ric relevance. Various elements of this parametrization have been already considered in the literature [13–17], but only as a useful coordinate ansatzes that are jus- tified because the involved expressions resemble FLRW parameters. In this article we provide an appealing the- oretical and covariant context for these expressions that could allow for new empiric or numeric results and/or a better understanding of existing work. 

The quasi–local scalar representation applied the LTB dust case leads to a complete description of the dynamics in the context of “fluid flow” (or “1+3”) evolution equa- tions that are fully general, but can be handled by simple numerical methods, and thus have a significant potential for empiric modeling of cosmological dust sources. This scalar representation also leads to a description of LTB dust models in the framework of spherical, non–linear, gauge invariant and covariant (GIC) perturbations over a FLRW background (in the context of the fluid flow perturbation formalism derived by Ellis and Bruni [32– 34]). The resulting perturbation framework is also con- sistent with the traditional perturbation approach [37], as the evolution equations of quasi–local scalars and their fluctuations generalize the evolution equations of linear perturbations of spherical dust sources in the comoving gauge [38]. 

The plan of the article is described below. We sum- marize in section II the metric, field equations, analytic solutions and basic regularity of LTB models in the con- ventional variables. In section III we introduce the rep- resentation of local covariant scalars and their evolution equations in the context of the covariant time slicing asso- ciated with a fluid flow approach to the dynamics of the models. The quasi–local scalars and their fluctuations are introduced and discussed in section IV, while section V deals with some theoretical and practical implications of these scalars. In particular, the quasi–local scalars lead naturally to an initial value approach in which ini- tial value functions are given by the new variables eval- uated at a fiducial (initial) hypersurface defined by the 1+3 time slicing. This approach leads to a FLRW–like form for the LTB metric, which has been suggested previ- ously (see [13, 14]) but as an ansatz without a theoretical context. The initial value approach also leads to simple scaling laws for all scalars and an intuitive gauge for the radial coordinate. Hence, in section VI we rewrite the analytic solutions of the models in terms of the above mentioned initial value parametrization, which leads in section VII to an elegant characterization of the coor- dinate locus of the curvature singularities that is sim- pler and more intuitive than that using conventional vari- ables. In section VIII we rewrite the Hellaby–Lake con- ditions [24, 39] to avoid shell–crossings in terms of the initial value functions, while in section IX we rewrite the analytic solutions in the same initial value approach, but using a suitably defined Omega parameter and the quasi– local expansion factor as initial value functions (which can be used also to parametrize the Hellaby–Lake condi- tions). 

In section X we address (as an application) an inter- esting theoretical issue raised by Hellaby and Krasinski (see [7, 10, 18]), namely: the possibility of mapping by an LTB model of a given density or velocity radial pro- file at one cosmic time to another density or velocity profile at a latter time. We show how the initial value parametrization of the analytic solutions greatly simpli- fies the understanding and handling of this problem. In section XI we derive three equivalent systems of fluid flow evolution equations for the quasi–local scalars and their fluctuations. Each of these systems provides a full description of the dynamics of the models, and all are effectively equivalent to systems of ordinary differential equations subjected to constraints, hence they are tech- nically more accessible than (and as fully general as) the system that arises for the local scalars in the 1+3 fluid flow framework. We explain how initial conditions can be set up and argue that these equations have a significant potential for numeric computation of quantities of ob- servational interest (null geodesics, red shift factor, area and luminosity distances, etc). In section XII we show how the quasi–local scalars, their fluctuations and their evolution equations can be understood in the framework of a gauge invariant and covariant formalism of spherical non–linear perturbations. 

In section XIII we derive a fourth system of fluid flow evolution equations, but now for the Omega and Hub- ble parameters introduced in section IX. These evolution equations lead to a dynamical system (like the one in [29]). We comment on the relation between the perturba- tion formalism of section XII and these parameters (and others proposed in the literature [16, 17]), thus provid- ing a theoretical context for quantities given as coordi- nate ansatzes on the basis of their resemblance to FLRW Omega and Hubble parameters. We discuss in section XIV various aspects of special LTB configurations, such as: closed models, models with simultaneous big–bang or collapsing singularities, simultaneous maximal expansion and a mixed elliptic/hyperbolic configuration. Our con- clusions and suggestions for future work are summarized in section XV. In Appendix A we discuss various regular- ity issues of the quasi–local scalars and their fluctuations, while Appendix B discusses the relation between quasi– local spatial curvature (which determines the kinematic class: parabolic, hyperbolic, elliptic) and local spatial curvature. 

Lemaitre–Tolman–Bondi (LTB) dust models [1, 2] are the spherically symmetric solutions of Einstein’s equa- tions characterized by the LTB line element and the 

## II. LTB DUST MODELS IN THE CONVENTIONAL VARIABLES. 

