  We derive the evolution equations for the spectra of the Universe.
  Here "spectra" means the eigenvalues of the Laplacian defined on a space,
which contain the geometrical information on the space.
  These equations are expected to be useful to analyze the evolution of the
geometrical structures of the Universe.
  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 the detailed analysis 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 nor the gravity theory. Then we investigate its time evolution with
the help of the evolution equations we derive.
  We also formulate the criteria for a good cosmological model in terms of the
spectral distance.
