### 4.17 Inhomogeneity: Perturbations

With a symmetric background, a mode decomposition is not only possible for matter fields but also for
geometry. The homogeneous modes can then be quantized as before, while higher modes are coupled as
perturbations implementing inhomogeneities [175]. As with matter Hamiltonians before, one can then also
deal with the gravitational part of the Hamiltonian constraint. In particular, there are terms with inverse
powers of the homogeneous fields, which receive corrections upon quantization. As with gradient terms in
matter Hamiltonians, there are several options for those corrections, which can be restricted by
relating them to the full Hamiltonian but also by requiring anomaly freedom at the effective
level.
This requires introducing the mode decomposition, analogous to symmetry conditions, at the quantum
level and splitting the full constraint into the homogeneous one plus correction terms. One can perform a
quantum mode decomposition by specializing the full theory to regular lattices defined using the
background model as the embedding space. This can be used to simplify basic operators of the full theory in
a way similar to symmetric models, although no restriction of classical degrees of freedom happens at this
level. For the simplest modes in certain gauges the Hamiltonian constraint operator becomes computable
explicitly [86], which is the first step in the derivation of effective constraints. From the Hamiltonian
constraint one obtains an effective expression in terms of discrete variables associated with the lattice state,
and in a subsequent continuum approximation one arrives at equations of the classical form but
including quantum corrections. There is thus no continuum limit at the quantum level, which
avoids difficulties similar to those faced by a Wheeler–DeWitt quantization of inhomogeneous
models.

Effective equations thus result in two steps, including the continuum approximation (see also
Section 6.4). The result contains corrections of the same types as in homogeneous models: effective
densities, higher powers of extrinsic curvature and quantum backreaction effects. In contrast to
homogeneous models, however, the relative dominance of these corrections is different. Corrections now
come from individual lattice sites and qualitatively agree with the homogeneous corrections, but are
evaluated in local lattice variables rather than global ones such as the total volume. Thus, the
arguments of effective densities as well as higher power functions such as are now
much smaller than they would be in an exactly homogeneous model. Both types of corrections
are affected differently by decreasing their arguments: effective-density corrections increase
for smaller arguments while higher-power corrections decrease. This makes effective-density
corrections much more relevant in inhomogeneous situations than they appear in homogeneous
ones [66].