MODERN COSMOLOGY

Explaining homogeneity and structure 151

references, see Gott and Liu [63]) are based on the quantum cosmology idea of
the wavefunction of the universe, taken to obey the Wheeler–de Witt equation
(a generalization to the cosmological context of the Schr ̈odinger equation) (see
e.g. [67]). This approach faces considerable technical problems, related to

• the meaning of time, because vanishing of the Hamiltonian of general
  relativity means that the wavefunction appears to be explicitly independent
  of time;


• divergences in the path-integrals often used to formulate the solutions to the
  Wheeler–de-Witt equation;


• the meaning of the wavefunction of the universe, in a context where
  probabilities are ill defined [56];


• the fundamentally important issue of the meaning of measurement in
  quantum theory (when does ‘collapse of the wavefunction’ take place, in
  a context where a classical ‘observer’ does not exist);


• the conditions which will lead to these quantum equations having classical-
  like behaviour at some stage in the history of the universe [65]; and


• the way in which this reduced set of equations, taken to be valid irrespective
  of the nature of the full quantum theory of gravity, relates to that as yet
  unknown theory.

The alternative is to work with the best current proposal for such a theory,
takenbymanytobeM-theory, which aims to unite the previously disparate
superstring theories into a single theory, with the previously separate theories
related to each other by a series of symmetries called dualities. There is a rapidly
growing literature onsuperstring cosmology,relating this theory to cosmology
[89]. In particular, much work is taking place on two approaches:

• Thepre big-bang proposal, where a ‘pre big-bang’ branch of the universe is
  related to a ‘post big-bang’ era by a duality:a(t)→ 1 /a(t),t→−t,and
  dimensional reduction results in a scalar field (a ‘dilaton’) occurring in the
  field equations (see Gasperini [58] for updated references).

This approach has major technical difficulties to solve, particularly related
to the transition from the ‘pre big-bang’ phase to the ‘post big-bang’ phase,
and to the transition from that phase to a standard cosmological expansion. In
additionally it faces fine-tuning problems related to its initial conditions. So this
too is very much a theory in the course of development, rather than a fully viable
proposal.

• Thebrane cosmology proposal, where the physical universe is confined to a
  four-dimensional ‘brane’ in a five-dimensional universe. The physics of this
  proposal are very speculative, and issues arise as to why the initial conditions
  in the 5D space had the precise nature so as to confine matter to this lower-
  dimensional subspace; and then the confinement problem is why they remain
  there.