Evolution of the density contrast in inhomogeneous dust models

With the help of families of density contrast indicators, we study the tend ency of gravitational systems to become increasingly lumpy with time. Depen ding upon their domain of definition, these indicators could be local or gl obal. We make a comparative study of these indicators in the context of inh omogeneous cosmological models of Lemaitre-Tolman and Szekeres. In particul ar, we look at the temporal asymptotic behaviour of these indicators and as k under what conditions, and for which class of models, they evolve monoton ically in time. We find that for the case of ever-expanding models, there is a larger class of indicators that grow monotonically with time,whereas the corresponding class for the recollapsing models is more restricted. Nevertheless, in the absence of decaying modes, indicators exist which grow monotonically with t ime for both ever-expanding and recollapsing models simultaneously. On the other hand, no such indicators may be found which grow monotonically if the decaying modes are allowed to exist. We also find the conditions for these indicators to be non-divergent at the initial singularity in both models. Our results can be of potential relevance for understanding structure forma tion in inhomogeneous settings and in debates regarding gravitational entro py and the arrow of time. In particular, the spatial dependence of turning points in inhomogeneous cosmologies may result in multiple density contrast arrows in recollapsing models over certain epochs. We also And that differ ent notions of asymptotic homogenization may be deduced, depending upon the density contrast indicators used.