Die Grenzen meiner Sprache bedeuten die Grenzen meiner Welt.

(The limits of my language mean the limits of my world.)

Ludwig Wittgenstein

Tractatus logico-philosophicus

While general relativity is very successful in describing gravitational interaction and the structure of space and time on large scales [308], quantum gravity is needed for 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 as well as 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 dominate. Singularity theorems illustrate that classical gravitational theory by itself cannot be sufficient to describe the development of the universe in a well-defined way [178]. After a finite time of backward evolution, the classical universe will collapse into a single point and energy densities will diverge. At this point, classic gravitational theory breaks down and cannot be used to determine what is happening. Quantum gravity, with its different dynamics on small scales, is expected to solve this problem.

The quantum description presents not only a modified dynamical behavior on small scales, but also a new conceptual setting. Rather than dealing with a classical spacetime manifold, we now have evolution equations for the wave function of a universe. This opens up 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.

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