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.
