Open FRW model in Loop QuantumCosmology is under consideration. The left and right invariant vector fields and holonomies along them are studied. It is shown that in the hyperbolic geometry of k = −1 itis possible to construct a suitable loop which provides us with quantum scalar constraint originally introduced by Vandersloot [19]. The quantum scalar constraint operator with negative cosmological constant is proved to be essentially self-adjoint. * lszulc@fuw.edu.pl 1 Strictly speaking the isotropic k = −1 model is derived from anisotropic Bianchi V

The basic idea of loop quantum cosmology (LQC) applies to every spatially homogeneous cosmological model; however only the spatially flat (so-called k = 0) case has been understood in detail in the literature thus far. In the closed (so-called k = 1) case certain technical difficulties have been the obstacle to development. In this work the difficulties are overcome, and a new LQC model of the spatially closed, homogeneous, isotropic universe is constructed. The topology of the spacelike section of the universe is assumed to be that of SU (2) or SO(3). Surprisingly, according to the new results achieved in this paper, the two cases can be distinguished from each other just by the local properties of the quantum geometry of the universe! The quantum Hamiltonian operator of the gravitational field takes the form of a difference operator, where the elementary step is the quantum of the 3-volume derived in the flat case by Ashtekar, Pawlowski and Singh. The mathematical properties of the operator are studied: it is essentially self-adjoint, bounded from above by 0, the 0 itself is not an eigenvalue, the eigenvectors form a basis. An estimate on the dimension of the spectral projection on any finite interval is provided.

A simplified theory of the diagonal Bianchi type I model coupled with a massless scalar field in loop quantum cosmology is constructed according to theμ scheme. Kinematical and physical sectors of the theory are under good analytical control as well as the scalar constraint operator. Although it is possible to compute numerically the nonsingular evolution of the three gravitational degrees of freedom, the naive implementation of theμ scheme to the diagonal Bianchi type I model is problematic. The lack of the full invariance of the theory with respect to the fiducial cell and fiducial metric scaling causes serious problems in the semiclassical limit of the theory. Because of this behavior it is very difficult to extract reasonable physics from the model. The weaknesses of the implementation of theμ scheme to the Bianchi I model do not imply limitations of theμ scheme in the isotropic case.

In the recent years the quantization methods of Loop Quantum Gravity have been successfully applied to the homogeneous and isotropic Friedmann-RobertsonWalker space-times. The resulting theory, called Loop Quantum Cosmology (LQC), resolves the Big Bang singularity by replacing it with the Big Bounce. We argue that LQC generates also certain corrections to field theoretical inflationary scenarios. These corrections imply that in the LQC the effective sonic horizon becomes infinite at some point after the bounce and that the scale of the inflationary potential implied by the COBE normalisation increases. The evolution of scalar fields immediately after the Bounce becomes modified in an interesting way. We point out that one can use COBE normalisation to establish an upper bound on the quantum of length of LQG.

This letter is motivated by the recent papers by Dittrich and Thiemann and, respectively, by Rovelli discussing the status of Quantum Geometry in the dynamical sector of Loop Quantum Gravity. Since the papers consider model examples, we also study the issue in the case of an example, namely on the Loop Quantum Cosmology model of space-isotropic universe. We derive the Rovelli-Thiemann-Ditrich partial observables corresponding to the quantum geometry operators of LQC in both Hilbert spaces: the kinematical one and, respectively, the physical Hilbert space of solutions to the quantum constraints. We find, that Quantum Geometry can be used to characterize the physical solutions, and the operators of quantum geometry preserve many of their kinematical properties.