Self-similar spherically symmetric cosmological models with two scalar fields

We shall study the qualitative properties of the class of self-similar sphe rically symmetric models with two non-interacting scalar fields with expone ntial potentials. Numerous scalar fields, and particularly scalar fields wi th exponential potentials, are known to arise in a variety of theories of t he fundamental interactions. In recent investigations of the cosmological i mplications of these theories a number of novel features has been obtained, including the important assisted inflationary behaviour. In order to study this inflationary scenario, we shall restrict our study here to the so-cal led 'spatially' self-similar models. We derive the governing system of evol ution equations for the model, which is an autonomous five-dimensional syst em of ordinary differential equations, in terms of appropriately defined bo unded variables. A monotonic function is obtained and the local stability o f the equilibria is determined, so that global properties of the physical p hase space can be deduced. Numerical integrations are also undertaken to co mplement the qualitative analysis and in order to investigate possible tran sient behaviour. From our dynamical analysis we find that, depending on the parameters of the models, in general the models begin expanding from a sta te with two scalar fields which are massless, and subsequently either recol lapse and again evolve towards a model with two massless scalar fields or c ontinue expanding towards the assisted inflationary solution. Hence we find that there exists an open set of models that undergo assisted inflation.