### 4.13 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 [120]. 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 modifications upon
quantization. As with gradient terms in matter Hamiltonians, there are several options for
those modifications which can only be restricted by relating them to the full Hamiltonian. This
would require introducing the mode decomposition, analogously to symmetry conditions, at the
quantum level and writing the full constraint operator as the homogeneous one plus correction
terms.
An additional complication compared to matter fields is that one is now dealing with infinitely many
coupled constraint equations since the lapse function is inhomogeneous, too. This function can itself
be decomposed into modes , with harmonics according to the symmetry, and each
amplitude is varied independently giving rise to a separate constraint. The main constraint arises from
the homogeneous mode, which describes how inhomogeneities affect the evolution of the homogeneous scale
factors.