By Parampreet Singh, Louisiana State University, USA

CQG Editorial Board member, Param Singh, Guest Edited the Applications of loop quantum gravity to cosmology focus issue in 2016 and 2017

A successful union of Einstein’s general relativity and quantum theory is one of the most fundamental problems of theoretical physics. Though a final theory of quantum gravity is not yet available, its lessons and techniques can already be used to understand  quantization of various spacetimes. Of these, cosmological spacetimes are of special interest. They provide a simpler yet a non-trivial and a highly rich setting to explore detailed implications of quantum gravitational theories. Various conceptual and technical difficulties encountered in understanding quantum dynamics of spacetime in  quantum gravity can be bypassed in such a setting. Further, valuable lessons can be learned for the quantization of more general spacetimes.

In the last decade, progress in loop quantum gravity has provided avenues which allow us to reliably answer various interesting questions about the quantum dynamics of spacetime in the cosmological setting. Quantum gravitational dynamics of cosmological spacetimes obtained using techniques of  loop quantum gravity leads to a novel picture where singularities of Einstein’s theory of general relativity are overcome and a new window opens to test loop quantum gravity effects through astronomical observations.

The scope of the Focus Issue: Applications of loop quantum gravity to cosmology, published last year in CQG, is to provide a snapshot of some of the rigorous and novel results on this research frontier in the cosmological setting.

Loop quantum gravity is a background independent, non-perturbative canonical quantization of gravity. Unlike gravitational phase space variables used in geometrodynamics – the spatial metric and the extrinsic curvature, it is   based on new canonical variables introduced by Ashtekar in 1986. The conjugate variables in the classical phase space in loop quantum gravity are three orthonormal triads and their momenta – the connection variables. These variables allow us to understand gravity, at least kinematically, in a similar way as the Yang-Mills theories. Further, there is a shift from the metric formulation of gravity to connection dynamics as in other gauge theories. As in Yang-Mills theories, the relevant physical variables for quantization turn out to be holonomies of connection (vector potential) and the fluxes of triads (electric field).

A mathematically precise formulation of loop quantum gravity was achieved in 1990’s. It overcame many hurdles one encounters towards constructing a background independent theory, with a high degree of mathematical sophistication. This set a platform to extract rigorous physical predictions and  led to some key insights on the physical nature  of the quantum spacetime. One of the fundamental results of loop quantum gravity is that the classical differential geometry of general relativity is replaced by a discrete quantum geometry. The discreteness of quantum geometry is a direct consequence of the quantization procedure using holonomies of connection which has many physical implications, in particular for black holes and for cosmological spacetimes.

In the last decade, in the setting of cosmological models, a significant progress has been made in finding reliable physics emerging from loop quantum gravity.  Various non-trivial and interesting questions are being answered, such as the following. Does loop quantum gravity result in a resolution of gravitational singularities? How generic is the singularity resolution? Do loop quantum gravity effects leave an imprint in astronomical observations such as those of cosmic microwave background? Are these predictions originating in the phase of the  very early Universe robust?

Extensive analytical, phenomenological and numerical investigations have now confirmed that singularities are resolved for various homogeneous spacetimes in loop quantum cosmology, a quantization of homogeneous spacetimes based on loop quantum gravity. This is a direct result of the discrete quantum geometry which bounds the curvature invariants by the Planck curvature scale. There is a steadily growing body of evidence for singularity resolution for various isotropic and anisotropic Bianchi spacetimes. Similar results are being obtained for black holes spacetimes as well as for inhomogeneous Gowdy spacetimes. Interestingly, singularity resolution as found in loop quantum cosmology also finds evidence in a  related group field theory approach.

Artistic impression of quantum nature of spacetime in loop quantum gravity. Copyright: Thomas Thiemann (Albert Einstein Institute & FAU Erlangen–Nuernberg); Milde Marketing Potsdam, Germany; Exozet Potsdam, Germany).

The emerging picture from various studies based on loop quantum gravity is that big bang of classical cosmology is replaced by a big bounce close to the Planck scale. The classical spacetime does not end at the singularities, but is extended beyond by non-perturbative loop quantum gravity effects. The bounce occurs without any fine tuning of initial conditions, or by violations of weak energy condition. It occurs for all types of matter and in presence of anisotropies. Further, it makes the inflationary phase quite natural to occur which is preceded by a rich and  interesting pre-inflationary dynamics.  This novel picture provides an avenue to reliably test predictions of loop quantum gravity in astronomical observations. A variety of resulting features of the quantum geometry are now being understood which can leave imprint in the cosmic microwave background, which includes predictions of power suppression at large angular scales.

The Focus Issue: Applications of loop quantum gravity to cosmology, brings together various state-of-the-art results on the above lines. The contributions are on various topics ranging from robust signatures of loop quantum gravity in cosmic microwave background, to various extensive studies on singularity resolution.

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