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It is a fundamental unsolved question in general relativity how tounambiguously characterize the effective collective dynamics of an ensemble offluid elements sourcing the local geometry, in the absence of exact symmetries.In a cosmological context this is sometimes referred to as the averagingproblem. At the heart of this problem in relativity is the non-uniqueness ofthe choice of foliation within which the statistical properties of the localspacetime are quantified, which can lead to ambiguity in the formulated averagetheory. This has led to debate in the literature on how to best construct andview such a coarse-grained hydrodynamic theory. Here, we address this ambiguityby performing the first quantitative investigation of foliation dependence incosmological spatial averaging. Starting from the aim of constructingslicing-independent integral functionals (volume, mass, entropy, etc.) as wellas average functionals (mean density, average curvature, etc.) defined onspatial volume sections, we investigate infinitesimal foliation variations andderive results on the foliation dependence of functionals and on extremalleaves. Our results show that one may only identify fully foliation-independentintegral functionals in special scenarios, requiring the existence ofassociated conserved currents. We then derive bounds on the foliationdependence of integral functionals for general scalar quantities under finitevariations within physically motivated classes of foliations. Our findingsprovide tools that are useful for quantifying, eliminating or constraining thefoliation dependence in cosmological averaging.

Splitting the spacetime: A systematic analysis of foliation dependence in cosmic averaging