Scale invariance in expanding and contracting universes from two-field models

We study cosmological perturbations produced by the most general two-derivative actions involving two scalar fields, coupled to Einstein gravity, with an arbitrary field space metric, that admit scaling solutions. For expanding universes, we find that scale-invariant adiabatic perturbation spectra can be produced for any equation of state parameter w that satisfies -1 \≤ w < -1/3, and that all the scaling solutions are attractors. For contracting universes, we show that scale-invariant adiabatic perturbations can be produced continuously as modes leave the horizon for any $w>-1/3$. The corresponding background solutions are unstable, which we argue is a universal feature of contracting models that yield scale-invariant spectra. At no point do we assume slow-roll. The presence of a nontrivial metric on field space is a crucial ingredient in our results.