In a recent paper Graham et al constructed oscillating and static universe models which are stable with respect to all classical perturbations. Here we show that such universes are quantummechanically unstable and can collapse by quantum tunneling to zero radius. We also present instantons describing nucleation of oscillating and static universes from nothing.

We discuss three candidate scenarios which seem to allow the possibility that the universe could have existed forever with no initial singularity: eternal inflation, cyclic evolution, and the emergent universe. The first two of these scenarios are geodesically incomplete to the past, and thus cannot describe a universe without a beginning. The third, although it is stable with respect to classical perturbations, can collapse quantum mechanically, and therefore cannot have an eternal past.

Oscillating solutions to the effective equations of Loop Quantum Cosmology have been suggested for the role of an 'eternal seed', providing a possible starting point for the emergent universe scenario. We investigate the stability of a particular model, sourced by a homogeneous massless scalar field and a negative cosmological constant, with respect to small perturbations and to quantum collapse. We find that the model has perturbatively stable and unstable solutions, with both types of solutions occupying significant regions of the parameter space. All solutions are unstable with respect to collapse by quantum tunneling to zero size. We discuss the possibility that the state resulting from the collapse is non-singular, so it may tunnel back to the oscillating regime. We argue that the universe is then likely to evolve to states of very large size with large particle occupation numbers.

In canonical quantum cosmology, the wave function of the universe lacks explicit time dependence. However, time evolution may be present implicitly through the semiclassical superspace variables, which themselves depend on time in classical dynamics. In this paper, we apply this approach to an oscillating universe model recently introduced by Graham et al. By extending the model to include a massless, minimally coupled scalar field φ which has little effect on the dynamics but can play the role of a "clock", we determine the decay rate of the oscillating universe.