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We investigate the averaging problem in cosmology as the problem ofintroducing a distance between spaces.  We first introduce the spectral distance, which is a measure of closenessbetween spaces defined in terms of the spectra of the Laplacian. Then we definea space S, the space of all spaces equipped with the spectral distance. Weargue that this space S can be regarded as a metric space and that it alsopossess other desirable properties. These facts make the space S a suitablearena for spacetime physics.  We apply the spectral framework to the averaging problem:  We sketch the model-fitting procedure in terms of the spectralrepresentation, and also discuss briefly how to analyze the dynamical aspectsof the averaging procedure with this scheme.  In these analyses, we are naturally led to the concept of the apparatus- andthe scale-dependent effective evolution of the universe. These observationssuggest that the spectral scheme seems to be suitable for the quantitativeanalysis of the averaging problem in cosmology.Comment: To appear in General Relativity and Gravitation. The development of a  talk given at South African Relativistic Cosmology Conference, February 1-5  1999, Department of Mathematics and Applied Mathematics, University of Cape  Town, South Afric

Spectral Representation and the Averaging Problem in Cosmology1