Loop Quantum Cosmology

Abstract

Quantum gravity is expected to be necessary in order to understand situations where classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e., the fact that the backward evolution of a classical space-time inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding space-time is then modified. One particular realization is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. Main effects are introduced into effective classical equations which allow to avoid interpretational problems of quantum theory. They give rise to new kinds of early universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function which allows to extend space-time beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of space-time arising in loop quantum gravity and its application to cosmology sheds new light on more general issues such as time.Electronic Supplementary MaterialSupplementary material is available for this article at 10.12942/lrr-2005-11.

Loop quantum cosmology and inhomogeneities

“…Explicit examples for such constructions can be found for reducing an anisotropic model to an isotropic one in [63] and for obtaining the spherically symmetric representation from the full one in [18]. A more general discussion is given in [11]. We can use the constructions presented here to shed more light on the role of inhomogeneities in these reductions.…”

Interacting Generalized Cosmic Chaplygin Gas in Loop Quantum Cosmology: A Singularity Free Universe

“…LQG is a theory trying to quantize the gravity with a non-perturbative and background independent method. The theory and principles of LQG when applied in the cosmological framework creates a new theoretical framework of Loop Quantum Cosmology(LQC) [37,38,39]. In this theory, classical space-time continuum is replaced by a discrete quantum geometry.…”

Quantum GowdyT3model: Schrödinger representation with unitary dynamics

“…In the following, to simplify the notation, we will let b k and b ÿk denote the variables b k t 0 and b ÿk t 0 , respectively, and collect them in the set of pairs f b k ; b ÿk g with k 2 Z ÿ f0g. 2 It is then straightforward to check that each of the considered pairs of variables decouples in the evolution, so that the evolution transformations are 2 2 block-diagonal in these coordinates. In more detail, time evolution t f ;t 0 maps b k ; b ÿk to a new pair b k t f ; b ÿk t f [seen as new data at t t 0 , related to the new configuration and momentum of the field as in Eq.…”