New variables for the Lema\^ıtre-Tolman-Bondi dust solutions

Abstract

We re-examine the Lemâitre-Tolman-Bondi (LTB) solutions with a dust source admitting symmetry centers. We consider as free parameters of the solutions the initial value functions: $Y_i$, ${\rho }_i$ and $\Ri$, obtained by restricting the curvature radius, $Y\equiv \sqrt{g_{\theta \theta }}$, the rest mass density, $\rho$, and the 3-dimensional Ricci scalar of the rest frames, $\mathbb{R}$, to an arbitrary regular Cauchy hypersurface, $\Ti$, marked by constant cosmic time ($t=t_i$). Using $Y_i$ to fix the radial coordinate and the topology (homeomorphic class) of $\Ti$, and scaling the time evolution in terms of an adimensional scale factor $y=Y/Y_i$, we show that the dynamics, regularity conditions and geometric features of the models are determined by ${\rho }_i$, $\Ri$ 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 center. In general, only models with initial density and curvature lumps evolve without shell crossing singularities, though special classes of initial conditions, associated with a simmultaneous big bang, allow for a regular evolution for initial density and curvature voids. Specific restrictions are found so that a regular evolution for $t>t_i$ is possible for initial voids. A step-by-step guideline is provided for using the new variables in the construction of LTB models and for plotting all relevant quantities.

36 pages, 41 graphs grouped into 12 figures. Revtex with BoxedEPSF macros for embedding figures

References in corpus (3)

• Matching Spherical Dust Solutions to Construct Cosmological Models
• Exact inhomogeneous cosmologies whose source is a radiation-matter mixture with consistent thermodynamics
• Relaxation versus collision times in the cosmological radiative era

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