Inflation in a closed universe
  To derive a power spectrum for energy density inhomogeneities in a closed
universe, we study a spatially-closed inflation-modified hot big bang model
whose evolutionary history is divided into three epochs: an early
slowly-rolling scalar field inflation epoch and the usual radiation and
non-relativistic matter epochs. (For our purposes it is not necessary to
consider a final dark energy dominated epoch.) We derive general solutions of
the relativistic linear perturbation equations in each epoch. The constants of
integration in the inflation epoch solutions are determined from de Sitter
invariant quantum-mechanical initial conditions in the Lorentzian section of
the inflating closed de Sitter space derived from Hawking's prescription that
the quantum state of the universe only include field configurations that are
regular on the Euclidean (de Sitter) sphere section. The constants of
integration in the radiation and matter epoch solutions are determined from
joining conditions derived by requiring that the linear perturbation equations
remain nonsingular at the transitions between epochs. The matter epoch power
spectrum of gauge-invariant energy density inhomogeneities is not a power law,
and depends on spatial wavenumber in the way expected for a generalization to
the closed model of the standard flat-space scale-invariant power spectrum. The
power spectrum we derive appears to differ from a number of other closed
inflation model power spectra derived assuming different (presumably non de
Sitter invariant) initial conditions.
