A canonical transformation and the tunneling probability for the birth of an asymptotically DeSitter universe with dust

In the present work, we study the quantum cosmology description of closed Friedmann-Robertson-Walker models in the presence of a positive cosmological constant and a generic perfect fluid. We work in the Schutz's variational formalism. If one uses the scale factor and its canonically conjugated momentum as the phase space variables that describe the geometrical sector of these models, one obtains Wheeler-DeWitt equations with operator ordering ambiguities. In order to avoid those ambiguities and simplify the quantum treatment of the models, we introduce new phase space variables. We explicitly demonstrate that the transformation leading from the old set of variables to the new one is canonical. In order to show that the above canonical transformations simplify the quantum treatment of those models, we consider a particular model where the perfect fluid is dust. We solve the Wheeler-DeWitt equation numerically using the Crank-Nicholson scheme and determine the time evolution of the initial wave function. Finally, we compare the results for the present model with the ones for another model where the only difference is the presence of a radiative perfect fluid, instead of dust.Comment: Revtex4, 18 pages, 2 EPS figure

Reply to "Comment on 'Quantization of FRW spacetimes in the presence of a cosmological constant and radiation'"

The Comment by Amore {\it et al.} [gr-qc/0611029] contains a valid criticism of the numerical precision of the results reported in a recent paper of ours [Phys. Rev. D {\bf 73}, 044022 (2006)], as well as fresh ideas on how to characterize a quantum cosmological singularity. However, we argue that, contrary to what is suggested in the Comment, the quantum cosmological models we studied show hardly any sign of singular behavior.Comment: 4 pages, accepted by Physical Review