Invariant characterization of the growing and decaying density modes in
  LTB dust models
  We obtain covariant expressions that generalize the growing and decaying
density modes of linear perturbation theory of dust sources by means of the
exact density perturbation from the formalism of quasi--local scalars
associated to weighed proper volume averages in LTB dust models. The relation
between these density modes and theoretical properties of generic LTB models is
thoroughly studied by looking at the evolution of the models through a
dynamical system whose phase space is parametrized by variables directly
related to the modes themselves. The conditions for absence of shell crossings,
as well as sign conditions on the modes, become interrelated fluid flow
preserved constraints that define phase space invariant subspaces. In the
general case (both density modes being nonzero) the evolution of phase space
trajectories exhibits the expected dominance of the decaying/growing in the
early/late evolution times defined by past/future attractors characterized by
asymptotic density inhomogeneity. In particular, the growing mode is also
dominant for collapsing layers that terminate in a future attractor associated
with a "Big Crunch" singularity, which is qualitatively different from the past
attractor marking the "Big Bang". Suppression of the decaying mode modifies the
early time evolution, with phase space trajectories emerging from an
Einstein--de Sitter past attractor associated with homogeneous conditions.
Suppression of the growing mode modifies the late time evolution as phase space
trajectories terminate in future attractors associated with homogeneous states.
General results are obtained relating the signs of the density modes and the
type of asymptotic density profile (clump or void). A critical review is given
of previous attempts in the literature to define these density modes for LTB
models.
