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