Loop Quantum Cosmology

TL;DR

Loop Quantum Cosmology (LQC) applies background-independent, non-perturbative loop quantization to homogeneous cosmologies, replacing the classical singularities with a quantum bounce arising from inverse-triad and holonomy corrections. By deriving a discrete quantum geometry and a difference equation for the universe's wave function, LQC connects minisuperspace models to the full loop quantum gravity framework and yields robust qualitative predictions, such as early-universe bounce and potential inflationary phases, largely independent of specific matter content but sensitive to quantization ambiguities. The framework extends to anisotropic and inhomogeneous settings, exploring isotropization, Bianchi IX dynamics, and perturbative inhomogeneities, while highlighting the need to relate reduced models to the full theory and to confront observational consequences via effective equations and cosmological perturbations. Overall, LQC provides a coherent, non-singular picture of quantum geometry in the early universe and charts a path toward connecting quantum gravity with observable 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.

Figures (10)

• Figure 1: Examples for bouncing solutions with positive curvature (left) or a negative potential (right, negative cosmological constant). The solid lines show solutions of effective equations with a bounce, while the dashed lines show classical solutions running into the singularity at $a=0$ where $\phi$ diverges.
• Figure 2: Example for a solution of $a(t)$ and $\phi(t)$ showing early loop inflation and later slow-roll inflation driven by a scalar which is pushed up its potential by loop effects. The left hand side is stretched in time so as to show all details. An idea of the duration of different phases can be obtained from Fig. \ref{['PushMov']}.
• Figure 3: Still of a Movie showing the initial push of a scalar $\phi$ up its potential and the ensuing slow-roll phase together with the corresponding inflationary phase of $a$. The movie is available from the online version LivRev of this article at http://relativity.livingreviews.org/Articles/lrr-2005-11/.
• Figure 4: Still of a Movie illustrating the Bianchi IX potential (\ref{['BIXPot']}) and the movement of its walls, rising toward zero $p^1$ and $p^2$ and along the diagonal direction, toward the classical singularity with decreasing volume $V=\sqrt{|p^1p^2p^3|}$. The contours are plotted for the function $W(p^1,p^2,V^2/(p^1p^2))$. The movie is available from the online version LivRev of this article at http://relativity.livingreviews.org/Articles/lrr-2005-11/.
• Figure 5: Still of a Movie illustrating the Bianchi IX potential in the anisotropy plane and its exponentially rising walls. Positive values of the potential are drawn logarithmically with solid contour lines and negative values with dashed contour lines. The movie is available from the online version LivRev of this article at http://relativity.livingreviews.org/Articles/lrr-2005-11/.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2: Generalized Wang Theorem