Ermakov–Lewis invariant in single field inflation

Abstract

We investigate the dynamical symmetry of the Sasaki–Mukhanov equation, which governs the evolution of primordial perturbations during inflation. By mapping the Sasaki–Mukhanov equation to a constant-frequency harmonic oscillator via an auxiliary Ermakov field satisfying the Ermakov–Pinney equation, we construct the associated sl(2, R) generators and derive the Ermakov–Lewis invariant. We apply this framework to de Sitter space and slow-roll inflation, demonstrating that the Bunch–Davies vacuum gives \(I_{EL} = 1/2\), conserved non-perturbatively at all orders in slow-roll.