Homoclinic structure of classical spacetimes emerging from the Hartle-Hawking model in quantum cosmology

We show that the dynamics of Lorentzian geometries, associated with the semiclassical solutions of the Wheeler-DeWitt equation in the Hartle-Hawking minisuperspace model with no-boundary conditions, is chaotic due to homoclinic phenomena, and its basic features can be topologically characterized by the transversal crossings of homoclinic cylinders in the phase space of the model. The invariant manifolds of the dynamics of the classical Hamiltonian are the periodic orbits of the center manifold, a fractal set of homoclinic orbits which have an oscillatory approach to the periodic orbits of the center manifold, and an infinite set of unstable periodic orbits with arbitrarily large periods. The latter orbits have the homoclinic orbits as an accumulation set. These invariant manifolds may be considered to describe classical universes, with initial conditions determined by the Hartle-Hawking no-boundary proposal. We discuss an estimate for the amplitudes of semiclassical states (corresponding to distinct classical solutions) contributing to the quantum state of the early universe.