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This paper considers the best way to fit an idealised exactly homogeneous and isotropic universe model to a realistic ('lumpy') universe; whether made explicit or not, some such approach of necessity underlies the use of the standard Robertson-Walker models as models of the real universe. Approaches based on averaging, normal coordinates and null data are presented, the latter offering the best opportunity to relate the fitting procedure to date obtainable by astronomical observations.

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We consider directly the equations by which matter imposes anisotropies on freely propagating background radiation, leading to a new way of using anisotropy measurements to limit the deviations of the Universe from a Friedmann-Robertson-Walker (FRW) geometry. This approach is complementary to the usual Sachs-Wolfe approach: the limits obtained are not as detailed, but they are more model-independent. We also give new results about combined matter-radiation perturbations in an almost-FRW universe, and a new exact solution of the linearised equations.PACS numbers: 98.80.Hw