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

TL;DR

This paper reveals a duality in scalar perturbations between expanding and contracting flat Friedmann universes, showing that solutions with reciprocal parameters produce identical perturbation spectra, generalizable to higher dimensions.

Contribution

It introduces a new duality relating density perturbations in expanding and contracting Friedmann cosmologies, extending the symmetry to arbitrary spacetime dimensions.

Findings

Expanding and contracting solutions with reciprocal parameters yield identical scalar perturbations.

The duality applies to both dominant and subdominant modes.

The symmetry generalizes to d-dimensional spacetimes.

Abstract

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 $\hat{\epsilon}=1/\epsilon$. 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 $\epsilon \equiv [(2d-5)+(d-1)w]/(d-2)$.