Conditions for generating scale-invariant density perturbations

We analyze the general conditions on the equation of state w required for quantum fluctuations of a scalar field to produce a scale-invariant spectrum of density perturbations, including the possibility of models which (in the four dimensional effective description) bounce from a contracting to an expanding phase. We show that there are only two stable cases: $w\approx -1$ and $w\gg 1.$ The first is utilized in inflationary models and the second in the ekpyrotic/cyclic scenario, assuming the perturbations propagate through the bounce. All other cases, including the $w\approx 0$ case considered by some authors, require extreme fine-tuning of initial conditions and/or the effective potential. For the ekpyrotic/cyclic $\left(w\gg 1\right)$ case, we also analyze the small deviations from scale invariance.