Spatial Curvature at the Sound Horizon
  The effect of spatial curvature on primordial perturbations is controlled by
$ \Omega_{K,0}/c_{s}^{2} $, where $ \Omega_{K,0} $ is today's fractional
density of spatial curvature and $ c_{s} $ is the speed of sound during
inflation. Here we study these effects in the limit $ c_{s}\ll 1 $. First, we
show that the standard cosmological soft theorems in flat universes are
violated in curved universes and the soft limits of correlators can have
non-universal contributions even in single-clock inflation. This is a
consequence of the fact that, in the presence of spatial curvature, there is a
gap between the spectrum of residual diffeomorphisms and that of physical
modes. Second, there are curvature corrections to primordial correlators, which
are not scale invariant. We provide explicit formulae for these corrections to
the power spectrum and the bispectrum to linear order in curvature in
single-clock inflation. We show that the large-scale CMB anisotropies could
provide interesting new constraints on these curvature effects, and therefore
on $ \Omega_{K,0}/c_{s}^{2} $, but it is necessary to go beyond our
linear-order treatment.
