The Phantom of the Cosmological Time-Crystals

I will discuss a recently proposed new cosmological phase where a scalar field moves exactly periodically in an expanding spatially-flat Friedmann universe. On average this phase has a vacuum or de Sitter equation of state and can be interesting to model Inflation and Dark Energy in a novel way. This phase corresponds to a limiting cycle of the equations of motion and can be considered as a cosmological realization of a general idea of a "time-crystal" introduced by Wilczek et. all in 2012. Recently we showed that this cosmological phase is only possible, provided the Null Energy Condition is violated and the so-called Phantom divide is crossed. Using methods from the dynamical systems, we proved that in a rather general class of single scalar field models called k-essence: i) this crossing causes infinite growth of quantum perturbations on short length-scales, and ii) exactly periodic solutions are only possible, provided the limiting cycle encircles a singularity in the phase plane. The configurations neighboring this singular curve in the phase space are linearly unstable on one side of the curve and superluminal on the other side. Moreover, the increment of the instability is infinitely growing for each mode by approaching the singularity, while for the configurations on the other side, the sound speed is growing without limit. We illustrated our general results by analytical and numerical studies of particular models proposed by Wilczek and collaborators. Finally I will briefly discuss systems where this idea of time-crystals may be realized.