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Abstract

It is a fundamental unsolved question in general relativity how to
unambiguously characterize the effective collective dynamics of an ensemble of
fluid elements sourcing the local geometry, in the absence of exact symmetries.
In a cosmological context this is sometimes referred to as the averaging
problem. At the heart of this problem in relativity is the non-uniqueness of
the choice of foliation within which the statistical properties of the local
spacetime are quantified, which can lead to ambiguity in the formulated average
theory. This has led to debate in the literature on how to best construct and
view such a coarse-grained hydrodynamic theory. Here, we address this ambiguity
by performing the first quantitative investigation of foliation dependence in
cosmological spatial averaging. Starting from the aim of constructing
slicing-independent integral functionals (volume, mass, entropy, etc.) as well
as average functionals (mean density, average curvature, etc.) defined on
spatial volume sections, we investigate infinitesimal foliation variations and
derive results on the foliation dependence of functionals and on extremal
leaves. Our results show that one may only identify fully foliation-independent
integral functionals in special scenarios, requiring the existence of
associated conserved currents. We then derive bounds on the foliation
dependence of integral functionals for general scalar quantities under finite
variations within physically motivated classes of foliations. Our findings
provide tools that are useful for quantifying, eliminating or constraining the
foliation dependence in cosmological averaging.