the choice of priors. For the strongest prior choice, we find odds of order 50:1 (200:1) in favour of a flat Universe when compared with a closed (open) model. We also report a robust, prior-independent lower limit to the number of Hubble spheres in the Universe, N_U > 5 (at 99% confidence). We forecast the accuracy with which future CMB and BAO observations will be able to constrain curvature, finding that a cosmic variance limited CMB experiment together with an SKA-like BAO observation will constrain curvature with a precision of about sigma ~ 4.5x10^{-4}. We demonstrate that the risk of 'model confusion' (i.e., wrongly favouring a flat Universe in the presence of curvature) is much larger than might be assumed from parameter errors forecasts for future probes. We argue that a 5-sigma detection threshold guarantees a confusion- and ambiguity-free model selection. Together with inflationary arguments, this implies that the geometry of the Universe is not knowable if the value of the curvature parameter is below |Omega_curvature| ~ 10^{-4}, a bound one order of magnitude larger than the size of curvature perturbations, ~ 10^{-5}. [abridged]