The Universe as an Oscillator

We apply the idea of using a matter time gauge in quantum gravity to quantum cosmology. For the Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe with dust and cosmological constant $\Lambda$, we show that the dynamics maps exactly to the simple harmonic oscillator in the dust-time gauge. For $\Lambda >0$ the oscillator frequency is imaginary, for $\Lambda<0$ it is real, and for $\Lambda=0$ the Universe is a free particle. This result provides (i) a simple demonstration of non-perturbative singularity avoidance in quantum gravity for all $\Lambda$, (ii) an exact Lorentzian Hartle-Hawking wave function, and (iii) gives the present age of the Universe as the characteristic decay time of the propagator.