The LQC evolution operator of FRW universe with positive cosmological constant

The self-adjointness of an evolution operator $\Theta_{\Lambda}$ corresponding to the model of flat FRW universe with massless scalar field and cosmological constant quantized in the framework of Loop Quantum Cosmology is studied in the case $\Lambda>0$. It is shown, that for \Lambda<\Lambda_c\approx 10.3 \lPl^{-2} the operator admits many self-adjoint extensions, each of the purely discrete spectrum. On the other hand for $\Lambda\geq\Lambda_c$ the operator is essentially self-adjoint, however the physical Hilbert space of the model does not contain any physically interesting states.Comment: RevTex4, 8 page

Geometric Characterizations of the Kerr Isolated Horizon

We formulate conditions on the geometry of a non-expanding horizon $\Delta$ which are sufficient for the space-time metric to coincide on $\Delta$ with the Kerr metric. We introduce an invariant which can be used as a measure of how different the geometry of a given non-expanding horizon is from the geometry of the Kerr horizon. Directly, our results concern the space-time metric at \IH at the zeroth and the first orders. Combained with the results of Ashtekar, Beetle and Lewandowski, our conditions can be used to compare the space-time geometry at the non-expanding horizon with that of Kerr to every order. The results should be useful to numerical relativity in analyzing the sense in which the final black hole horizon produced by a collapse or a merger approaches the Kerr horizon.Comment: 11 pages, relevance of the results for the numerical relativity explained, mistakes correcte

Effective dynamics of the hybrid quantization of the Gowdy T^3 universe

The quantum dynamics of the linearly polarized Gowdy T^3 model (compact inhomogeneous universes admitting linearly polarized gravitational waves) is analyzed within Loop Quantum Cosmology by means of an effective dynamics. The analysis, performed via analytical and numerical methods, proves that the behavior found in the evolution of vacuum (homogeneous) Bianchi I universes is preserved qualitatively also in the presence of inhomogeneities. More precisely, the initial singularity is replaced by a big bounce which joins deterministically two large classical universes. In addition, we show that the size of the universe at the bounce is at least of the same order of magnitude (roughly speaking) as the size of the corresponding homogeneous universe obtained in the absence of gravitational waves. In particular, a precise lower bound for the ratio of these two sizes is found. Finally, the comparison of the amplitudes of the gravitational wave modes in the distant future and past shows that, statistically (i.e., for large samples of universes), the difference in amplitude is enhanced for nearly homogeneous universes, whereas this difference vanishes in inhomogeneity dominated cases. The presented analysis constitutes the first systematic effective study of an inhomogeneous system within Loop Quantum Cosmology, and it proves the robustness of the results obtained for homogeneous cosmologies in this context.Comment: 21 pages, 11 figures, RevTex4-1 + BibTe

Physical evolution in Loop Quantum Cosmology: The example of vacuum Bianchi I

We use the vacuum Bianchi I model as an example to investigate the concept of physical evolution in Loop Quantum Cosmology (LQC) in the absence of the massless scalar field which has been used so far in the literature as an internal time. In order to retrieve the system dynamics when no such a suitable clock field is present, we explore different constructions of families of unitarily related partial observables. These observables are parameterized, respectively, by: (i) one of the components of the densitized triad, and (ii) its conjugate momentum; each of them playing the role of an evolution parameter. Exploiting the properties of the considered example, we investigate in detail the domains of applicability of each construction. In both cases the observables possess a neat physical interpretation only in an approximate sense. However, whereas in case (i) such interpretation is reasonably accurate only for a portion of the evolution of the universe, in case (ii) it remains so during all the evolution (at least in the physically interesting cases). The constructed families of observables are next used to describe the evolution of the Bianchi I universe. The performed analysis confirms the robustness of the bounces, also in absence of matter fields, as well as the preservation of the semiclassicality through them. The concept of evolution studied here and the presented construction of observables are applicable to a wide class of models in LQC, including quantizations of the Bianchi I model obtained with other prescriptions for the improved dynamics.Comment: RevTex4, 22 pages, 4 figure