### 4.20 Summary

Loop cosmology is a phenomenological description of quantum effects in cosmology, obtained in the
framework of a background independent and non-perturbative quantization. There are different types of
quantum corrections to classical equations: effective densities in matter Hamiltonians or the gravitational
part of the Hamiltonian constraint, higher powers of extrinsic curvature, and quantum backreaction effects.
While the former two correspond directly to discreteness effects of quantum geometry, the latter is a
genuine quantum effect. Effective density corrections have a non-perturbative component as they contain
inverse powers of the Planck length and thus the gravitational constant, while purely perturbative
corrections arise from extrinsic curvature terms. Quantum backreaction has not yet been studied
systematically except for special models.
These corrections are responsible for a surprising variety of phenomena, which all improve the
behavior in classical cosmology. Nevertheless, they were not motivated by phenomenology but were
derived through background-independent quantization. In most models and applications the
corrected equations introduce different geometrical effects separately, rather than proceeding with
complete effective equations including all quantum corrections. In this sense, such studies are
phenomenological since they isolate specific effects. While some situations have already been supported
by the rigorous effective equations discussed in Section 6, such a justification remains to be
worked out in general; see [110] for a general discussion. Once all corrections are included, one
has complete effective equations in a strict sense. This has been achieved so far only in one
class of models; see Section 6.3. Most currently available models are incomplete but provide a
wide range of perturbative calculations and have passed several non-trivial tests for consistent
behavior.

Details of the derivation in cosmological models and of their technical origin will now be reviewed in
Section 5, before we come to precise effective equations in Section 6 and a discussion of the link to the full
theory in Section 7.