Gravitational entropy of local cosmic voids

We undertake a non-perturbative study of the evolution of the "gravitational entropy" proposed by Clifton, Ellis and Tavakol (CET) on local expanding cosmic CDM voids of $\sim 50-100$ Mpc size described as spherical under-dense regions with negative spatial curvature, whose dynamics is determined by Lemaitre-Tolman-Bondi (LTB) dust models asymptotic to three different types of FLRW background: $\Lambda$CDM, Einstein de Sitter and "open" FLRW with $\Lambda=0$ and negative spatial curvature. By assuming generic nearly spatially flat and linear initial conditions at the last scattering time, we examine analytically and numerically the CET entropy evolution into a fully non-linear regime in our present cosmic time and beyond. Both analytic and numerical analysis reveal that the late time CET entropy growth is determined by the amplitude of initial fluctuations of spatial curvature at the last scattering time. This entropy growth decays to zero in the late asymptotic time range for all voids, but at a faster rate in voids with $\Lambda$CDM and open FLRW backgrounds. However, only for voids in a $\Lambda$CDM background this suppression is sufficiently rapid for the CET entropy itself to reach a terminal equilibrium (or "saturation") value. The CET gravitational temperature vanishes asymptotically if $\Lambda=0$ and becomes asymptotically proportional to $\Lambda$ for voids in a $\Lambda$CDM background. In the linear regime of the LTB evolution our results coincide, qualitatively and quantitatively, with previous results based on linear perturbation theory.