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 [46]. This poses the mathematical problem as to 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 less symmetries and the relations between them. This allows 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 those complicated operators, the prime example being the Hamiltonian constraint which dictates the dynamics of the theory, the link between model and the full theory is not always clear-cut. Nevertheless, one can try different versions in the model in explicit ways and see what implications this has, so again the robustness issue arises. This has already been applied to issues such as the semiclassical limit and general properties of quantum dynamics. Thus, general ideas which are required for this new, background independent quantization scheme, can be tried in a rather simple context in explicit ways to see how those constructions work in practice.

At the same time, there are possible phenomenological consequences in the physical systems being studied, which is 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 in 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 such a way, as illustrated later, a partial view of the full theory and its properties emerges also from a physical, not just mathematical perspective.

With this wide range of problems being investigated we can 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 [56, 55, 199, 50, 69, 51, 96]. For complementary applications of loop quantum gravity to cosmology see [140, 141, 2, 114, 152, 1].

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