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We study a recently proposed new cosmological phase where a scalar fieldmoves periodically in an expanding spatially-flat Friedmann universe. Thisphase corresponds to a limiting cycle of the equations of motion and can beconsidered as a cosmological realization of a "time-crystal". We show that thisphase is only possible, provided the Null Energy Condition is violated and theso-called Phantom divide is crossed. We prove that in general k-essence models:i) this crossing causes infinite growth of quantum perturbations on shortscales, and ii) exactly periodic solutions are only possible, provided thelimiting cycle encircles a singularity in the phase plane. The configurationsneighboring this singular curve in the phase space are linearly unstable on oneside of the curve and superluminal on the other side. Moreover, the incrementof the instability is infinitely growing for each mode by approaching thesingularity, while for the configurations on the other side, the sound speed isgrowing without limit. We illustrate our general results by analytical andnumerical studies of a particular class of such k-essence models.

The Phantom of the New Oscillatory Cosmological Phase