On the curvature of the present-day universe

We discuss the effect of curvature and matter inhomogeneities on the averaged
scalar curvature of the present-day universe. Motivated by studies of averaged
inhomogeneous cosmologies, we contemplate on the question of whether it is
sensible to assume that curvature averages out on some scale of homogeneity,
as implied by the standard concordance model of cosmology, or whether
the averaged scalar curvature can be largely negative today, as required for
an explanation of dark energy from inhomogeneities. We confront both
conjectures with a detailed analysis of the kinematical backreaction term and
estimate its strength for a multi-scale inhomogeneous matter and curvature
distribution. Our main result is a formula for the spatially averaged scalar
curvature involving quantities that are all measurable on regional (i.e. up to
100 Mpc) scales. We propose strategies to quantitatively evaluate the formula,
and pinpoint the assumptions implied by the conjecture of a small or zero
averaged curvature. We reach the conclusion that the standard concordance
model needs fine tuning in the sense of an assumed equipartition law for
curvature in order to reconcile it with the estimated properties of the averaged
physical space, whereas a negative averaged curvature is favoured, independent
of the prior on the value of the cosmological constant.