Examination of the nature of the Bianchi type cosmological singularities
  We present quantum (and classical) Bianchi I model, with free massless scalar field, of the Universe. Our model may be treated as the simplest prototype of the quantum BKL (Belinskii-Khalatnikov-Lifshitz) scenario. The quantization is done by making use of the nonstandard Loop Quantum Cosmology (LQC). Since the method is quite new, we present in details its motivation and the formalism. To make the nonstandard method easily understandable, we include its application to the FRW model. In the nonstandard LQC, we first solve the Hamiltonian constraint of the theory at the classical level and find elementary observables. Physical compound observables are defined in terms of elementary ones. We find that classical Big Bang singularity is replaced by quantum Big Bounce transition due to modification of classical theory by holonomy around a loop with finite size. The energy density of matter fields at the Big Bounce depends on a free parameter {\lambda}, which value is expected to be determined from future cosmological observations. The phase space is divided into two distinct regions: Kasner-like and Kasner-unlike. We use the elementary observables to quantize volume and directional volume operators in both cases. Spectra of these operators are bounded from below and discrete, and depend on {\lambda}. The discreteness may imply a foamy structure of spacetime at semiclassical level. At the quantum level an evolution of the model is generated by the so-called true Hamiltonian. This enables introducing a time parameter valued in the set of all real numbers. 
Piotr Dzierzak
gr-qc