Spectral evolution of the Universe

We derive the evolution equations for the spectra of the Universe. Here “spectra” means the eigenvalues of the Laplacian on a space, which contain the geometrical information about the space. These spectral evolution equations are expected to be useful to analyze the time development of the geometrical structures of the Universe. In particular, these equations are indispensable for the investigation of the time evolution of the spectral distance between two geometries. As an application, we investigate the time development of the spectral distance between two universes (the real Universe and its model universe) that are very close to each other; it is the first necessary step for the detailed study of the model-fitting problem in cosmology with the spectral scheme. We find out a universal formula for the spectral distance between two very close universes, which turns out to be independent of the detailed form of the distance or the gravity theory. Then we investigate its time evolution with the help of the spectral evolution equations. We also formulate the criteria for a good cosmological model in terms of the spectral distance.