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/p, we show that an expanding solution characterized by Ε=3(1 + w)/2 produces the same scalar perturbations as a contracting solution with Ε̂=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).