Loop quantum cosmology is based on quantum Riemannian geometry, or loop quantum gravity [254, 23, 293, 256], which is an attempt at a non-perturbative and background-independent quantization of general relativity. This means that no assumption of small fields or the presence of a classical background metric are made, both of which are expected to be essential close to classical singularities at which the gravitational field diverges and space degenerates. In contrast to other approaches to quantum cosmology, there is a direct link between cosmological models and the full theory [46, 96], as we will describe later in Section 7. With cosmological applications we are thus able to test several possible constructions and draw conclusions for open issues in the full theory. At the same time, of course, we can learn about physical effects, which have to be expected from properties of the quantization and can potentially lead to observable predictions. Since the full theory is not completed yet, however, an important issue in this context is the robustness of those applications to choices in the full theory and quantization ambiguities.

The full theory itself is, understandably, extremely complex and thus requires approximation schemes for direct applications. Loop quantum cosmology is based on symmetry reduction, in the simplest case to isotropic geometries [54]. This poses the mathematical problem of how the quantum representation of a model and its composite operators can be derived from that of the full theory, and in which sense this can be regarded as an approximation with suitable correction terms. Research in this direction currently proceeds by studying symmetric models with fewer symmetries and the relationships between them. This allows one to see what role anisotropies and inhomogeneities play in the full theory.

While this work is still in progress, one can obtain full quantizations of models by using basic features, as they can already be derived from the full theory together with constructions of more complicated operators in a way analogous to what one does in the full theory (see Section 5). For these complicated operators, the prime example being the Hamiltonian constraint, which dictates the dynamics of the theory, the link between a model and the full theory is not always clear-cut. Nevertheless, one can try different versions of specific Hamiltonian constraints in the model in explicit ways and see what implications this has; the robustness issue arises again. This has already been applied to issues such as the semiclassical limit and general properties of quantum dynamics as described in Section 6. Thus, general ideas, which are required for this new, background-independent quantization scheme, can be tried in a rather simple context, in explicit ways, in order to see how those constructions work in practice.

At the same time, there are possible phenomenological consequences to the physical systems being studied, which are the subject of Section 4. In fact, it turned out, rather surprisingly, that already very basic effects, such as the discreteness of quantum geometry (and other features briefly reviewed in Section 3, for which a reliable derivation from the full theory is available), have very specific implications for early-universe cosmology. While quantitative aspects depend on quantization ambiguities, there is a rich source of qualitative effects, which work together in a well-defined and viable picture of the early universe. In this way, as later illustrated, a partial view of the full theory and its properties emerges from a physical as well as a mathematical perspective.

With this wide range of problems being investigated, we must keep our eyes open to input from all sides. There are mathematical-consistency conditions in the full theory, some of which are identically satisfied in the simplest models (such as the isotropic model, which has only one Hamiltonian constraint and thus a trivial constraint algebra). They are being studied in different, more complicated models and also in the full theory directly. Since the conditions are not easy to satisfy, they put stringent bounds on possible ambiguities. From physical applications, on the other hand, we obtain conceptual and phenomenological constraints, which can be complementary to those obtained from consistency checks. All this contributes to a test and better understanding of the background-independent framework and its implications.

Other reviews of loop quantum cosmology at different levels can be found in [64, 63, 297, 58, 99, 59, 141, 67, 145, 15, 110]. For complementary applications of loop quantum gravity to cosmology see [204, 205, 2, 167, 222, 1].

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