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 $ϕ(x)$, potential $V(ϕ)$ and constant equation of state $w=p/ρ$, we show that an expanding solution characterized by $ε=3(1+w)/2$ produces the same scalar perturbations as a contracting solution with $\hatε=1/ε$. 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 $ε\equiv [(2d-5)+(d-1)w]/(d-2)$.

9 pages, 2 figures, 1 table