Inflation with Violation of the Null Energy Condition

Inflation may have been driven by a component which violated the Null-Energy Condition, thereby leading to super inflation. We provide the formalism to study cosmological perturbations when such a component is described by a scalar field with arbitrary Lagrangian. Since the background curvature grows with time, gravitational waves always have a blue spectrum. We apply our formalism to the case of phantom inflation with an exponential potential (whose polelike inflationary stage is an attractor for inhomogeneous cosmological models for any value of the potential slope) as an example. We finally compare the predictions of super inflation with those of standard inflation stressing the role of gravitational waves