  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 phi on M, to
construct a Lorentzian metric on M that depends upon phi. 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 field theory when f is chosen
so that the resultant metric on M has the form of a
Friedmann-Lemaitre-Robertson-Walker metric, and the Lagrangian has a
particularly simple form. It will be shown that the scalar-scalar theory
determined by this 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. I conclude the paper with a discussion of how
probabilities can be assigned to the various universes of a multiverse. This is
accomplished by using the action of the universes, with universes having action
closer to zero being more likely than universes with large positive values for
their action. In order to assure that universe models similar to our own
universe are likely to exist I found it necessary to introduce a second scalar
field on M, and to modify the original Lagrangian. In the end my theory has
three scalar fields, two on the manifold M and one on the cotangent bundle of
M.
