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 phi(x), potential V(phi) and constant equation of state w=p/rho, we show that an expanding solution characterized by epsilon=3(1+w)/2 produces the same scalar perturbations as a contracting solution with (epsilon) over cap =1/epsilon. 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 epsilon=[(2d-5)+(d-1)w]/(d-2).