The main purpose of this book is to explore the structure of supergravity theories at the classical level. Where appropriate we take a general D-dimensional viewpoint, usually with special emphasis on D = 4. Readers can consult the Contents for a detailed list of the topics treated, so we limit ourselves here to a few comments to guide readers. We have tried to organize the material so that readers of varying educational backgrounds can begin to read at a point appropriate to their background. Part I should be accessible to readers who have studied relativistic field theory enough to appreciate the importance of Lagrangians, actions, and their symmetries. Part II describes the differential geometric background and some basic physics of the general theory of relativity. The basic supergravity theories are presented in Part III using techniques developed in earlier chapters. In Part IV we discuss complex geometry and apply it to matter couplings in global N = 1 supersymmetry. In Part V we begin a systematic derivation of N = 1 matter-coupled supergravity using the conformal compensator method. The going can get tough on this subject. For this reason we present the final physical action and transformation rules and some basic applications in two separate short chapters in Part VI. Part VII is devoted to a systematic discussion of N = 2 supergravity, including a short chapter with the results needed for applications. Two major applications of supergravity, classical solutions and the AdS/CFT correspondence, are discussed in Part VIII in considerable detail. It should be possible to understand these chapters without full study of earlier parts of the book.

In approaching the issue of how the universe started, it is common cause that we have to face up to the unsolved problem of quantum gravity: the domain where Einstein's theory of gravity is expected to break down because quantum effects become so dominant that they affect the very nature of space and time. Comparing the gravitational constants of nature with those from quantum theory leads to the Planck length ℓP ≈ 10-33cm, which is taken to be the characteristic scale at which quantum gravity dominates. By contrast, most (but not all) variant classical gravitational theories modify GR at low energies (see Chapter 14).

Quantum gravity processes are presumed to have dominated the very earliest times, preceding inflation: the geometry and quantum state that provide the initial data for any inflationary epoch themselves are usually assumed to come from the as yet unknown quantum gravity theory. There are many theories of the quantum origin of the universe, but none has attained dominance. The problem is that we do not have a good theory of quantum gravity (Rovelli, 2004, Weltmann, Murugan and Ellis, 2010), so all these attempts are essentially different proposals for extrapolating known physics into the unknown. A key issue is whether quantum effects can remove the initial singularity and make possible universes without a beginning.

In addition, the weakness of the gravitational force implies that it will be very difficult, though perhaps not impossible, to observationally test theories of quantum gravity.

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.