### 4.15 Inhomogeneities

Allowing for inhomogeneities inevitably means taking a big step from finitely many degrees of freedom
to infinitely many ones. There is no straightforward way to cut down the number of degrees of freedom to
finitely many ones while being more general than in the homogeneous context. One possibility would be to
introduce a small-scale cutoff such that only finitely many wave modes arise, e.g., through a lattice, as is
indeed done in some coherent state constructions [269]. This is in fact expected to happen in a discrete
framework such as quantum geometry, but would, at this stage of defining a model, simply be introduced by
hand.
For the analysis of inhomogeneous situations there are several different approximation schemes:

- Use only isotropic quantum geometry and in particular its effective description, but couple it
to inhomogeneous matter fields. Some problems to this approach include backreaction effects
that are ignored (which is also the case in most classical treatments) and that there is no direct
way to check modifications used, in particular for gradient terms of the matter Hamiltonian.
So far, this approach has led to a few indications of possible effects.
- Start with the full constraint operator, write it as the homogeneous one plus correction terms
from inhomogeneities, and derive effective classical equations. This approach is more ambitious
since contact to the full theory is realized. Several correction terms for different metric modes
in linearized perturbation theory have been computed [84, 90, 91, 92], although complete
effective equations are not yet available. These equations already show the potential not only
for phenomenological applications in cosmology but also to test fundamental issues such as the
anomaly problem and covariance; see Section 6.5.4.
- There are inhomogeneous symmetric models, such as the spherically-symmetric one or
Einstein–Rosen waves, which have infinitely many kinematical degrees of freedom but can
be treated explicitly. Contact to the full theory is present here as well, through the
symmetry-reduction procedure of Section 7. This procedure itself can be tested by studying
those models with a complexity between homogeneous ones and the full theory, but results
can also be used for physical applications involving inhomogeneities. Many issues that are of
importance in the full theory, such as the anomaly problem, also arise here and can thus be
studied more explicitly.