  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.
