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

TL;DR

This work identifies a precise duality between expanding and contracting FRW cosmologies with a single scalar field and constant $\epsilon$, showing that an expanding solution with $\epsilon$ yields the same scalar perturbation spectrum as a contracting solution with $\hat{\epsilon}=1/\epsilon$ for both dominant and subdominant modes. The authors derive this using Mukhanov's variables $u$ and $v$, establish the invariance of the relevant combinations under $\epsilon\to1/\epsilon$, and decompose perturbations into growth and decay modes with explicit spectral indices. They also demonstrate that tensor perturbations are not dual-invariant, thereby breaking the degeneracy, and extend the framework to $d$ spacetime dimensions. The results have potential implications for inflationary and cyclic/ekpyrotic cosmologies and highlight connections to other cosmological dualities and observational signatures.

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 $ε\equiv [(2d-5)+(d-1)w]/(d-2)$.

Figures (2)

• Figure 1: Penrose diagrams for spatially-flat FRW universes with: (a) $0<\epsilon<1$, expanding; (b) $1<\epsilon<\infty$, expanding; (c) $0<\epsilon<1$, contracting; (d) $1<\epsilon<\infty$, contracting. The left edge of each diagram is the world line of a comoving observer at the origin; curved lines represent other comoving world lines and spatial hypersurfaces. The Hubble horizon is a curve connecting the $90^{o}$ vertex to the lightlike boundary, but the precise curve depends on $\epsilon$. For illustration, we have shown the horizon for $\epsilon=0$ in (a, c) and for $\epsilon=2$ in (b, d). In this paper, we focus on cases (a) and (d), in which comoving scales exit the Hubble horizon.
• Figure 2: The dominant and subdominant scalar spectral indices (a) as a function of $w$ and (b) as a function of $\textrm{ln}\epsilon$. Note especially the symmetry of (b).