We examine the radial asymptotic behavior of spherically symmetric Lemaître-Tolman-Bondi dust models by looking at their covariant scalars along radial rays, which are spacelike geodesics parametrized by proper length , orthogonal to the 4-velocity and to the orbits of SO(3). By introducing quasi-local scalars defined as integral functions along the rays, we obtain a complete and covariant representation of the models, leading to an initial value parametrization in which all scalars can be given by scaling laws depending on two metric scale factors and two basic initial value functions. Considering regular "open" LTB models whose space slices allow for a diverging , we provide the conditions on the radial coordinate so that its asymptotic limit corresponds to the limit as → ∞. The "asymptotic state" is then defined as this limit, together with asymptotic series expansion around it, evaluated for all metric functions, covariant scalars (local and quasi-local) and their fluctuations. By looking at different sets of initial conditions, we examine and classify the asymptotic states of parabolic, hyperbolic and open elliptic models admitting a symmetry center. We show that in the radial direction the models can be asymptotic to any one of the following spacetimes: FLRW dust cosmologies with zero or negative spatial curvature, sections of Minkowski flat space (including Milne's space), sections of the Schwarzschild-Kruskal manifold or self-similar dust solutions.

Abstract. We obtain an elegant and useful description of the dynamics of Szekeres dust models (in their full generality) by means of "quasi-local" scalar variables constructed by suitable integral distributions that can be interpreted as weighed proper volume averages of the local covariant scalars. In terms of these variables, the field equations and basic physical and geometric quantities are formally identical to their corresponding expressions in the spherically symmetric LTB dust models. Since we can map every Szekeres model to a unique LTB model, rigorous results valid for the latter models can be readily generalized to a nonspherical Szekeres geometry. The new variables lead naturally to an initial value formulation in which all scalars are expressed as scaling laws in terms of their values at an arbitrary initial space slice. These variables also yield a significant simplification of numerical work, since the fluid flow evolution equations become a system of autonomous ordinary differential equations subjected to algebraic constraints containing the information on the deviations from spherical symmetry. As an example of how this formalism can be applied, we show that spherical symmetry is stable against small dipole-like perturbations. This new approach to the dynamics of the Szekeres solutions has an enormous potential for dealing with a wide variety of theoretical issues and for constructing non-spherical models of cosmological inhomogeneities to fit observational data.

We undertake a comprehensive and rigorous analytic study of the evolution of radial profiles of covariant scalars in regular Lemaître–Tolman–Bondi (LTB) dust models. We consider specifically the phenomenon of ‘profile inversions’ in which an initial clump profile of density, spatial curvature or the expansion scalar might evolve into a void profile (and vice versa). Previous work in the literature on models with density void profiles and/or allowing for density profile inversions is given full generalization, with some erroneous results corrected. We prove rigorously that if an evolution without shell crossings is assumed, then only the ‘clump to void’ inversion can occur in density profiles, and only in hyperbolic models or regions with negative spatial curvature. The profiles of spatial curvature follow similar patterns as those of the density, with ‘clump to void’ inversions only possible for hyperbolic models or regions. However, profiles of the expansion scalar are less restrictive, with profile inversions necessarily taking place in elliptic models. We also examine radial profiles in special LTB configurations: closed elliptic models, models with a simultaneous big bang singularity, as well as a locally collapsing elliptic region surrounded by an expanding hyperbolic background. The general analytic statements that we obtain allow for setting up the right initial conditions to construct fully regular LTB models with any specific qualitative requirements for the profiles of all scalars and their time evolution. The results presented can be very useful in guiding future numerical work on these models and in revising previous analytic work on all their applications.

Inhomogeneous cosmological models are able to fit cosmological observations without dark energy under the assumption that we live close to the "center" of a very large-scale under-dense region. Most studies fitting observations by means of inhomogeneities also assume spherical symmetry, and thus being at (or very near) the center may imply being located at a very special and unlikely observation point. We argue that such spherical voids should be treated only as a gross first approximation to configurations that follow from a suitable smoothing out of the non-spherical part of the inhomogeneities on angular scales. In this Letter we present a toy construction that supports the above statement. The construction uses parts of the Szekeres model, which is inhomogeneous and anisotropic thus it also addresses the limitations of spherical inhomogeneities. By using the thin-shell approximation (which means that the Israel-Darmois continuity conditions are not fulfilled between the shells) we construct a model of evolving cosmic structures, containing several elongated supercluster-like structures with underdense regions between them, which altogether provides a reasonable coarse-grained description of cosmic structures. While this configuration is not spherically symmetric, its proper volume average yields a spherical void profile of 250 Mpc that roughly agrees with observations. Also, by considering a non-spherical inhomogeneity, the definition of a "center" location becomes more nuanced, and thus the constraints placed by fitting observations on our position with respect to this location become less restrictive.

A numerical approach is considered for spherically symmetric spacetimes that generalize Lemaître-Tolman-Bondi dust solutions to nonzero pressure ("LTB spacetimes"). We introduce quasi-local (QL) variables that are covariant LTB objects satisfying evolution equations of Friedman-Lemaître-Robertson-Walker (FLRW) cosmologies. We prove rigorously that relative deviations of the local covariant scalars from the QL scalars are non-linear, gauge invariant and covariant perturbations on a FLRW formal "background" given by the QL scalars. The dynamics of LTB spacetimes is completely determined by the QL scalars and these exact perturbations. Since LTB spacetimes are compatible with a wide variety of "equations of state", either single fluids or mixtures, a large number of known solutions with dark matter and dark energy sources in a FLRW framework (or with linear perturbations) can be readily examined under idealized but non-trivial inhomogeneous conditions. Coordinate choices and initial conditions are derived for a numerical treatment of the perturbation equations, allowing us to study non-linear effects in a variety of phenomena, such as gravitational collapse, non-local effects, void formation, dark matter and dark energy couplings and particle creation. In particular, the embedding of inhomogeneous regions can be performed by a smooth matching with a suitable FLRW solution, thus generalizing the Newtonian "top hat" models that are widely used in astrophysical literature. As examples of the application of the formalism, we examine numerically the formation of a black hole in an expanding Chaplygin gas FLRW universe, as well as the evolution of density clumps and voids in an interactive mixture of cold dark matter and dark energy. PACS numbers: 12.60.Jv, 14.80.Ly, 95.30.Cq, 95.30.Tg, 95.35.+d, 98.35.Gi