Eternal inflation with Liouville cosmology

We present a concrete holographic realization of the eternal inflation in (1+1)$(1+1)$-dimensional Liouville gravity by applying the philosophy of the FRW/CFT correspondence proposed by Freivogel, Sekino, Susskind and Yeh (FSSY). The dual boundary theory is nothing but the old matrix model describing the two-dimensional Liouville gravity coupled with minimal model matter fields. In Liouville gravity, the flat Minkowski space or even the AdS space will decay into the dS space, which is in stark contrast with higher-dimensional theories, but the spirit of the FSSY conjecture applies with only minimal modification. We investigate the classical geometry as well as some correlation functions to support our claim. We also study an analytic continuation to the time-like Liouville theory to discuss possible applications in (1+3)$(1+3)$-dimensional cosmology along with the original FSSY conjecture, where the boundary theory involves the time-like Liouville theory. We show that the decay rate in the (1+3)$(1+3)$ dimension is more suppressed due to the quantum gravity correction of the boundary theory.