Spectral Evolution of the Universe

We derive the evolution equations for the spectra of a space (“Universe”). Here the spectra mean the eigenvalues of the Laplacian on a space, which contain the geometrical information on the space. These spectral evolution equations are expected to be useful to analyze the time evolution of the geometrical structures of the Universe. In particular, it is indispensable to investigate the time evolution of the spectral distance between two spaces, which is a measure of closeness between two geometries defined in terms of the spectra. As an application, we investigate the time evolution of the spectral distance between two Universes that are very close to each other; it is the first necessary step for analyzing the time evolution of the geometrical discrepancies between the real Universe and its model. 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 spectral distance nor 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.