  The Universe is homogeneous and isotropic on large scales, so on those scales
it is usually modelled as a Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW)
space-time. The non-linearity of the Einstein field equations raises concern
over averaging over small-scale deviations form homogeneity and isotropy, with
possible implications on the applicability of the FLRW metric to the Universe,
even on large scales. Here I present a technique, based on the multiple-scales
method of singular perturbation theory, to handle the small-scale
inhomogeneities consistently. I obtain a leading order effective Einstein
equation for the large-scale space-time metric, which contains a back-reaction
term. The derivation relies on a series of consistency conditions, that ensure
that the growth of deviations from the large-scale space-time metric do not
grow unboundedly; criteria for their satisfiability are discussed, and it is
shown that they are indeed satisfied if matter is non-relativistic on small
scales. The analysis is performed in harmonic gauge, and conversion to other
gauges is discussed. I estimate the magnitude of the back-reaction term
relative to the critical density of the Universe in the example of an NFW halo,
and find it to be of the order of a few percent. In this example, the
back-reaction term is interpreted as a contribution of the energy-density of
gravitational potential energy, averaged over the small-scale, to the total
energy-momentum tensor.
