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Buchert coarse-graining and the classical energy conditions

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.