  In this paper I shall show how notions of Finsler geometry can be used to
construct a new type of geometry using a scalar field, f, on the cotangent
bundle of a differentiable manifold, M. This new geometry will be called
Lorentzian Cofinsler geometry. This geometry will enable me to use the second
vertical derivatives of f, along with the differential of the scalar field, phi
on M, to construct a Lorentzian metric tensor on M, that depends upon phi. f
will be chosen so that the resultant metric on M has the form of a FLRW metric,
with the t equal constant slices being flat. When the Horndeski Lagrangians are
evaluated for this choice of geometry the quartic and quintic Lagrangians are
of third order, but reduce to non-degenerate second-order Lagrangians plus a
divergence. Upon varying phi in these "scalarized" Horndeski Lagrangians,
equations will be obtained which admit self-inflating universe solutions,
provided that the coefficient functions appearing in the Horndeski Lagrangians
are chosen suitably. This approach is also used to study solutions of the most
general conformally invariant scalar-tensor field theory which is flat space
compatible (i.e., such that the Lagrangians of the field theory are
well-defined when either the space is flat or the scalar field is constant).
There too the coefficient functions can be chosen to give self-inflating
universes. Arguments will be presented to show that it is possible to construct
model universes that begin explosively, and then settle down to a period of
much quieter acceleration which either continues forever, or stops and
collapses to the models original, pre-expansion, state.
