1209.1962 (Roberto A. Sussman)

Averaging and exact perturbations in LTB dust models    [PDF]

Roberto A. Sussman

We introduce a scalar weighed average ("q-average") acting on concentric comoving domains in spherically symmetric Lemaitre-Tolman-Bondi (LTB) dust models. The resulting averaging formalism allows for an elegant coordinate independent dynamical study of the models, providing as well a valuable theoretical insight on the properties of scalar averaging in inhomogeneous spacetimes. The q-averages of those covariant scalars common to FLRW models (the "q-scalars") identically satisfy FLRW evolution laws and determine for every domain a unique FLRW background state. All curvature and kinematic proper tensors and their invariant contractions are expressible in terms of the q-scalars and their linear and quadratic local fluctuations, which convey the effects of inhomogeneity through the ratio of Weyl to Ricci curvature invariants and the magnitude of radial gradients. We define also non-local fluctuations associated with the intuitive notion of a "contrast" with respect to FLRW reference averaged values assigned to a whole domain or time slice. The q-averages of local and non-local quadratic fluctuations are directly and exactly related to statistical correlation moments of the density and Hubble expansion scalar. The evolution equations for the q-scalars and suitably defined perturbations completely determine the dynamics of the models without the back-reaction correlation terms of Buchert's average. A rigorous formalism of exact spherical non-linear perturbations can be defined over a formal FLRW background state associated to the q-scalars, recovering the standard results of linear perturbation theory in the appropriate limit. We briefly explore the possible application of this formalism to open theoretical issues, such as the relation between the growth of inhomogeneity and a definition of gravitational entropy.

View original: http://arxiv.org/abs/1209.1962