Back-reaction and effective acceleration in generic LTB dust models

Abstract

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 Lemaître–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