Density Perturbations and Primordial Non-Gaussianities in a Closed
  Universe
  The spatial curvature of the universe is not yet known. Even though at
present the Universe is very close to being essentially flat and most
signatures of curvature appear to have been diluted by inflation, if the number
of e-foldings during inflation is close to the minimum necessary to explain the
horizon problem, the curvature of the universe may have left imprints in the
cosmic microwave background (CMB) that may be observable, especially at large
angles. Motivated by general results on quantum cosmology and using effective
field theory techniques, we develop a general approach for analytically
computing the power spectrum of density perturbations for a closed universe.
Following a Hamiltonian formalism we determine the corresponding Bunch-Davis
vacuum, find analytic expressions for two-point functions and higher
correlators, expanding in terms of $S^3$ harmonics. In particular we
concentrate on potential implications for observable non-Gaussianities. We
consider cubic interactions as well as higher derivative ones to explore the
consequence of a speed of sound $c_s\neq 1$. For large multipoles curvature
effects are negligible and reproduce the known results for the flat case.
However, they depart from the flat space result for relatively small
multipoles. In this limit non-Gaussianities may lead to potentially observable
values of $f_{\rm NL}$. In particular we find terms in $f_{\rm NL}$ that are
absent in the flat space case, which are important at large scales and may be
observable even if a long period of inflation dilutes the curvature. We compare
our results with previous discussions in the literature.
