eng Max Planck Institute for Gravitational Physics (Albert Einstein Institute) Living Reviews in Relativity 1433-8351 2008-07-02 11 4 10.12942/lrr-2008-4 lrr-2008-4 article Loop Quantum Cosmology Martin Bojowald 1 Institute for Gravitational Physics and Geometry, The Pennsylvania State University, University Park, PA 16802, U.S.A. Quantum gravity is expected to be necessary in order to understand situations in which 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 spacetime 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 spacetime is then modified. One particular theory 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. The main effects are introduced into effective classical equations, which allow one to avoid the 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 an extension of quantum spacetime 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 spacetime arising in loop quantum gravity and its application to cosmology sheds light on more general issues, such as the nature of time. http://www.livingreviews.org/lrr-2008-4 Ashtekar variables, Big Bang, canonical quantum gravity, cosmology, effective equations, equations of motion, inflation, quantum cosmology, singularities, symmetry, time, difference equations