  Several recent proposals to embed inflation into high-energy physics rely on
inflationary dynamics characterized by a strongly non-geodesic motion in
negatively curved field space. This naturally leads to a transient instability
of perturbations on sub-Hubble scales, and to their exponential amplification.
Supported by first-principle numerical computations, and by the analytical
insight provided by the effective field theory of inflation, we show that the
bispectrum is enhanced in flattened configurations, and we argue that an
analogous result holds for all higher-order correlation functions. These
``hyper non-Gaussianities'' thus provide powerful model-independent constraints
on non-standard inflationary attractors motivated by the search for ultraviolet
completions of inflation.
