Quantum cosmology as automorphic dynamics

We revisit pure quantum cosmology in three dimensions. The Wheeler-DeWitt equation can be solved perturbatively and the dynamics reduces to a particle on moduli space. Its time evolution is equivalent to the TT̄ deformation. Focusing on spacetimes with torus slices, we show that inflationary cosmologies correspond to particle trajectories in Artin's billiard. The resulting automorphic dynamics is developed both from a first and second quantized perspectives. Our main application is to give an interpretation for the Hartle-Hawking state which is here the analytic continuation of the Maloney-Witten partition function. We obtain its spectral decomposition and an exact representation as an average involving the M\"obius function.