Averaging over cosmic structure: Cosmological backreaction and the gauge problem

Abstract

The observation that accelerated cosmic expansion appears to start around the time that nonlinear cosmic structure is appearing seems like an extraordinary coincidence, unless the acceleration is somehow driven by the emergence of the structure. That has given rise to the controversial concept of a gravitational backreaction through which inhomogeneity becomes a driver of accelerated expansion. The standard route when studying strongly inhomogeneous cosmological models is to take either a perturbative approach or a spatial averaging approach. Here we argue that because backreaction is in fact a nonlinear multiscale phenomenon, perturbative approaches may have a limited validity. With respect to the currently proposed averaging approaches we here show that they lead to gauge dependent backreaction and hence ambiguous estimates of its magnitude. In the present study, we formalise inhomogeneous cosmic evolution within the framework of foliations of spacetime. Fixing a foliation amounts to making a gauge choice. Addressing the correspondence between the metric tensor and the foliation allows us to clarify the implications of choosing a foliation for the representation of equivalent cosmologies. It is important to note that, within the context of backreaction, this formalism allows us to discuss the vagaries of averaging in the framework of spacetime foliations. It reveals that spatial averaging can induce artificial, i.e. gauge dependent, backreaction terms that arise from any specific choice of gauge. Averaging methods presented so far all encounter this problem. However, within our foliation framework, we can produce a gauge invariant method of averaging by invoking the gauge-invariant Bardeen formalism for cosmological perturbation theory. We demonstrate that this implies the gauge invariance of the averaging procedure. This makes it applicable to standard cosmological simulations.