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.

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Loop Quantum Cosmology: A Status Report

Using techniques from loop quantum gravity, the standard theory of cosmological perturbations was recently generalized to encompass the Planck era. We now apply this framework to explore pre-inflationary dynamics. The framework enables us to isolate and resolve the true trans-Planckian difficulties, with interesting lessons both for theory and observations. Specifically, for a large class of initial conditions at the bounce, we are led to a self consistent extension of the inflationary paradigm over the 11 orders of magnitude in density and curvature, from the big bounce to the onset of slow roll. In addition, for a narrow window of initial conditions, there are departures from the standard paradigm, with novel effects—such as a modification of the consistency relation between the ratio of the tensor to scalar power spectrum and the tensor spectral index, as well as a new source for non-Gaussianities—which could extend the reach of cosmological observations to the deep Planck regime of the early universe. Communicated by P Singh

http://iopscience.iop.org/article/10.1088/0264-9381/30/8/085014/pdf

The pre-inflationary dynamics of loop quantum cosmology: Confronting quantum gravity with observations

The problem of time in quantum gravity occurs because ‘time’ is taken to have a dierent meaning in each of general relativity and ordinary quantum theory. This incompatibility creates serious problems with trying to replace these two branches of physics with a single framework in regimes in which neither quantum theory nor general relativity can be neglected, such as in black holes or in the very early universe. Strategies for resolving the Problem of Time have evolved somewhat since Kucha r and Isham’s well-known reviews from the early 90’s. These come in the following divisions I) time before quantization, such as hidden time or matter time. II) Time after quantization, such as emergent semiclassical time.

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The problem of time in quantum gravity

In order to make predictions in quantum cosmology one has to resolve the problem of the operator ordering in the hamiltonian. This is equivalent to defining a differential operator on superspace, the infinite dimensional manifold of all 3-metrics and matter field configurations on a 3-surface S. We propose that this operator should be the laplacian in a natural metric. There remains a difficulty however because the metric connection on superspace depends on the metric in a nonlinear manner. This nonlinearity would be inconsistent with the interpretation of the lapse function as a Lagrange multiplier. It may cancel out if an equal number of fermion degrees of freedom are included. The probability measure on superspace would be the measure associated with this metric. In situations in which the wave function can be interpreted in terms of classical solutions by the WKB approximation, this choice of measure implies that the probability of finding the 3-metric and matter field configuration in a given region of superspace is proportional to the proper time that the solutions spend in that region. As an application we compute the probability distribution of the density parameter Ω in a minisuperspace model which is in the quantum state defined by a path integral over compact 4-geometries. If we restrict attention to a fixed value of the density, we find that the probability distribution is entirely concentrated at Ω = 1 .

Operator Ordering and the Flatness of the Universe

The Problem of Time occurs because the ‘time’ of GR and of ordinary Quantum Theory are mutually incompatible notions. This is problematic in trying to replace these two branches of physics with a single framework in situations in which the conditions of both apply, e.g. in black holes or in the very early universe. Emphasis in this Review is on the Problem of Time being multi-faceted and on the nature of each of the eight principal facets. Namely, the Frozen Formalism Problem, Configurational Relationalism Problem (formerly Sandwich Problem), Foliation Dependence Problem, Constraint Closure Problem (formerly Functional Evolution Problem), Multiple Choice Problem, Global Problem of Time, Problem of Beables (alias Problem of Observables) and Spacetime Reconstruction/Replacement Problem. Strategizing in this Review is not just centred about the Frozen Formalism Problem facet, but rather about each of the eight facets. Particular emphasis is placed upon A) relationalism as an underpinning of the facets and as a selector of particular strategies (especially a modification of Barbour relationalism, though also with some consideration of Rovelli relationalism). B) Classifying approaches by the full ordering in which they embrace constrain, quantize, find time/history and find observables, rather than only by partial orderings such as “Dirac-quantize”. C) Foliation (in)dependence and Spacetime Reconstruction for a wide range of physical theories, strategizing centred about the Problem of Beables, the Patching Approach to the Global Problem of Time, and the role of the question-types considered in physics. D) The Halliwell- and Gambini–Porto–Pullin-type combined Strategies in the context of semiclassical quantum cosmology.

Problem of time in quantum gravity

A practical way to deal with the problem of time in quantum cosmology and quantum gravity is proposed. The main tool is effective equations, which mainly restrict explicit considerations to semiclassical regimes but have the crucial advantage of allowing the consistent use of local internal times in non-deparameterizable systems. Different local internal times are related merely by gauge transformations, thereby enabling relational evolution through turning points of non-global internal times. The main consequence of the local nature of internal time is the necessity of its complex valuedness, reminiscent of more general than non-unitarity of evolution defined for finite ranges of time. By several general arguments, the consistency of this setting is demonstrated. Finally, we attempt an outlook on the nature of time in highly quantum regimes. The focus of this paper is on conceptual issues.

An effective approach to the problem of time

The effective approach to quantum dynamics allows a reformulation of the Dirac quantization procedure for constrained systems in terms of an infinite-dimensional constrained system of classical type. For semiclassical approximations, the quantum constrained system can be truncated to finite size and solved by the reduced phase space or gauge-fixing methods. In particular, the classical feasibility of local internal times is directly generalized to quantum systems, overcoming the main difficulties associated with the general problem of time in the semiclassical realm. The key features of local internal times and the procedure of patching global solutions using overlapping intervals of local internal times are described and illustrated by two quantum mechanical examples. Relational evolution in a given choice of internal time is most conveniently described and interpreted in a corresponding choice of gauge at the effective level and changing the internal clock is, therefore, essentially achieved by a gauge transformation. This article complements the conceptual discussion in [M. Bojowald, P. A. H\"ohn, and A. Tsobanjan, Classical Quantum Gravity 28, 035006 (2011).].

Effective approach to the problem of time: general features and examples

An effective formalism for quantum constrained systems is presented which allows manageable derivations of solutions and observables, including a treatment of physical reality conditions without requiring full knowledge of the physical inner product. Instead of a state equation from a constraint operator, an infinite system of constraint functions on the quantum phase space of expectation values and moments of states is used. The examples of linear constraints as well as the free non-relativistic particle in parameterized form illustrate how standard problems of constrained systems can be dealt with in this framework.

Effective Constraints for Quantum Systems

A practical way to deal with the problem of time in quantum cosmology and quantum gravity is proposed. The main tool is effective equations, which mainly restrict explicit considerations to semiclassical regimes but have the crucial advantage of allowing the consistent use of local internal times in non-deparameterizable systems. Different local internal times are related merely by gauge transformations, thereby enabling relational evolution through turning points of non-global internal times. The main consequence of the local nature of internal time is the necessity of its complex-valuedness, reminiscent of but more general than non-unitarity of evolution defined for finite ranges of time. By several general arguments, the consistency of this setting is demonstrated. Finally, we attempt an outlook on the nature of time in highly quantum regimes. The focus of this note is on conceptual issues.

An effective formalism for quantum constrained systems is presented which allows manageable derivations of solutions and observables, including a treatment of physical reality conditions without requiring full knowledge of the physical inner product. Instead of a state equation from a constraint operator, an infinite system of constraint functions on the quantum phase space of expectation values and moments of states is used. The examples of linear constraints as well as the free non-relativistic particle in parametrized form illustrate how standard problems of constrained systems can be dealt with in this framework.

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Artur Tsobanjan

Papers

An effective approach to the problem of time

Effective approach to the problem of time: general features and examples

Effective Constraints for Quantum Systems

An effective approach to the problem of time

Effective constraints for quantum systems