Title:

Cosmological solutions of eleven dimensional supergravity

The first part of this thesis introduces, in succeeding chapters: i) KaluzaKlein theory, illustrated by a five dimensional example showing how the d=5 Lagrangian can give, up to a multiplicative constant, a d=4 Lagrangian. The equations of motion are derived and a Kasner type solution given for a time dependent version. ii) Eleven dimensional supergravity. The Lagrangian for the bosonic sector is given together with the field equations derived from it. iii) Dimensional reduction and spontaneous compactification. This explains how the fields present in a theory with greater than four dimensions may give classical solutions in which the metric describes a product space (the FreundRubin mechanism). One factor of the product representing ordinary spacetime, the other the extra dimensions. Symmetry aspects are discussed. iv) Cosmology and inflation. A description of standard hot cosmology is given, it being presumed that at 'late' times the extra dimensions have been frozen out and this description is approximately valid. The standard approach to inflation resulting from a phase transition is given to illustrate how different the d=11 supergravity approach given here is. v) KaluzaKlein cosmology, illustrating how spontaneous compactification can be formulated as a time dependent mechanism (dynamical compactification) for the early universe in a gravitational model with perfect fluid matter. Examples of how the scale factors, describing the size of the ordinary and extra dimensions, vary with time are given from my own calculations. I then go on to show that for particular solutions of d=11 supergravity there exists a 4form field F which has components on the internal space and that the existence of this F does not require or imply the existence of any Killing spinors. The internal space is not necessarily an Einstein space, an example being Q". The conditions which F must satisfy are derived. Combining supergravity and dynamical opacification time dependent (cosmological) equations of the RobertsonWalker type are formulated for several models with different topologies. Suitable forms for F are chosen for each model. The Ricci tensor is in general nonisotropic in the internal space, but must be diagonal. The equations are then solved numerically with and without the new F field present. The solutions are studied to select those which give possible inflationary behaviour. It is shown that for the FreundRubin ansatz alone it is impossible to get sufficient expansion. A study of how the various components of the new type of F contribute to the energymomentum tensor follows. This leads to necessary conditions to be imposed on F for inflation. The various models are studied to determine whether these conditions are fulfilled. In certain cases there is a class of solutions for which an arbitrarily large expansion of the ordinary dimensions can be obtained. In these cases analytic asymptotic solutions are given. Even with the necessary conditions satisfied other features of the model, the form of F and the topology of the internal space, may give solutions in which some of the extra dimensions expand. The arbitrarily large expansion of the ordinary space can solve the horizon problem, but the associated rapid contraction of the extra dimensions is likely to be unphysical. To solve the flatness problem would require the present day density to be much larger than observed. If a 4index tensor is similarly used to give the energymomentun tensor in a nonsupersymmetric gravity theory in greater than eleven dimensions the inflationary outlook is more hopefull because the contraction of the extra dimensions can be reduced by choosing more extra dimensions. Next follows a digression to study chiral d=10 supergravity showing that inflationary type solutions with five dimensions expanding exist. In a static case it is possible to get one more dimension to be compact by the presence of a 1form field. This method has been extended to a dynamical case. Some discussion and comments are then given. Some technical details appear in appendices together with a note on supersymmetry and orientation of the internal space.
