Cosmological billiards arise as a map of the solution to the Einstein equations, when the most general symmetry of the metric tensor is implemented, under the Belinskii, Khalatnikov and Lifshitz (BKL) paradigm, for which points are spatially decoupled in the asymptotical limit close to the cosmological singularity. Cosmological billiards in $4=3+1$ dimensions for the case of pure gravity are analyzed for those features, for which the content of Weyl reflections in the BKL maps requires the definition of a three-dimensional restricted phase space. The role of Poincaré sections in these processes is outlined. The quantum regime is investigated within this framework: as a result, one-epoch BKL eras are found to be the most probable configuration at which the wave functions have to be evaluated; furthermore, BKL eras containing $n\gg 1$ epochs are shown to be a less probable configuration for the wave functions. This description of the dynamics allows one to gain information about the connections between the statistical characterization of the maps which imply the different symmetry-quotienting mechanisms and the characterization of the semiclassical limit of the wave functions, for which evidence is produced for the phenomenon of “scars,” here for the first time outlined for the wave function of the universe in cosmological billiards, analyzed for the lowest silver ratios, and compared with the implications of a Farey map. The connections between the classical BKL probabilties and the quanum BKL probabilities for the “scarred” wave functions of the universe are provided and compared within different expansions according to different limits of the BKL statistics

• Received 17 April 2013

DOI:https://doi.org/10.1103/PhysRevD.88.104014