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Self-similar spherically symmetric cosmological models with two scalar fields

We shall study the qualitative properties of the class of self-similar spherically symmetric models with two non-interacting scalar fields with exponential potentials. Numerous scalar fields, and particularly scalar fields with exponential potentials, are known to arise in a variety of theories of the fundamental interactions. In recent investigations of the cosmological implications 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-called `spatially' self-similar models. We derive the governing system of evolution equations for the model, which is an autonomous five-dimensional system of ordinary differential equations, in terms of appropriately defined bounded variables. A monotonic function is obtained and the local stability of the equilibria is determined, so that global properties of the physical phase space can be deduced. Numerical integrations are also undertaken to complement the qualitative analysis and in order to investigate possible transient behaviour. From our dynamical analysis we find that, depending on the parameters of the models, in general the models begin expanding from a state with two scalar fields which are massless, and subsequently either recollapse and again evolve towards a model with two massless scalar fields or continue expanding towards the assisted inflationary solution. Hence we find that there exists an open set of models that undergo assisted inflation.