On the covariance of scalar averaging and backreaction in relativistic
  inhomogeneous cosmology
  We introduce a generalization of the 4-dimensional averaging window function
of Gasperini, Marozzi and Veneziano (2010) that may prove useful for a number
of applications. The covariant nature of spatial scalar averaging schemes to
address the averaging problem in relativistic cosmology is an important
property that is implied by construction, but usually remains implicit. We
employ here the approach of Gasperini et al. for two reasons. First, the
formalism and its generalization presented here are manifestly covariant.
Second, the formalism is convenient for disentangling the dependencies on
foliation, volume measure, and boundaries in the averaged expressions entering
in scalar averaging schemes. These properties will prove handy for simplifying
expressions, but also for investigating extremal foliations and for comparing
averaged properties of different foliations directly. The proposed
generalization of the window function allows for choosing the most appropriate
averaging scheme for the physical problem at hand, and for distinguishing
between the role of the foliation itself and the role of the volume measure in
averaged dynamic equations. We also show that one particular window function
obtained from this generalized class results in an averaging scheme
corresponding to that of a recent investigation by Buchert, Mourier and Roy
(2018) and, as a byproduct, we explicitly show that the general equations for
backreaction derived therein are covariant.
