  Penrose conjectured a connection between entropy and Weyl curvature of the
Universe. This is plausible, as the almost homogeneous and isotropic Universe
at the onset of structure formation has negligible Weyl curvature, which then
grows (relative to the Ricci curvature) due to the formation of large-scale
structure and thus reminds us of the second law of thermodynamics. We study two
scalar measures to quantify the deviations from a homogeneous and isotropic
space-time, the relative information entropy and a Weyl tensor invariant, and
show their relation to the averaging problem. We calculate these two quantities
up to second order in standard cosmological perturbation theory and find that
they are correlated and can be linked via the kinematic backreaction of a
spatially averaged universe model.
