  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.
