On spherical dust fluctuations: the exact vs. the perturbative approach
  We examine the relation between the dynamics of Lema\^{\i}tre-Tolman-Bondi
(LTB) dust models (with and without $\Lambda$) and the dynamics of dust
perturbations in two of the more familiar formalisms used in cosmology: the
metric based Cosmological Perturbation Theory (CPT) and the Covariant Gauge
Invariant (GIC) perturbations. For this purpose we recast the evolution of LTB
models in terms of a covariant and gauge invariant formalism of local and
non-local "exact fluctuations " on a Friedmann-Lema\^{\i}tre-Robertson-Walker
(FLRW) background defined by suitable averages of covariant scalars. We examine
the properties of these fluctuations, which can be defined for a confined
comoving domain or for an asymptotic domain extending to whole time slices. In
particular, the non-local density fluctuation provides a covariant and precise
definition for the notion of the "density contrast ". We show that in their
linear regime these LTB exact fluctuations (local and non-local) are fully
equivalent to the conventional cosmological perturbations in the
synchronous-comoving gauge of CPT and to GIC perturbations. As an immediate
consequence, we show the time-invariance of the spatial curvature perturbation
in a simple form. The present work may provide important theoretical
connections between the exact and perturbative (linear or no-linear) approach
to the dynamics of dust sources in General Relativity.
