Monte Carlo studies of quantum cosmology by the generalized Lefschetz thimble method

Quantum cosmology aims at elucidating the beginning of our Universe. Back in early 80's, Vilenkin and Hartle-Hawking put forward the "tunneling from nothing" and "no boundary" proposals. Recently there has been renewed interest in this subject from the viewpoint of defining the oscillating path integral for Lorentzian quantum gravity using the Picard-Lefschetz theory. Aiming at going beyond the mini-superspace and saddle-point approximations, we perform Monte Carlo calculations using the generalized Lefschetz thimble method to overcome the sign problem. In particular, we confirm that either Vilenkin or Hartle-Hawking saddle point becomes relevant if one uses the Robin boundary condition depending on its parameter. We also clarify some fundamental issues in quantum cosmology, such as an issue related to the integration domain of the lapse function and an issue related to reading off the real geometry from the complex geometry obtained at the saddle point.