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

• Received 23 July 2018

DOI:https://doi.org/10.1103/PhysRevD.98.086002