Inflation in a closed universe

TL;DR

This paper develops a model for the power spectrum of energy density inhomogeneities in a closed universe during inflation, incorporating quantum initial conditions and epoch transitions, revealing a non-power-law spectrum.

Contribution

It provides a detailed derivation of the matter epoch power spectrum in a closed universe with de Sitter invariant initial conditions, differing from previous models.

Findings

Matter epoch power spectrum is not a simple power law.

Power spectrum depends on spatial wavenumber as expected for a closed universe.

Derived spectrum differs from other closed inflation models with different initial conditions.

Abstract

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.