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.