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.