In the previous chapter, cosmological models which drop the isotropy assumption of FLRW models were considered; here we drop the homogeneity assumption. Of course, perturbed FLRW models also satisfy neither assumption, but they are treated only in perturbation theory. Here we aim to study models in the fully nonlinear theory. Inhomogeneous models have been applied both globally (as shown by the use of LTB models in Chapter 15) and to model localized inhomogeneities and, e.g. their fully nonlinear effects on observation via lensing (as shown by the use of Swiss cheese models in Chapter 16). In the global context, issues such as whether inflation could remove inhomogeneity, or whether hierarchical models could fit the data, can be examined: these are essential to judging the robustness of the assumptions of the standard model.

For example, the evidence cited as support for the standard model can be well fitted by nonstandard models, as we have seen in Chapter 15. Thus one can legitimately ask, what is the largest family of cosmological models that can fit the observations? One can then try to devise observational tests to eliminate as many of them as one can.

One may also wonder why we look for exact models of structure formation, when the perturbative theory is so successful? The inflationary paradigm coupled with the perturbation theory of FLRW models has offered the first viable explanation of the observed degree of inhomogeneity in the universe (see Chapter 10). However, the galaxies, clusters and voids we observe now have values of (e.g.) δρ/ρ outside the perturbative regime.