Cosmological time crystals from Einstein-cubic gravities

By including appropriate Riemann-cubic invariants, we find that the dynamics of classical time crystals can be straightforwardly realized in Einstein gravity on the FLRW metric. The time reflection symmetry is spontaneously broken in the two vacua with the same scale factor a, but opposite a˙. The tunneling from one vacuum to the other provides a robust mechanism for bounce universes; it always occurs for systems with positive energy density. For a suitable matter energy-momentum tensor we also construct cyclic universes. Cosmological solutions that resemble the classical time crystals can be constructed in massive gravity.