Loop quantum cosmology (LQC) is the result of applying principles of loop quantum gravity (LQG) to cosmological settings. The distinguishing feature of LQC is the prominent role played by the quantum geometry effects of LQG. In particular, quantum geometry creates a brand new repulsive force which is totally negligible at low spacetime curvature but rises very rapidly in the Planck regime, overwhelming the classical gravitational attraction. In cosmological models, while Einstein's equations hold to an excellent degree of approximation at low curvature, they undergo major modifications in the Planck regime: for matter satisfying the usual energy conditions, any time a curvature invariant grows to the Planck scale, quantum geometry effects dilute it, thereby resolving singularities of general relativity. Quantum geometry corrections become more sophisticated as the models become richer. In particular, in anisotropic models, there are significant changes in the dynamics of shear potentials which tame their singular behavior in striking contrast to older results on anisotropies in bouncing models. Once singularities are resolved, the conceptual paradigm of cosmology changes and one has to revisit many of the standard issues—e.g. the 'horizon problem'—from a new perspective. Such conceptual issues as well as potential observational consequences of the new Planck scale physics are being explored, especially within the inflationary paradigm. These considerations have given rise to a burst of activity in LQC in recent years, with contributions from quantum gravity experts, mathematical physicists and cosmologists. The goal of this review is to provide an overview of the current state of the art in LQC for three sets of audiences: young researchers interested in entering this area; the quantum gravity community in general and cosmologists who wish to apply LQC to probe modifications in the standard paradigm of the early universe. In this review, effort has been made to streamline the material so that each of these communities can read only the sections they are most interested in, without loss of continuity.

/pdf/loop-quantum-cosmology-a-status-report-1zx4841jrd.pdf

Loop Quantum Cosmology: A Status Report

This is a first version of the introductory lectures on loop quantum gravity that I will give at the quantum gravity school in Zakopane. The theory is presented in self-contained form, without emphasis on its derivation from classical general relativity. Dynamics is given in the covariant form. Some applications are described, including the recent derivation of de Sitter cosmology from full quantum gravity.

http://inspirehep.net/record/889853/files/QGQGS%202011_003.pdf

Zakopane lectures on loop gravity

Cosmological perturbations are generally described by quantum fields on (curved but) classical space-times. While this strategy has a large domain of validity, it can not be justified in the quantum gravity era where curvature and matter densities are of Planck scale. Using techniques from loop quantum gravity, the standard theory of cosmological perturbations is extended to overcome this limitation. The new framework sharpens conceptual issues by distinguishing between the true and apparent trans-Planckian difficulties and provides sufficient conditions under which the true difficulties can be overcome within a quantum gravity theory. In a companion paper, this framework is applied to the standard inflationary model, with interesting implications to theory as well as observations.

Extension of the quantum theory of cosmological perturbations to the Planck era

A brief overview of loop quantum cosmology of homogeneous isotropic models is presented with emphasis on the origin of and subtleties associated with the resolution of big bang and big crunch singularities. These results bear out the remarkable intuition that John Wheeler had. Discussion is organized at two levels. The the main text provides a bird’s eye view of the subject that should be accessible to non-experts. Appendices address conceptual and technical issues that are often raised by experts in loop quantum gravity and string theory.

Loop Quantum Cosmology: An Overview

A fully consistent linear perturbation theory for cosmology is derived in the presence of quantum corrections as they are suggested by properties of inverse volume operators in loop quantum gravity. The underlying constraints present a consistent deformation of the classical system, which shows that the discreteness in loop quantum gravity can be implemented in effective equations without spoiling space-time covariance. Nevertheless, nontrivial quantum corrections do arise in the constraint algebra. Since correction terms must appear in tightly controlled forms to avoid anomalies, detailed insights for the correct implementation of constraint operators can be gained. The procedures of this article thus provide a clear link between fundamental quantum gravity and phenomenology.

Anomaly freedom in perturbative loop quantum gravity

Linear cosmological perturbation theory is pivotal to a theoretical understanding of current cosmological experimental data provided eg by cosmic microwave anisotropy probes A key issue in that theory is to extract the gauge-invariant degrees of freedom which allow unambiguous comparison between theory and experiment When one goes beyond first (linear) order, the task of writing the Einstein equations expanded to nth order in terms of quantities that are gauge-invariant up to terms of higher orders becomes highly non-trivial and cumbersome This fact has prevented progress for instance on the issue of the stability of linear perturbation theory and is a subject of current debate in the literature In this series of papers we circumvent these difficulties by passing to a manifestly gauge-invariant framework In other words, we only perturb gauge-invariant, ie measurable quantities, rather than gauge variant ones Thus, gauge invariance is preserved non-perturbatively while we construct the perturbation theory for the equations of motion for the gauge-invariant observables to all orders In this first paper we develop the general framework which is based on a seminal paper due to Brown and Kuchař as well as the relational formalism due to Rovelli In the second, companion, paper we apply our general theory to FRW cosmologies and derive the deviations from the standard treatment in linear order As it turns out, these deviations are negligible in the late universe, thus our theory is in agreement with the standard treatment However, the real strength of our formalism is that it admits a straightforward and unambiguous, gauge-invariant generalization to higher orders This will also allow us to settle the stability issue in a future publication

/pdf/manifestly-gauge-invariant-general-relativistic-perturbation-bz0ch5fane.pdf

Manifestly gauge-invariant general relativistic perturbation theory: I. Foundations

In our companion paper we identified a complete set of manifestly gauge-invariant observables for general relativity This was possible by coupling the system of gravity and matter to pressureless dust which plays the role of a dynamically coupled observer The evolution of those observables is governed by a physical Hamiltonian and we derived the corresponding equations of motion Linear perturbation theory of those equations of motion around a general exact solution in terms of manifestly gauge invariant perturbations was then developed In this paper we specialise our previous results to an FRW background which is also a solution of our modified equations of motion We then compare the resulting equations with those derived in standard cosmological perturbation theory (SCPT) We exhibit the precise relation between our manifestly gauge-invariant perturbations and the linearly gauge-invariant variables in SCPT We find that our equations of motion can be cast into SCPT form plus corrections These corrections are the trace that the dust leaves on the system in terms of a conserved energy momentum current density It turns out that these corrections decay, in fact, in the late universe they are negligible whatever the value of the conserved current We conclude that the addition of dust which serves as a test observer medium, while implying modifications of Einstein's equations without dust, leads to acceptable agreement with known results, while having the advantage that one now talks about manifestly gauge-invariant, that is measurable, quantities, which can be used even in perturbation theory at higher orders

Manifestly Gauge-Invariant General Relativistic Perturbation Theory: II. FRW Background and First Order

In our companion paper we identified a complete set of manifestly gauge-invariant observables for general relativity. This was possible by coupling the system of gravity and matter to pressureless dust which plays the role of a dynamically coupled observer. The evolution of those observables is governed by a physical Hamiltonian and we derived the corresponding equations of motion. Linear perturbation theory of those equations of motion around a general exact solution in terms of manifestly gauge-invariant perturbations was then developed. In this paper we specialize our previous results to an FRW background which is also a solution of our modified equations of motion. We then compare the resulting equations with those derived in standard cosmological perturbation theory (SCPT). We exhibit the precise relation between our manifestly gauge-invariant perturbations and the linearly gauge-invariant variables in SCPT. We find that our equations of motion can be cast into SCPT form plus corrections. These corrections are the trace that the dust leaves on the system in terms of a conserved energy–momentum current density. It turns out that these corrections decay; in fact, in the late universe they are negligible whatever the value of the conserved current. We conclude that the addition of dust which serves as a test observer medium, while implying modifications of Einstein's equations without dust, leads to acceptable agreement with known results, while having the advantage that one now talks about manifestly gauge-invariant, that is measurable, quantities, which can be used even in perturbation theory at higher orders.

/pdf/manifestly-gauge-invariant-general-relativistic-perturbation-59wiqjalwh.pdf

Manifestly gauge-invariant general relativistic perturbation theory: II. FRW background and first order

Linear cosmological perturbation theory is pivotal to a theoretical understanding of current cosmological experimental data provided e.g. by cosmic microwave anisotropy probes. A key issue in that theory is to extract the gauge invariant degrees of freedom which allow unambiguous comparison between theory and experiment. When one goes beyond first (linear) order, the task of writing the Einstein equations expanded to n'th order in terms of quantities that are gauge invariant up to terms of higher orders becomes highly non-trivial and cumbersome. This fact has prevented progress for instance on the issue of the stability of linear perturbation theory and is a subject of current debate in the literature. In this series of papers we circumvent these difficulties by passing to a manifestly gauge invariant framework. In other words, we only perturb gauge invariant, i.e. measurable quantities, rather than gauge variant ones. Thus, gauge invariance is preserved non perturbatively while we construct the perturbation theory for the equations of motion for the gauge invariant observables to all orders. In this first paper we develop the general framework which is based on a seminal paper due to Brown and Kuchar as well as the realtional formalism due to Rovelli. In the second, companion, paper we apply our general theory to FRW cosmologies and derive the deviations from the standard treatment in linear order. As it turns out, these deviations are negligible in the late universe, thus our theory is in agreement with the standard treatment. However, the real strength of our formalism is that it admits a straightforward and unambiguous, gauge invariant generalisation to higher orders. This will also allow us to settle the stability issue in a future publication.

S. Hofmann

Papers

Manifestly gauge-invariant general relativistic perturbation theory: I. Foundations

Manifestly Gauge-Invariant General Relativistic Perturbation Theory: II. FRW Background and First Order

Manifestly gauge-invariant general relativistic perturbation theory: II. FRW background and first order

Manifestly Gauge-Invariant General Relativistic Perturbation Theory : I. Foundations

明白なゲージ不変一般相対論的摂動論:ii.frw背景と一次数