Abstract
Our Universe is nearly spatially flat, but this does not mean that it is exactly spatially flat. In this paper we derive general quadratic actions for cosmological perturbations in nonflat models from the Horndeski theory. This allows us to study how the spatial curvature influences the behavior of cosmological perturbations in the early Universe described by some general scalar-tensor theory. We show that a tiny spatial curvature at the onset of inflation is unlikely to yield large [or $\mathcal{O}(1)$] effects on the primordial spectra even if one modifies gravity. We also argue that nonsingular cosmological solutions in the Horndeski theory are unstable in spatially open cases as well as in flat cases.
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