  The subject of cosmological backreaction in General Relativity is often
approached by coordinate-dependent and metric-based analyses. We present in
this letter an averaging formalism for the scalar parts of Einstein's equations
that is coordinate-independent and only functionally depends on a metric. This
formalism is applicable to general 3+1 foliations of spacetime for an arbitrary
fluid with tilted flow. We clarify the dependence on spacetime foliation and
argue that this dependence is weak in cosmological settings. We also introduce
a new set of averaged equations that feature a global cosmological time despite
the generality of the setting, and we put the statistical nature of effective
cosmologies into perspective.
