General theory of cosmological perturbations in open and closed universes from the Horndeski action

TL;DR

This paper derives general quadratic actions for cosmological perturbations in non-flat universes within the Horndeski theory, showing that tiny spatial curvature has limited impact on primordial spectra and that non-singular solutions are unstable.

Contribution

It provides the first comprehensive derivation of quadratic actions for perturbations in non-flat Horndeski models, analyzing the effects of spatial curvature on early universe dynamics.

Findings

Tiny spatial curvature minimally affects primordial spectra.

Non-singular solutions are unstable in open and flat Horndeski models.

Spatial curvature influences perturbation behavior but does not lead to large primordial spectrum modifications.

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 non-flat 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 ${\cal O}(1)$) effects on the primordial spectra even if one modifies gravity. We also argue that non-singular cosmological solutions in the Horndeski theory are unstable in spatially open cases as well as in flat cases.