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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 N (x) is inhomogeneous, too. This function can itself be decomposed into modes sum nNnYn(x), with harmonics Yn(x) according to the symmetry, and each amplitude Nn 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.


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