  A new class of inflationary attractors characterized by a strongly
non-geodesic motion has been discovered and explored in the past few years. I
describe how they naturally arise in negatively curved field space, allowing to
inflate on potentials that are steep in Planck units, albeit without
alleviating the ever-present naturalness issue of inflation. In these
scenarios, fluctuations often experience a transient tachyonic instability,
which can be described by a single-field effective field theory with an
imaginary speed of sound. Independently of the precise ultraviolet origin of
the latter, this leaves a peculiar imprint in the form of a high-level of
primordial non-Gaussianities of flattened type for all higher-order correlation
functions. On small scales, a transient phase of strongly non-geodesic motion
provides a mechanism to generate primordial black holes and can leave specific
signatures in the form of oscillations in the frequency profile of the
stochastic gravitational wave background.
