Problem of time in slightly inhomogeneous cosmology

The Problem of Time (PoT) is a multi-faceted conceptual incompatibility between various areas of Theoretical Physics. Whilst usually stated as between GR and QM, in fact 8/9ths of it is already present at the classical level. Thus we adopt a ‘top down ’ classical and then quantum approach. I consider a local resolution to the Problem of Time that is Machian, which was previously realized for relational triangle and minisuperspace models. This resolution has three levels: classical, semiclassical and combined semiclassical–histories–records. This article’s specific model is a slightly inhomogeneous cos-mology considered for now at the classical level. This is motivated by how the inhomogeneous fluctuations that underlie structure formation – galaxies and CMB hotspots – might have been seeded by quantum cosmological fluctuations, as magnified by some inflationary mechanism. In particular, I consider perturbations about S3 case of this up to second order – which has a number of parallels with the Halliwell-Hawking model but has a number of conceptual differences and useful upgradings. The article’s main features are that the elimination part of the model’s thin sandwich is straightforward, but the modewise constraints manage to be second-class. Thus the elimination part only arises as an intermediate geometry between superspace and Riem. The reduced geometries have surprising singularities influenced by the matter content of the universe, though the N-body problem anticipates these with its collinear singularities. I also give a ‘basis set ’ of Kuchař beables for this model arena.