Gravitational entropy and the cosmological no-hair conjecture

The gravitational entropy and no-hair conjectures seem to predict contradictory future states of our Universe. The growth of the gravitational entropy is associated with the growth of inhomogeneity, while the no-hair conjecture argues that a universe dominated by dark energy should asymptotically approach a homogeneous and isotropic de Sitter state. The aim of this paper is to study these two conjectures. The investigation is based on the Simsilun simulation, which simulates the universe using the approximation of the Silent Universe. The Silent Universe is a solution to the Einstein equations that assumes irrotational, nonviscous, and insulated dust, with vanishing magnetic part of the Weyl curvature. The initial conditions for the Simsilun simulation are sourced from the Millennium simulation, which results with a realistically appearing but relativistic at origin simulation of a universe. The Simsilun simulation is evolved from the early universe ($t=25\mathrm{Myr}$) until far future ($t=1000\mathrm{Gyr}$). The results of this investigation show that both conjectures are correct. On global scales, a universe with a positive cosmological constant and nonpositive spatial curvature does indeed approach the de Sitter state. At the same time it keeps generating the gravitational entropy.

Figure 1

The evolution of the volume averages of the density field normalized by the present-day density (upper most panel); expansion rate normalized by the present-day expansion rate (second upper panel); shear normalized by the one-sixth of the present-day expansion rate [cf. (27)] and multiplied by $-1$ so that is can be presented in the log-y plot (second lower panel); the Weyl curvature normalized by the one-sixth of the present-day density (lower most panel). As seen, asymptotically the system approaches the de Sitter state, i.e. ${\rho }_{\mathcal{D}}\to 0$, ${\mathrm{\Sigma }}_{\mathcal{D}}\to 0$, ${\mathcal{W}}_{\mathcal{D}}\to 0$, and $\mathrm{\Theta }\to \sqrt{3\mathrm{\Lambda }}$. Also, the product of the shear and Weyl curvature is negative $\mathrm{\Sigma }\mathcal{W}<0$ which as follows from (27) should imply a non-negative rate of change of the gravitational entropy.

Figure 2

The rate of change of the gravitational entropy within the whole domain of the Simsilun simulation. Upper panel: change of rate of the HBM gravitational entropy; lower panel: change of rate of the CET gravitational entropy.

Figure 3

The rate of change of the CET gravitational entropy for a single overdense cell (upper panel), and a single underdense cell (lower panel).