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

Understand

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.