  We propose a novel cosmological scenario, in which standard inflation is
replaced by an expanding phase with a drastic violation of the Null Energy
Condition (NEC): \dot H >> H^2. The model is based on the recently introduced
Galileon theories, that allow NEC violating solutions without instabilities.
The unperturbed solution describes a Universe that is asymptotically Minkowski
in the past, expands with increasing energy density until it exits the regime
of validity of the effective field theory and reheats. This solution is a
dynamical attractor and the Universe is driven to it, even if it is initially
contracting. The study of perturbations of the Galileon field reveals some
subtleties, related to the gross violation of the NEC and it shows that
adiabatic perturbations are cosmologically irrelevant. The model, however,
suggests a new way to produce a scale invariant spectrum of isocurvature
perturbations, which can later be converted to adiabatic: the Galileon is
forced by symmetry to couple to the other fields as a dilaton; the effective
metric it yields on the NEC violating solution is that of de Sitter space, so
that all light scalars will automatically acquire a nearly scale-invariant
spectrum of perturbations.
