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 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 } \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt \hbox { $ > $ } } 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 independently of the equation of state of dark energy with a precision of about \sigma \sim 4.5 \times 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 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 _ { \kappa } | \sim 10 ^ { -4 } .
This bound is one order of magnitude larger than what one would naively expect from the size of curvature perturbations , \sim 10 ^ { -5 } .