General Relativistic Evolution Equations for Density Perturbations in
  Closed, Flat and Open FLRW Universes
  It is shown that the decomposition theorems of York, Stewart and Walker for
symmetric spatial second-rank tensors, such as the perturbed metric tensor and
perturbed Ricci tensor, and the spatial fluid velocity vector imply that, for
open, flat or closed Friedmann-Lemaitre-Robertson-Walker universes, there are
exactly two, unique, independent gauge-invariant quantities which describe the
true, physical perturbations to the energy density and particle number density.
Using these two new quantities, evolution equations for cosmological density
perturbations and for entropy perturbations, adapted to non-barotropic
equations of state for the pressure, are derived. Density perturbations evolve
adiabatically if and only if the particle number density does not contribute to
the pressure. Local density perturbations do not affect the global expansion of
the universe. The new perturbation theory has an exact non-relativistic limit
in a non-static universe. The gauge problem of cosmology has thus been solved.
