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
\cite{Geshnizjani:2011dk} 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 $\omega_k \propto k^0$ and $k^3$ remain viable.
