A new duality relating density perturbations in expanding and contracting Friedmann cosmologies

Abstract

For a 4-dimensional spatially-flat Friedmann-Robertson-Walker universe with a scalar field $\varphi (x)$, potential $V(\varphi )$ 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 $\hat{\epsilon }=1/\epsilon$. 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 $\epsilon \equiv [(2d-5)+(d-1)w]/(d-2)$.

9 pages, 2 figures, 1 table