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 [Formula Presented] potential [Formula Presented] and constant equation of state [Formula Presented] we show that an expanding solution characterized by [Formula Presented] produces the same scalar perturbations as a contracting solution with [Formula Presented] 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 [Formula Presented].