  So-called "Buchert averaging" is actually a coarse-graining procedure, where
fine detail is "smeared out" due to limited spatio-temporal resolution. For
technical reasons, (to be explained herein), "averaging" is not really an
appropriate term, and I shall consistently describe the process as a
"coarse-graining". Because Einstein gravity is nonlinear the coarse-grained
Einstein tensor is typically not equal to the Einstein tensor of the
coarse-grained spacetime geometry. The discrepancy can be viewed as an
"effective" stress-energy. To keep otherwise messy technical issues firmly
under control, I shall work with conformal-FLRW (CFLRW) cosmologies. These
CFLRW-based models are particularly tractable, and are also particularly
attractive observationally: the CMB is not distorted. In this CFLRW context one
can prove some rigorous theorems regarding the interplay between Buchert
coarse-graining, tracelessness of the effective stress-energy, and the
classical energy conditions.
