Cosmological perturbations and stability of nonsingular cosmologies with limiting curvature

We revisit nonsingular cosmologies in which the limiting curvature hypothesis is realized. We study the cosmological perturbations of the theory and determine the general criteria for stability. For the simplest model, we find generic Ostrogradski instabilities unless the action contains the Weyl tensor squared with the appropriate coefficient. When considering two specific nonsingular cosmological scenarios (one inflationary and one genesis model), we find ghost and gradient instabilities throughout most of the cosmic evolution. Furthermore, we show that the theory is equivalent to a theory of gravity where the action is a general function of the Ricci and Gauss-Bonnet scalars, and this type of theory is known to suffer from instabilities in anisotropic backgrounds. This leads us to construct a new type of curvature-invariant scalar function. We show that it does not have Ostrogradski instabilities, and it avoids ghost and gradient instabilities for most of the interesting background inflationary and genesis trajectories. We further show that it does not possess additional new degrees of freedom in an anisotropic spacetime. This opens the door for studying stable alternative nonsingular very early Universe cosmologies.