In this paper, I shall show how the notions of Finsler geometry can be used to construct a similar geometry using a scalar field, f , on the cotangent bundle of a differentiable manifold M . This will enable me to use the second vertical derivatives of f , along with the differential of a scalar field φ on M , to construct a Lorentzian metric on M that depends upon φ . I refer to a field theory based upon a manifold with such a Lorentzian structure as a scalar–scalar field theory. We shall study such a theory when f is chosen so that the resultant metric on M has the form of a Friedmann–Lemaître–Robertson–Walker metric, and the Lagrangian has a particularly simple form. It will be shown that the scalar–scalar theory determined by the Lagrangian can generate self-inflating universes, which can be pieced together to form multiverses with non-Hausdorff topologies, in which the global time function multifurcates at t = 0. Some of the universes in these multiverses begin explosively, and then settle down to a period of much quieter accelerated expansion, which can be followed by a collapse to its original, pre-expansion state. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.