Intermediate homogenization of the Universe and the problem of gravitational entropy

This paper studies intermediate homogenization of inhomogeneous cosmological models. It shows that spherically symmetric models, regardless of the equation of state, can undergo intermediate homogenization; i.e., a model can approach a homogeneous and isotropic state (which acts as a saddle point) from a relatively wide range of initial inhomogeneous conditions. The homogenization is not permanent-just temporary. Eventually, the model evolves toward a future inhomogeneous state. We also looked at the problem of the gravitational entropy. All definitions of entropy that we checked give decreasing gravitational entropy during the homogenization process. Thus, we should either accept that gravitational entropy can decrease or try to define it in other ways than just via density gradients, as these decrease during homogenization.