  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.
