Tunnelling geometries I. Analyticity, unitarity and instantons in quantum cosmology

We present a theory of tunnelling geometries originating from the no-boundary quantum state of Hartle and Hawking. We reformulate the no-boundary wavefunction in the representation of true physical variables and calculate it in the one-loop approximation. For this purpose a special technique is developed, which reduces the formalism of complex tunnelling geometries to the real ones, and also the method of collective variables is applied, separating the macroscopic collective degrees of freedom from the microscopic modes. The quantum distribution of Lorentzian universes, defined on the space of such collective variables, incorporates the probabilty conservation and represents the partition function of quasi-DeSitter gravitational instantons weighted by their Euclidean effective action. They represent closed compact manifolds obtained by the procedure of doubling the Euclidean spacetime which nucleates the Lorentzian universes. The over-Planckian behaviour of their distribution is determined by the anomalous scaling of the theory on such instantons, which serves as a criterion for the high-energy normalizability of the no-boudary wavefunction and the validity of the semiclassical expansion.