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[en] We examine numerically and qualitatively the LemaItre-Tolman-Bondi (LTB) inhomogeneous dust solutions as a three-dimensional dynamical system characterized by six critical points. One of the coordinates of the phase space is an average density parameter, (Ω), which behaves as the ordinary Ω in Friedman-LemaItre-Robertson-Walker (FLRW) dust spacetimes. The other two coordinates, a shear parameter and a density contrast function, convey the effects of inhomogeneity. As long as shell crossing singularities are absent, this phase space is bounded or it can be trivially compactified. This space contains several invariant subspaces which define relevant particular cases, such as 'parabolic' evolution, FLRW dust and the Schwarzschild-Kruskal vacuum limit. We examine in detail the phase-space evolution of several dust configurations: a low-density void formation scenario, high-density re-collapsing universes with open, closed and wormhole topologies, a structure formation scenario with a black hole surrounded by an expanding background and the Schwarzschild-Kruskal vacuum case. Solution curves (except regular centers) start expanding from a past attractor (source) in the plane (Ω) = 1, associated with self-similar regime at an initial singularity. Depending on the initial conditions and specific configurations, the curves approach several saddle points as they evolve between this past attractor and two other possible future attractors: perpetually expanding curves terminate at a line of sinks at (Ω) = 0, while collapsing curves reach maximal expansion as (Ω) diverges and end up in a sink that coincides with the past attractor and is also associated with self-similar behavior
[en] We consider a magnetized degenerate gas of fermions as the matter source of a homogeneous but anisotropic Bianchi I spacetime with a Kasner--like metric. We examine the dynamics of this system by means of a qualitative and numerical study of Einstein-Maxwell field equations which reduce to a non-linear autonomous system. For all initial conditions and combinations of free parameters the gas evolves from an initial anisotropic singularity into an asymptotic state that is completely determined by a stable attractor. Depending on the initial conditions the anisotropic singularity is of the cigar or plate types
[en] We provide a thorough examination of the conditions for the existence of back-reaction and an 'effective' acceleration (in the context of Buchert's averaging formalism) in regular generic spherically symmetric Lemaitre-Tolman-Bondi (LTB) dust models. By considering arbitrary spherical comoving domains,we verify rigorously the fulfillment of these conditions expressed in terms of suitable scalar variables that are evaluated at the boundary of every domain. Effective deceleration necessarily occurs in all domains in (a) the asymptotic radial range of models converging to a FLRW background (b) the asymptotic time range of non-vacuum hyperbolic models (c) LTB self-similar solutions and (d) near a simultaneous big bang. Accelerating domains are proven to exist in the following scenarios: (i) central vacuum regions(ii) central (non-vacuum) density voids (iii) the intermediate radial range of models converging to a FLRW background (iv) the asymptotic radial range of models converging to a Minkowski vacuum and (v) domains near and or intersecting a non-simultaneous big bang. All these scenarios occur in hyperbolic models with negative averaged and local spatial curvature though scenarios (iv) and (v) are also possible in low density regions of a class of elliptic models in which the local spatial curvature is negative but its average is positive. Rough numerical estimates between -0.003 and -0.5 were found for the effective deceleration parameter. While the existence of accelerating domains cannot be ruled out in models converging to an Einstein-de Sitter background and in domains undergoing gravitational collapse the conditions for this are very restrictive. The results obtained may provide important theoretical clues on the effects of back-reaction and averaging in more general non-spherical models. Communicated by L Andersson (paper)
[en] A numerical approach is considered for spherically symmetric spacetimes that generalize Lemaitre-Tolman-Bondi dust solutions to nonzero pressure ('LTB spacetimes'). We introduce quasilocal (QL) variables that are covariant LTB objects satisfying evolution equations of Friedman-Lemaitre-Robertson-Walker (FLRW) cosmologies. We prove rigorously that relative deviations of the local covariant scalars from the QL scalars are nonlinear, 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 nontrivial inhomogeneous conditions. Coordinate choices and initial conditions are derived for a numerical treatment of the perturbation equations, allowing us to study nonlinear effects in a variety of phenomena, such as gravitational collapse, nonlocal 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.
[en] We undertake a comprehensive and rigorous analytic study of the evolution of radial profiles of covariant scalars in regular LemaItre-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.
[en] We introduce a weighed scalar average formalism (‘q-average’) for the study of the theoretical properties and the dynamics of spherically symmetric Lemaître–Tolman–Bondi (LTB) dust models. The ‘q-scalars’ that emerge by applying the q-averages to the density, Hubble expansion and spatial curvature (which are common to FLRW models) are directly expressible in terms of curvature and kinematic invariants and identically satisfy FLRW evolution laws without the back-reaction terms that characterize Buchert's average. The local and non-local fluctuations and perturbations with respect to the q-average convey the effects of inhomogeneity through the ratio of curvature and kinematic invariants and the magnitude of radial gradients. All curvature and kinematic proper tensors that characterize the models are expressible as irreducible algebraic expansions on the metric and 4-velocity, whose coefficients are the q-scalars and their linear and quadratic local fluctuation. All invariant contractions of these tensors are quadratic fluctuations, whose q-averages are directly and exactly related to statistical correlation moments of the density and Hubble expansion scalar. We explore the application of this formalism to a definition of a gravitational entropy functional proposed by Hosoya et al (2004 Phys. Rev. Lett. 92 141302–14). We show that a positive entropy production follows from a negative correlation between fluctuations of the density and Hubble scalar, providing a brief outline on its fulfilment in various LTB models and regions. While the q-average formalism is specially suited for LTB (and Szekeres) models, it may provide a valuable theoretical insight into the properties of scalar averaging in inhomogeneous spacetimes in general. (paper)
[en] We examine the exact perturbations that arise from the q-average formalism that was applied in the preceding article (part I) to Lemaître–Tolman–Bondi (LTB) models. By introducing an initial value parametrization, we show that all LTB scalars that take an FLRW ‘look-alike’ form (frequently used in the literature dealing with LTB models) follow as q-averages of covariant scalars that are common to FLRW models. These q-scalars determine for every averaging domain a unique FLRW background state through Darmois matching conditions at the domain boundary, though the definition of this background does not require an actual matching with an FLRW region (Swiss cheese-type models). Local perturbations describe the deviation from the FLRW background state through the local gradients of covariant scalars at the boundary of every comoving domain, while non-local perturbations do so in terms of the intuitive notion of a ‘contrast’ of local scalars with respect to FLRW reference values that emerge from q-averages assigned to the whole domain or the whole time slice in the asymptotic limit. We derive fluid flow evolution equations that completely determine the dynamics of the models in terms of the q-scalars and both types of perturbations. A rigorous formalism of exact spherical nonlinear perturbations is defined over the FLRW background state associated with the q-scalars, recovering the standard results of linear perturbation theory in the appropriate limit. We examine the notion of the amplitude and illustrate the differences between local and non-local perturbations by qualitative diagrams and through an example of a cosmic density void that follows from the numeric solution of the evolution equations. (paper)
[en] We re-examine the Lemaitre-Tolman-Bondi (LTB) dust solutions by considering as free parameters the initial value functions: Yi, ρi, (3)Ri, obtained by restricting the curvature radius, Y ≡ √ g θθ , the rest mass density, ρ, and the three-dimensional Ricci scalar of the rest frames, (3)R, to an arbitrary regular Cauchy hypersurface, Ti, marked by constant cosmic time (t = ti). Assuming the existence of symmetry centres, we use Yi to fix the radial coordinate and the topology (homeomorphic class) of Ti, while the time evolution is described in terms of an adimensional scale factor y Y/Yi. We show that the dynamics, regularity conditions and geometric features of the models are determined by ρi, (3)Ri and by suitably constructed volume averages and contrast functions expressible in terms of invariant scalars defined in Ti. These quantities lead to a straightforward characterization of initial conditions in terms of the nature of the inhomogeneity of Ti, as density and/or curvature overdensities ('lumps') and underdensities ('voids') around a symmetry centre. In general, only models with initial density and curvature lumps evolve without shell-crossing singularities, though special classes of initial conditions, associated with a simultaneous big bang, allow for a regular evolution for initial density and curvature voids. Specific restrictions are found so that a regular evolution for t ≥ ti is possible for initial voids. A brief guideline is provided for using the new variables in the construction of LTB models and for plotting all relevant quantities
[en] We derive a class of non-static inhomogeneous dust solutions in gravity described by the Lemaître–Tolman–Bondi (LTB) metric. The field equations are fully integrated for all parameter subcases and compared with analogous subcases of LTB dust solutions of GR. Since the solutions do not admit regular symmetry centres, we have two possibilities: (i) a spherical dust cloud with angle deficit acting as the source of a vacuum Schwarzschild-like solution associated with a global monopole, or (ii) fully regular dust wormholes without angle deficit, whose rest frames are homeomorphic to the Schwarzschild–Kruskal manifold or to a 3d torus. The compatibility between the LTB metric and generic ansatzes furnishes an ‘inverse procedure’ to generate LTB solutions whose sources are found from the geometry. While the resulting fluids may have an elusive physical interpretation, they can be used as exact non-perturbative toy models in theoretical and cosmological applications of theories. (paper)
[en] 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 weighted 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 and sign conditions on the modes become interrelated fluid flow preserved constraints that define sub-cases of LTB models as 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. (paper)