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

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. By considering two classes of phenomenological priors for the curvature parameter, we show that, given the current data, the probability that the Universe is spatially infinite lies between 67 and 98 per cent, depending on the choice of priors. For the strongest prior choice, we find odds of the order of 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 per cent confidence). We forecast the accuracy with which future cosmic microwave background (CMB) and baryonic acoustic oscillation (BAO) observations will be able to constrain curvature, finding that a cosmic variance-limited CMB experiment together with an Square Kilometer Array (SKA)-like BAO observation will constrain curvature independently of the equation of state of dark energy with a precision of about σ ~ 4.5 × 10^-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 error forecasts for future probes. We argue that a 5σ 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 |Ω_κ| ~ 10^-4. This bound is one order of magnitude larger than what one would naively expect from the size of curvature perturbations, ~10^-5.