Positive cosmological constant in loop quantum cosmology

The k = 0 Friedmann-Lemaitre-Robertson-Walker model with a positive cosmological constant and a massless scalar field is analyzed in detail. If one uses the scalar field as relational time, new features arise already in the Hamiltonian framework of classical general relativity: In a finite interval of relational time, the universe expands out to infinite proper time and zero matter density. In the deparametrized quantum theory, the true Hamiltonian now fails to be essentially self-adjoint both in the Wheeler-DeWitt approach and in loop quantum cosmology. Irrespective of the choice of the self-adjoint extension, the big-bang singularity persists in the Wheeler-DeWitt theory while it is resolved and replaced by a big bounce in loop quantum cosmology. Furthermore, the quantum evolution is surprisingly insensitive to the choice of the self-adjoint extension. This may be a special case of a yet-to-be discovered general property of a certain class of symmetric operators that fail to be essentially self-adjoint.