Homogeneous Loop Quantum Cosmology Research Articles

A Brief Overview of Results about Uniqueness of the Quantization in Cosmology

The purpose of this review is to provide a brief overview of recent conceptual developments regarding possible criteria to guarantee the uniqueness of the quantization in a variety of situations that are found in cosmological systems. These criteria impose certain conditions on the representation of a group of physically relevant linear transformations. Generally, this group contains any existing symmetry of the spatial sections. These symmetries may or may not be sufficient for the purpose of uniqueness and may have to be complemented with other remaining symmetries that affect the time direction or with dynamical transformations that are, in fact, not symmetries. We discuss the extent to which a unitary implementation of the resulting group suffices to fix the quantization—a demand that can be seen as a weaker version of the requirement of invariance. In particular, a strict invariance under certain transformations may eliminate some physically interesting possibilities in the passage to the quantum theory. This is the first review in which this unified perspective is adopted to discuss otherwise different uniqueness criteria proposed either in homogeneous loop quantum cosmology or in the Fock quantization of inhomogeneous cosmologies.

Hybrid Loop Quantum Cosmology: An Overview

Loop Quantum Gravity is a nonperturbative and background independent program for the quantization of General Relativity. Its underlying formalism has been applied successfully to the study of cosmological spacetimes, both to test the principles and techniques of the theory and to discuss its physical consequences. These applications have opened a new area of research known as Loop Quantum Cosmology. The hybrid approach addresses the quantization of cosmological systems that include fields. This proposal combines the description of a finite number of degrees of freedom using Loop Quantum Cosmology, typically corresponding to a homogeneous background, and a Fock quantization of the field content of the model. In this review we first present a summary of the foundations of homogeneous Loop Quantum Cosmology and we then revisit the hybrid quantization approach, applying it to the study of Gowdy spacetimes with linearly polarized gravitational waves on toroidal spatial sections, and to the analysis of cosmological perturbations in preinflationary and inflationary stages of the Universe. The main challenge is to extract predictions about quantum geometry effects that eventually might be confronted with cosmological observations. This is the first extensive review of the hybrid approach in the literature on Loop Quantum Cosmology.

Kinematical uniqueness of homogeneous isotropic LQC

In a paper by Ashtekar and Campiglia, invariance under volume preserving residual diffeomorphisms has been used to single out the standard representation of the reduced holonomy-flux algebra in homogeneous loop quantum cosmology (LQC). In this paper, we use invariance under all residual diffeomorphisms to single out the standard kinematical Hilbert space of homogeneous isotropic LQC for both the standard configuration space , as well as for the Fleischhack one . We first determine the scale invariant Radon measures on these spaces, and then show that the Haar measure on is the only such measure for which the momentum operator is hermitian w.r.t. the corresponding inner product. In particular, the measure is forced to be identically zero on in the Fleischhack case, so that for both approaches, the standard kinematical LQC-Hilbert space is singled out.

Uniqueness of measures in loop quantum cosmology

In Ashtekar and Campiglia [Classical Quantum Gravity 29, 242001 (2012)], residual diffeomorphisms have been used to single out the standard representation of the reduced holonomy-flux algebra in homogeneous loop quantum cosmology (LQC). We show that, in the homogeneous isotropic case, unitarity of the translations with respect to the extended ℝ-action (exponentiated reduced fluxes in the standard approach) singles out the Bohr measure on both the standard quantum configuration space ℝBohr as well as on the Fleischhack one (ℝ⊔ℝBohr). Thus, in both situations, the same condition singles out the standard kinematical Hilbert space of LQC.

Why are the effective equations of loop quantum cosmology so accurate?

We point out that the relative Heisenberg uncertainty relations vanish for noncompact spaces in homogeneous loop quantum cosmology. As a consequence, for sharply peaked states quantum fluctuations in the scale factor never become important, even near the bounce point. This shows why quantum backreaction effects remain negligible and explains the surprising accuracy of the effective equations in describing the dynamics of sharply peaked wave packets. This also underlines the fact that minisuperspace models---where it is global variables that are quantized---do not capture the local quantum fluctuations of the geometry.

Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models

The mathematical structure of homogeneous loop quantum cosmology is analyzed, starting with and taking into account the general classification of homogeneous connections not restricted to be Abelian. As a first consequence, it is seen that the usual approach of quantizing Abelian models using spaces of functions on the Bohr compactification of the real line does not capture all properties of homogeneous connections. A new, more general quantization is introduced which applies to non-Abelian models and, in the Abelian case, can be mapped by an isometric, but not unitary, algebra morphism onto common representations making use of the Bohr compactification. Physically, the Bohr compactification of spaces of Abelian connections leads to a degeneracy of edge lengths and representations of holonomies. Lifting this degeneracy, the new quantization gives rise to several dynamical properties, including lattice refinement seen as a direct consequence of state-dependent regularizations of the Hamiltonian constraint of loop quantum gravity. The representation of basic operators - holonomies and fluxes - can be derived from the full theory specialized to lattices. With the new methods of this article, loop quantum cosmology comes closer to the full theory and is in a better position to produce reliable predictions when all quantum effects of the theory are taken into account.

Lattice loop quantum cosmology: scalar perturbations

We study the scalar modes of linear perturbations in loop quantum cosmology. This is done on a lattice where each cell is taken to be homogeneous and isotropic and can be quantized via the standard homogeneous loop quantum cosmology techniques. The appropriate interactions between nearby cells are included in the Hamiltonian in order to obtain the correct physics. It is shown that the quantum theory is anomaly free: the scalar and diffeomorphism constraint operators weakly commute with the Hamiltonian. Finally, the effective theory encoding the leading order quantum gravity corrections is derived and is shown to give the same holonomy-corrected effective equations that have been obtained in previous studies.Communicated by P R L V Moniz.

Geometric perspective on singularity resolution and uniqueness in loop quantum cosmology

We reexamine the issue of singularity resolution in homogeneous loop quantum cosmology from the perspective of geometrical entities such as expansion rate and the shear scalar. These quantities are very reliable measures of the properties of spacetime and can be defined not only at the classical and effective level, but also at an operator level in the quantum theory. From their behavior in the effective constraint surface and in the effective loop quantum spacetime, we show that one can severely restrict the ambiguities in regularization of the quantum constraint and rule out unphysical choices. We analyze this in the flat isotropic model and the Bianchi-I spacetimes. In the former case we show that the expansion rate is absolutely bounded only for the so-called improved quantization, a result which synergizes with uniqueness of this quantization as proved earlier. Surprisingly, for the Bianchi-I spacetime, we show that out of the available choices, the expansion rate and shear are bounded for only one regularization of the quantum constraint. It turns out that only for this choice, the theory exhibits quantum gravity corrections at a unique scale, and is physically viable.

Semiclassical states in homogeneous loop quantum cosmology

Semiclassical states in homogeneous loop quantum cosmology (LQC) are constructed in two different ways. In the first approach, we firstly construct an exponentiated annihilation operator. Then a kind of semiclassical (coherent) state is obtained by solving the eigenequation of that operator. Moreover, we use these coherent states to analyse the semiclassical limit of the quantum dynamics. It turns out that the Hamiltonian constraint operator employed currently in homogeneous LQC has a correct classical limit with respect to the coherent states. In the second approach, the other kind of semiclassical state is derived from the mathematical construction of coherent states for compact Lie groups due to Hall.

Homogeneous loop quantum cosmology: the role of the spin connection

Homogeneous cosmological models with non-vanishingintrinsic curvature require a special treatment when they are quantized with loop quantum cosmological methods. Guidance from the full theory which is lost in this context can be replaced by two criteria for an acceptable quantization, admissibility of a continuum approximation and local stability. A quantization of the corresponding Hamiltonian constraints is presented and shown to lead to a locally stable, non-singular evolution compatible with almost classical behaviour at large volume. As an application, the Bianchi IX model and its modified behaviour close to its classical singularity is explored.