On spatial volume averaging in Lema\^{\i}tre--Tolman--Bondi dust models. Part I: back reaction, spacial curvature and binding energy

TL;DR

This paper analytically investigates conditions under which the back reaction term in Lemaître-Tolman-Bondi dust models is positive, linking it to spatial curvature and binding energy, with implications for effective cosmic acceleration.

Contribution

It introduces proper volume and quasi-local averages to analyze back reaction in LTB models, establishing when it is positive based on curvature and domain properties.

Findings

Back reaction is positive in hyperbolic domains with negative curvature.

Positive back reaction can occur even with inner elliptic collapsing regions.

Conditions for positive back reaction are more restrictive when positive curvature decreases asymptotically.

Abstract

We provide a comprehensive analytic study (rigorous and qualitative) of the conditions for the existence of a a positive kinematic back reaction term $\QQ>0$, in the context of Buchert's scalar averaging formalism applied to spherically symmetric Lema\^itre-Tolman-Bondi (LTB) dust solutions in which averaging domains are given as spherical comoving regions containing a symmetry center. We introduce proper volume and quasi-local average functionals and functions in order to examine the conditions for $\QQ\geq 0$, and in the process we also explore the relation between back reaction, spatial curvature and binding energy for a wide variety of LTB configurations. The back reaction term is positive for all "hyperbolic" regular domains with negative spatial curvature, either in the full radial range or in the radial asymptotic range. This result is also valid if these domains contain an inner "elliptic" region with positive curvature undergoing local collapse. For some cases in which positive spatial curvature decreases asymptotically, the conditions for a positive back reaction can still be met but seem to be more restrictive. Since $\QQ>0$ is a necessary condition for a positive "effective" acceleration that would mimic the effect of dark energy (in the context of Buchert's formalism), we examine this issue in LTB models in a follow up paper (part II).