Relative information entropy and Weyl curvature of the inhomogeneous Universe

The observed arrow of time is commonly attributed to the growth of entropy, and in a cosmological setting this raises the question of how to define and quantify an entropy associated with the gravitational field itself Following Penrose’s Weyl Curvature Hypothesis (WCH), one expects gravitational entropy to be closely related to the Weyl tensor, vanishing in the early, almost FLRW universe and increasing as structure forms. Among the various proposals for a gravitational entropy, the relative information entropy introduced by Hosoya, Buchert and Morita (HBM) (Hosoya et al. 2004) stands out for its direct link to inhomogeneities of the matter density and to the Buchert averaging formalism. Previous work by Li et al. (Li et al. 2012, Li et al. 2015) showed, at linear order in scalar perturbations on a dust FLRW background, a formal correspondence between the HBM entropy, the kinematical backreaction and the averaged Weyl scalar invariant W. The aim of this thesis is to test the robustness and generality of this correspondence beyond that restricted setting. After reviewing the motivation for gravitational entropy and several existing proposals, we establish the necessary formalism: the 1+3 covariant split, Buchert averaging, and cosmological perturbation theory in the synchronous comoving gauge. We derive covariant expressions for the electric and magnetic parts of the Weyl tensor, for W and for its time derivative, in terms of the expansion, shear and extrinsic curvature of the spatial slices. We then re-derive the linear relation found by Li et al. and extend the analysis in two directions: (i) to linear perturbations including tensor modes, and (ii) to the next-to-leading order in the non-linear perturbation expansion (third order in the density contrast). At linear order we find that the HBM functional remains insensitive to tensor modes, depending only on the scalar density perturbation, while the Weyl scalar does carry tensor contributions through both its electric and magnetic parts. These tensor contributions, however, decay rapidly in the dust regime and are subdominant at late times. At higher order, we show that the linear equality between the HBM entropy and a combination of the averaged Weyl scalar and the kinematical backreaction does not extend straightforwardly. Finally, by expressing both the HBM gravitational entropy and the time evolution of the average of the Weyl scalar invariant and of the kinematical backreaction in terms of the extrinsic curvature, we find no simple link between the first two. Instead, we identify a closer structural similarity between the entropy production rate and the kinematical backreaction, up to additional curvature terms. This suggests that, in the regimes explored here, the HBM proposal is more naturally tied to the backreaction mechanism than to a monotonic growth of a Weyl-based gravitational entropy in the sense envisaged by the WCH. The linear correspondence with the Weyl scalar therefore, appears as a special feature of first-order scalar perturbations, rather than a generic property of the HBM functional at the non-linear level.