1 Introduction

While general relativity is very successful in describing the gravitational interaction and the structure of space and time on large scales [205], quantum gravity is needed for the small-scale behavior. This is usually relevant when curvature, or in physical terms energy densities and tidal forces, becomes large. In cosmology this is the case close to the Big Bang, and also in the interior of black holes. We are thus able to learn about gravity on small scales by looking at the early history of the universe.

Starting with general relativity on large scales and evolving backward in time, the universe becomes smaller and smaller and quantum effects eventually become important. That the classical theory by itself cannot be sufficient to describe the history in a well-defined way is illustrated by singularity theorems [123] which also apply in this case: After a finite time of backward evolution the classical universe will collapse into a single point and energy densities diverge. At this point, the theory breaks down and cannot be used to determine what is happening there. Quantum gravity, with its different dynamics on small scales, is expected to solve this problem.

The quantum description does not only present a modified dynamical behavior on small scales but also a new conceptual setting. Rather than dealing with a classical space-time manifold, we now have evolution equations for the wave function of a universe. This opens a vast number of problems on various levels from mathematical physics to cosmological observations, and even philosophy. This review is intended to give an overview and summary of the current status of those problems, in particular in the new framework of loop quantum cosmology.

"Loop Quantum Cosmology" Martin Bojowald http://www.livingreviews.org/lrr-2005-11

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