Birth of de Sitter universe from a time crystal universe

We show that a simple subclass of Horndeski theory can describe a time crystal universe. The time crystal universe can be regarded as a baby universe nucleated from a flat space, which is mediated by an extension of Giddings-Strominger instanton in a 2-form theory dual to the Horndeski theory. Remarkably, when a cosmological constant is included, de Sitter universe can be created by tunneling from the time crystal universe. It gives rise to a past completion of an inflationary universe.