Averaging in spherically symmetric cosmology

Abstract

The averaging problem in cosmology is of fundamental importance. When applied to study cosmological evolution, the theory of macroscopic gravity (MG) can be regarded as a long-distance modification of general relativity. In the MG approach to the averaging problem in cosmology, the Einstein field equations on cosmological scales are modified by appropriate gravitational correlation terms. We study the averaging problem within the class of spherically symmetric cosmological models. That is, we shall take the microscopic equations and effect the averaging procedure to determine the precise form of the correlation tensor in this case. In particular, by working in volume-preserving coordinates, we calculate the form of the correlation tensor under some reasonable assumptions on the form for the inhomogeneous gravitational field and matter distribution. We find that the correlation tensor in a Friedmann-Lemaître-Robertson-Walker (FLRW) background must be of the form of a spatial curvature. Inhomogeneities and spatial averaging, through this spatial curvature correction term, can have a very significant dynamical effect on the dynamics of the Universe and cosmological observations; in particular, we discuss whether spatial averaging might lead to a more conservative explanation of the observed acceleration of the Universe (without the introduction of exotic dark matter fields). We also find that the correlation tensor for a non-FLRW background can be interpreted as the sum of a spatial curvature and an anisotropic fluid. This may lead to interesting effects of averaging on astrophysical scales. We also discuss the results of averaging an inhomogeneous Lemaître-Tolman-Bondi solution as well as calculations of linear perturbations (that is, the backreaction) in an FLRW background, which support the main conclusions of the analysis.

Observational constraints on the averaged universe

“…The solutions to the macroscopic field equations (2), with an FLRW ansatz for the macroscopic metric, have recently been studied in [24,[34][35][36][37], where it was found that the extra terms involving the correlation tensor Z α µ β[γ νσ] take the same form in the macroscopic field equations that a spatial curvature curvature term takes in Einstein's equations. In fact, for a spatially flat macroscopic metric, and with spatial correlations only, it can be shown that extra terms in Eq.…”

Averaged Lemaître–Tolman–Bondi dynamics

“…Various combinations of the parameters t B0 , r c and r E were checked, but even though one of them reproduces roughly the same value as the H D (z D ) analysis, the physical significance of this term remains doubtful. curvature term is also known from Macroscopic Gravity [8][9][10]. At first order in E one has R D ∝ a −2 D for the averaged curvature which is similar to the FLRW case.…”

“…A very general manner to deal with the averaging problem relies on the exact covariant Macroscopic Gravity formalism by Zalaletdinov [7]. Macroscopic gravity has been applied to spherically symmetric cosmology by Coley et al in [8][9][10], observational aspects of the resulting dynamics were discussed in [11]. A result of this averaging is the appearance of an additional spatial curvature term in the dynamical equations.…”

Averaging the Schwarzschild spacetime

“…The latter is significant for cosmology because the governing equations for the model evolution differ from the Friedmann equations by the additional backreaction term which depends on the scale factor in the same manner as the spatial curvature [8,11]. This means that the dynamical curvature parameter is decoupled from the geometrical value of the curvature, distorting the luminosity distance-redshift relation.…”