Can non-local or higher derivative theories provide alternatives to inflation?

The standard mechanism for producing the observed scale-invariant power spectrum from adiabatic vacuum fluctuations relies on first order derivative of fields in the action for curvature perturbations. It has been proven [1] that, under this ansatz, any theory of early universe that matches cosmological observations should include a phase of accelerated expansion (i.e. inflation) or it has to break at least one of the following tenets of classical general relativity: Null Energy Conditions (NEC), subluminal signal propagation, or sub-Planckian energy densities. We extend this proof to a large class of theories with higher (spatial) derivative or non-local terms in the action. Interestingly, only theories in the neighborhood of Lifshitz points with w(k) proportional to k(0) and k(3) remain viable.