New duality relating density perturbations in expanding and contracting Friedmann cosmologies

For a 4-dimensional spatially flat Friedmann-Robertson-Walker universe with a scalar field φ(x), potential V(φ) and constant equation of state w=p/ρ, we show that an expanding solution characterized by ε=3(1+w)/2 produces the same scalar perturbations as a contracting solution with ^ε=1/ε. The same symmetry applies to both the dominant and subdominant scalar perturbation modes. This result admits a simple physical interpretation and generalizes to d spacetime dimensions if we define ε≡[(2d-5)+(d-1)w]/(d-2).